Cher iY ; ' - > < Se is " 4 7 oa i - Ps 4 oa 4 Lh * ' Re vr : Piss ee ea a Fa tre gpa, Malm. a K TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. - * ad st ie a Pay neat + ‘ ] T Ran SAC PION 5 OF THE mO Y Aber oocinET Y OF EDINBURGH. VOL. XXXIITI. EDINBURGH: PUBLISHED BY ROBERT GRANT & SON, 107 PRINCES STREET, AND WILLIAMS & NORGATE, 14 HENRIETTA STREET, COVENT GARDEN, LONDON. MDCCCLXXXVIII. Parr I. published ’ . May 13, 1887. Parr TF, * : : . April 30, 1888. Parr Thy, i ; : . October 20, 1888. WATE VLE. IX. CONTENTS. PART L. (1885-86.) . The Atomic Weight of Tungsten. By Joun WADDELL, B.A., D.Sc., . On Dew. By Joun AITKEN, F.RBS.E., . On the Foundations of the Kinetic Theory of Gases. By Professor Tait, Sec. R.S.E., The Eggs and Larve of Teleosteans. By J. T. CunnincHaAM, B.A., F.R.S.E. (Plates L-VIL), . On the Fructification of some Ferns from the Carboniferous For- mation. By Rosert Kinston, F.R.S.E., F.G.S. (Plates VIIL, Exe), ; ; : . On the Colours of Thin Plates. By the Right Hon. Lorp RAYLEIGH, Hon. F.R.S.E. (Plate X.), On the Electrical Properties of Hydrogenised Palladium. By CarGitL G. Kwnott, D.Sc. (Edin.), F.R.S.E., Professor of Physics, Imperial University, Tokayo, Japan. (Plate XI), The Electrical Resistance of Nickel at High Temperatures. By CarGitL G. Knorr, D.Sc. (Edin.), F.R.S.E., Professor of Physics, Imperial University, Tokayo, Japan. (Plate XII.), . The Formation of the Germinal Layers in Teleostet. By GEORGE Brook, F.R.S.E., F.L.S., Lecturer on Comparative Embryology in the University of Edinburgh. (Plates XIII.-XV.), PAGE 97 137 187 199 vl CONTENTS. PAGE X. On the Structure of Suberites domuncula, Olivi (O. S.), together with a Note on peculiar Capsules found on the Surface of Spongelia. By J. ArtHurR Tuomson, F.R.S.E. (Plates XVI, RNVIE) 5 ; . 241 XI. The Reproductive Organs of Bdellostoma, and a Teleostean Ovum from West Coast of Africa. By J. T. Cunnineuam, B.A., F.R.S.E., ; . 247 PART II. (1886-87.) XII. On the Foundations of the Kinetic Theory of Gases. Il. By Professor Tait, Sec. R.S.E., . : . ea XIUL Tables for facilitating the Computation of Differential Refraction in Position, Angle, and Distance. By the Hon. Lorp M‘LAReEn, F.R.S.E., : ; ; - - 278 XIV. Ona Class of Alternating Functions. By THomas Murr, LL.D., F.R.S.E., 309 XV. Expansion of Functions in terms of Linear, Cylindric, Spherical, and Allied Functions. By P. ALEXANDER, M.A. Communi- cated by Dr T. Murr, F.R.S.E., , : i 813 XVI. On Cases of Instability in Open Structures. By E. Sane, LL.D., E.R.S.E., 4 : . oe XVII. On the Fossil Flora of the Radstock Series of the Somerset and Bristol Coal Field (Upper Coal Measures). Parts I., II. By Rosert Kipston, F.R.S.E., F.G.S. (Plates X VIII.- bo Gh 8 i Rs ; ; », a8 XVIII. A Diatomaceous Deposit from North Tolsta, Lewis. By JOHN RATTRAY, M.A., B.Sc., F.R.S.E., of H.M. “ Challenger ” Commission, Edinburgh. (Plate XXIX.), . . 419 XIX. On the Minute Structure of the Eye in certain Cymothoide. By Frank E. Bepparp, M.A., F.R.S.E., F.Z.S., Prosector to the Zoological Society, and Lecturer on Biology at Guy’s Hospital. (Plate XXX.),. é . 443 CONTENTS. XX. Report on the Pennatulida dredged by H.M.S. “ Porcupine.” By A. Mitnes MarsHatt, M.D., D.Sc., M.A., F.R.S., Beyer Professor of Zoology in the Owens College ; and G. H. Fow er, B.A., Ph.D., Berkeley Fellow of the Owens College, Manchester. Communicated by Joun Murray, Esq. (Plates XXXI., XXXITI.), XXI. On the Determination of the Curve, on one of the Coordinate Planes, which forms the Outer Limit of the Positions of the Point of Contact of an Ellipsoid which always touches the three Planes of Reference. By G. Puarr, Docteur és-sciences. Communi- cated by Professor Talr, XXII. On the Partition of Energy between the Translatory and Rotational Motions of a Set of Non-Homogeneous Elastic Spheres. By Professor W. BURNSIDE, : : : XXIII. A Contribution to our Knowledge of the Physical Properties of Methyl-Alcohol. By W. Dirrmar, F.R.SS. Lond. & Edin., and CHArLes A. Fawsitt. (Plate XX XIII), XXIV. On the Thermal Conductivity of Iron, Copper, and German Silver. By A. Cricuton MircHetit, Esq. Communicated, with an Introduction, by Professor Tarr, Sec. R.S.E. (Plates OOS DOC XXV. Critical Experiments on the Chloroplatinate Method for the De- termination of Potassium, Rubidium, and Ammonium ; and a Redetermination of the Atomic Weight of Platinum. By W. Dirrmar, F.R.S.E., and Joun M‘Arruour, F.R.S.E., PART III. (1887-88.) XXVI. The Polycheta Sedentaria of the Firth of Forth. By J. T. CunnincHaM, B.A., Fellow of University College, Oxford, F.R.S.E., Superintendent of the Granton Marine Laboratory; and G. A. Ramace, Vans Dunlop Scholar in Edinburgh Uni- versity. (Plates XX XVI-XLVIL), Vil PAGE 453 465 501 535 635 viii CONTENTS. APPENDIX — PAGE The Council of the Society, : : . 688 Alphabetical List of the Ordinary Fellows, : , . 689 List of Honorary Fellows, ; > 2, List of Ordinary Fellows Elected during Session 1886-87, . S08 Laws of the Society, : i AOS The Keith, Makdougall-Brisbane, Neill, and Victoria Jubilee Prizes, : ‘ : 5 : : . ie Awards of the Keith, Makdougall-Brisbane, Neill, and Victoria Jubilee Prizes, : ; : 1) ele Proceedings of Statutory General Meeting, » heal List of Public Institutions and Individuals entitled to receive Copies of the Transactions and Proceedings, : . | 25 Index, , ; : : : : . hell Purwted 25 FEB aes 96 JUL 1897 TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. VOL. XXXIII. PART I.—FOR THE SESSION 1885-86. Va. IX. XI . The Eggs and Larve of Teleosteans. . Ow the Fructification of some Ferns from the Carboniferous Formation. . On the Colours of Thin Plates. CONTENTS. . The Atomic Weight of Tungsten. By Joun Wanpett, B.A., D.Sc., . On Dew. By Mr Joun Aitken, . On the Foundations of the Kinetic Theory of Gases. By Professor Tart, . By J. T. Cunnineuam, B.A. (Plates I. to VIL), By Rozert Kinston, F.R.S,E., F.G.8. (Plates VIIL, IX.), By Lorp Raytrieu. (Plate X.), . On the Electrical Properties of Hydrogenised Palladium. By Carcuu G. Knorr, D.Sc. (Edin.), F.R.S.E., Professor of Physics, Imperial University, Tokayo, : Japan. (Plate XI.), . The Electrical Resistance of Nickel at High Temperatures. By Careiun G. Knorr, D.Sc. (Edin.), F.R.S.E., Professor of Physics, Imperial University, Tokayo, Japan. (Plate XII.), y é § - The Formation of the Germinal Layers in Teleostei. By Grorae Brook, F.L.S. Lecturer on Comparative Embryology in the University of Edinburgh. (Plates XIII. to XV.), On the Structure of Suberites domuncula, Olivi (O .S.), together with a Note on peculiar Capsules found on the Surface of Spongelia. By J. ARTHUR THOMSON. (Plates XVI., XVII.), : ’ . The Reproductive Organs of Bdellostoma and Teleostean Ovum, from West Coast of , Africa, By J. T. Cunnineuam, B.A., [Issued May 31, 1887.) 171 187 199 241 247 vy NG arn TRANSACTIONS. I.—The Atomic Weight of Tungsten. By JoHN WADDELL, B.A., D.Sc. (Read 7th June 1886.) From the results obtained by BERzELIuS in his experiments with tungsten, the number 189°28 is calculated as the atomic weight of that metal. Many later investigations have been made, in which uniformly a lower figure was arrived at, in the majority of cases very nearly 184, which has therefore been regarded as the atomic weight. Some of the determinations were made with a special view to the support or overthrow of a theory. Dumas, for instance, wished to discover whether the atomic weight of tungsten was exactly double that of molybdenum. In such cases it would be natural to suppose that special care would be taken to secure purity of materials. Dumas obtained the numbers 96 and 184 as the atomic weights of molybdenum and tungsten, but he seems to have distrusted his results ; for he remarks—“ Is it necessary, however, to conclude from this discussion, that some simple ratios of the kind which one used to admit between molybdenum and tungsten cannot exist? I do not think so” (Ann. Chim. Phys., [3] 55, 144). In studying the literature of the subject, I felt that no security was afforded by the various experimenters of the purity of the compounds made use of by them. SCHEIBLER (Jour. Prac. Chem., 83, 278), apparently considers a single recrystallisation of the sodium salt all that is necessary. This salt was used as a starting point, and though the further working up into barium metatungstate may have tended to purification, ScHEIBLER certainly does not prove that he has thus freed his tungsten from molybdenum and silicon. BERNOULLI (Pogg. Ann., 111, 573) considered that he had obtained pure tungstic acid, though he gives no indication of an attempt to free from molybdenum. He prepared his tungstic acid from wolfram, which contained silica and niobic acid, from which VOL. XXXII. PART 1. 2 DR JOHN WADDELL ON the tungstic acid was separated by solution in ammonia. He found that the reduced metal, when ignited in a current of chlorine, was entirely changed into a volatile chloride. This test proves the absence of silica, which would have formed a residue, but has no bearing whatever upon the question, whether or no molybdenum were present; for molybdenum pentachloride, and tungsten hexachloride are produced under precisely the same conditions, and are very similar in appearance and characteristics. Similar criticisms could be made on all the investigations to which I have referred. It appeared to be too readily assumed that recrystallisation would insure purity, no certainty being afforded either by a trustworthy mode of separation of all possible impurities, or by tests showing the absence of admixture. In my work, I kept specially in view the elimination of silicon and molyb- denum, at the same time using all precautions to free from other impurities as well. The reason for paying special attention to silicon and molybdenum, was that these are very similar in many respects to tungsten, and are moreover very liable to exist in ores of that metal. The presence of both was proved in some scheelite with which preliminary experiments were made, and their elimination was found to be no very easy matter. The material used as starting point in the subsequent investigation was commercial tungsten, a dark grey powder, containing 94-98 per cent. of the metal. The impurity was supposed by the manufacturers to be chiefly lower oxides of tungsten, and in particular molybdenum was said to be absent. The employ- ment of metallic tungsten as starting point is not to be recommended, owing to the great difficulty of oxidation. The most convenient material, doubtless, is sodium tungstate, which can be obtained readily enough. In my work, how- ever, some interesting facts with regard to oxidation were brought into prominence, for I tried several methods, the results of which are here given. The first mode of oxidation tested was continued boiling with aqua regia, but at the end of a week a great deal of the metal was still unacted upon, and the method was considered unsatisfactory. Another plan tried was mixing the metal with nitre, and throwing the mixture, small portions at a time, into a red hot crucible. Moderately good results were thus obtained; and the process would, I think, work fairly well on the large scale where proper furnaces and crucibles are available. On the small scale, the chief objection is that the crucible itself is attacked. A third method furnished good results, and considerable attention was devoted to the determination of the most favourable conditions. The process consisted in the ignition of a mixture of potassium chlorate, sugar, and metal. After an ex- tended number of trials, the proportions found most satisfactory were 9: 3: 5. THE ATOMIC WEIGHT OF TUNGSTEN. 3 This is the most expeditious method of oxidation, for 200 grammes of metal can readily be ignited in one operation, and two-thirds of the total quantity are oxidised in a few seconds. I found it most convenient to mould the materials into a cone, sufficient water having been added to make this operation possible. The cone having been placed on a large iron tray was lighted at the apex, so that the propagation of the combustion was downward. By this precaution, loss of material caused by spurting was in large measure avoided. The objection to this mode of oxidationis the great difficulty experienced in re- covering tungstic acid from the great bulk of salt. If, however, appliances are at hand for evaporation, with strong hydrochloric acid on an oil bath and long continued drying at 120°, the process might be made fairly successful, otherwise the difficulties encountered in recovering the tungstic acid far outweigh the advantage gained by rapid oxidation. The method which I found most convenient and satisfactory was the igni- tion of the metal in a porcelain tube, a current of air being passed over the red hot mass. The metal must form only a thin layer, else the oxidised material will obstruct the passage of air, for tungstic acid occupies nearly four times the volume of metallic tungsten. In my experiments, sulphurous fumes were produced in considerable quantity, evidently showing the presence of sulphide in the commercial tungsten. The vapours were absorbed by caustic soda solution, and in this solution traces of molybdenum were found. After about six hours’ heating, the metal was, as a rule, to a great extent oxidised, and had acquired a green colour. If the operation be long continued, say for thirty hours, the compound produced possesses a canary-yellow colour, and is practically tungstic acid. . The green colour usually obtained was doubtless due to the presence of partially oxidised and totally unoxidised material. In ordinary cases complete oxidisation was not considered necessary, for in subsequent operations the small amount which had escaped the action of the hot air was easily elimin- ated. The greater part of the tungstate obtained was prepared from the im- pure tungstic acid just mentioned. The mode of treatment was as follows :— The partially oxidised mass was fused in a platinum basin with one half of its weight of sodium carbonate. The fusion was complete in a few minutes. The fused mass after cooling was disintegrated with water, and the tungstate dissolved out, leaving the unoxidised metal as residue. It was found that the platinum basin was somewhat attacked, and subsequent examination proved the presence of lead in the tungsten residue. The solution of tungstate was boiled in a large silver basin with addition of ammonia carbonate, the latter being employed to separate out any silica or iron and aluminium hydrates possibly present. A very slight precipitate formed and was filtered off. The filtrate was again treated with ammonium carbonate, the process being repeated 4 DR JOHN WADDELL ON till no more precipitate was produced. The liquid was then evaporated to dryness, and thus a mixture of sodium tungstate and carbonate was obtained. For my purpose the presence of carbonate was no disadvantage, and I did not crystallise out the tungstate. As I have already remarked, the method of purification hitherto almost universally adopted was recrystallisation. I employed the method of fractional precipitation as more likely to give a decided test. In the case of all the precipitations being alike, the probability of purity would nearly amount to a certainty, for it is extremely unlikely that the proportions of the impurities would be the same in each of the precipitates. In case of a difference appearing in the precipitations, it is natural to suppose that those most widely separated would exhibit the greatest dissimilarity, and that the middle frac- tionations would be similar and practically pure. It was known that the great bulk of my salt was tungstate. Any impurities more readily precipitated than tungsten ought to be concentrated in the first fractionation, while anything less readily precipitated would be chiefly found in the last portion. The question presented itself in what form it was best to precipitate the tungsten. From the soluble tungstate it was possible to throw down either an insoluble tungstate or tungstic acid. As I wished to determine the atomic weight of the metal by reduction of tungstic acid, it is evident that if the former method were employed it would be well to produce a tungstate which could be easily decomposed and changed to tungstic acid. Such a compound is mercurous tungstate, which loses its mercury on ignition, From some experiments tried with mercurous nitrate as precipitant, I decided that this method of fractionation was not so feasible as the precipitation of tungstic acid direct by means of hydrochloric acid. Before proceeding with the fractionation, however, I freed the sodium tungstate so far as possible from molybdenum. The method employed was that recommended by Rose. Sufficient tartaric acid was added to a solution of the alkaline salt to prevent the precipitation of tungstic acid on acidification with hydrochloric acid. A stream of hydrogen sulphide was then passed through the solution, and an appreciable though small precipitate of molyb- denum sulphide was thus obtained. The filtered solution containing about 300 grammes of solid tungstate had a blue colour, owing to the presence of a small quantity of one of the lower oxides of tungsten. The liquid was decolorised by the passage of a current of air, and was then ready for fractional precipitation. It is to be noted that though sufficient tartaric acid had been added to the solution to prevent precipitation of tungstic acid by a small quantity of dilute hydrochloric acid, yet a considerable excess of the latter was capable of producing quite a pre- THE ATOMIC WEIGHT OF TUNGSTEN. 5 cipitate. This fact is important, otherwise the method must have been greatly modified. It is further to be noted that the precipitation was gradual, hence there was ample opportunity for the liquid to be well mixed, and the precipi- tation was therefore not of a local character. The mode of procedure was the following :— The liquid was boiled in a porcelain basin, and to it a measured quantity of pure hydrochloric acid was added. The boiling was then continued till the precipitate was formed in sufficient quantity, when the contents of the basin were removed to a large beaker and allowed to settle. The supernatant liquid was after some time decanted, and the precipitate washed once or twice by decantation. The precipitate was set aside for future use, the separate decantations were united and evaporated to the bulk of the original solution, and acid added as before. This process was repeated till eleven fractionations were obtained. The first of these had a dark green colour, probably because the current of air had not thoroughly oxidised the liquid. The subsequent precipitates, as far as the seventh, were pale yellow; while the remaining fractionations were dirty green, and not so finely divided as those which preceded. These differences were, I think, caused by the fact that for the final precipitations the liquid required to be boiled down to small bulk, in order to obtain a reasonable quantity of tungstic acid. The tartaric acid under these circumstances probably exercises a reducing action, and in the small quantity of liquid the precipitate was in all likelihood aggregated by the continued boiling. The third precipitate was purified, and used for estimation of the atomic weight. The precipitate was washed several times by decantation, and then repeatedly on a filter. It was then dissolved in pure ammonia, and after filtration reprecipitated by addition of pure hydrochloric acid. The solution in ammonia had a triple purpose. It insured the oxidation of the precipitate in case any lower oxide of tungsten were present. It separated any slight trace of impurity not soluble in ammonia. It aided the washing from sodium salts, for the solution and reprecipitation presented fresh surfaces to the action of the wash water. The washing (which was with water containing a little hydrochloric acid in order to prevent the precipitate running through the filter) was continued till the filtrate yielded no residue on evaporation, and a test portion of the precipitate gave only a slight indication of sodium by the spectro- scope. The precipitate was afterwards dried and ignited in a current of air. The tungstic acid thus obtained had a beautiful pale canary-yellow colour, and was quite uniform in appearance. It was reduced in a porcelain tube by a current of pure hydrogen, the temperature being gradually raised from below dull redness to the highest obtainable by a strong blast in a Fletcher’s furnace. Assuming that tungstic acid has the composition expressed by the formula 6 DR JOHN WADDELL ON WO,, and that it is reduced to metallic tungsten by ignition in a stream of hydrogen, the weight of oxygen lost, compared with that of the tungsten left behind, gives all the data required for determining the atomic weight of the latter. As a mean of three experiments made with the precipitate described, I obtained the number 184°5. The tenth and eleventh precipitates were united, and treated in the same way as the third fractionation. It was difficult to wash, and the tungstic acid had a greenish tinge, and altogether did not appear so pure as what had been before obtained. During the reduction there was a slight volatilisation. The atomic weight, estimated in this sample, was 183-7. The seventh precipitate was subjected to similar treatment. Its appearance and behaviour were quite satisfactory, and the atomic weight calculated was 184. I made a number of other determinations which need not be described. Some of them were made with tungstic acid not freed from molybdenum by sulphuretted hydrogen. In a number of cases a slight volatilisation was observed, and in these the atomic weight estimated was low. So uniformly was the volatilisation noticed when the number obtained was below 184, that there is every reason to believe that where no volatilisation was observed none actually occurred. The uniformity in result was greatly in favour of purity in the tungstic acid; but as silica is with difficulty precipitated from a silicate by means of hydro- chloric acid, and as tungstic acid exhibits the same characteristics, I thought it advisable, if possible, to prove the absence of silica independently. This I did by Marignac’s method of separation, which consists in fusing with hydrogen potassium sulphate. Tungstic acid forms a tungstate under these circum- stances, while silica remains unaltered, and is left undissolved when the fused mass is treated with water. As no residue was left after solution, the absence of silica was established. ° Subjoined is a table of the estimations described above :— | Fraction. WO,. W left. O, lost. Atom weight. Remarks, it LB ETs 14006 VI1L5 2891 184:55 No volatilisation. ‘9900 "7855 "2045 184:37 os 11479 ‘9110 "2369 184:59 My VIL. 9894 7847 2047 184-00 ‘i X. 4°5639 3°6201 ‘9438 183°69 Slight volatilisation. Of the above determinations, those denoted III. are the most trustworthy, from the fact that there are three concordant estimations. The mean of these is 184°5, which is the atomic weight of tungsten calculated from oxygen equal 16. This reduced to 0=15'96 gives W =184:04. I made a couple of determinations of the specific gravity of two specimens THE ATOMIC WEIGHT OF TUNGSTEN. G of metallic tungsten, the first A had not been specially freed from molybdenum, the second B was a portion of III. The determinations were made in a specific gravity bottle of 10 ¢.c. capacity. The bottle was weighed empty, and full of distilled water which had been boiled to expel air. The bottle was then dried and weighed again, then some tungsten was introduced, and another weighing was made. The metal was afterwards covered with water, and the bottle placed under an air-pump, in order to extract the air enclosed in the powder. The bottle was then filled with water, and a weighing again made, and as the water evaporated slowly, but perceptibly, weighings were taken when the meniscus in the capillary touched two fixed marks scratched on the stopper. How nearly the readings agreed, is shown in the table below. | Specimen. | Wt. tungsten. Reading. ee Specific gravity. A 1:0353 Upper mark 0566 18249 Lower mark ‘0566 18:249 B ‘7187 Upper mark 0382 LOaia | Lower mark 0383 18°765 | Most reliance should be placed upon the result obtained with B, because of the assured purity of the metal, and because a first trial would be liable to experimental errors. All such errors tend to give a low specific gravity. It was not thought necessary to make another determination, because the number agrees so well with those obtained by the best authorities. The highest determination is that given by Roscoz, viz., 19:13; the majority of experimenters give figures lying between 18 and 19, while a number so low as 16°54 has been obtained. The density seems to depend upon the method of preparing the metal. My work has been confirmatory of the commonly accepted atomic weight and specific gravity of tungsten ; its chief value lies in the fact that the subject was attacked in a way so far as I know not hitherto attempted, and the corro- borative evidence is therefore all the more trustworthy. My thanks are due to Prof. Crum Brown and Dr Gisson for valuable suggestions and kindly assistance. . ie oo 4 Fa bs bad et as - 4 y Lh. i (ae fe UN J: i tied A az : =e ay oY Lil Sey nie | LBS Galt nN niet - ne > im (3 my ie - paw reeks ri nia a eyebace als ihe rea ote . _~ tack mk eRe. ry: 7 r ” =o : , er ~ % “ i ee et vir. 7 : 1 ; 4 = 4 a é — i _ > = i iin dain Greniaita ate a j 4 — tavern: eres i =" ? oul et» . , ~ ‘ ’ ip) } . j ya =| ? < 4 ran : i ' — “ =) a ! * : ‘ > basi : , , is, 4 wy AG dy ; vs iy et ol suis Dk =f ae male | tru cube a ; vets, haart ine 4 ghug, Inamgmel Salth..og rasan Hy wunea ta ui Te Be Sits are ayes, — an ui ale } sh slag ee a ae I _ tual a ee 1 ngalates: ma ; : * ' esse iad sal sonal vicke epad te wh oedl . Besiwide os 1 nak ea : iad be rages a alti +3 j } iA ad | aa eg AT Le seg aw SE Case ae ea r —a a ” Sd ostuegias c ‘ a, at | ool ot vanes % Ate een, ty yin ns hag anal eae IIl.—On Dew. By Mr Joun AITKEN. (Read 21st December 1885.) The immense amount that has been written on the subject of dew renders it extremely difficult for one to state anything regarding it which has not been previously expressed in some form: It has been examined over and over by minds of every type, and from every point of view; so that every possible explanation of the different phenomena seems to have been given, and so many passing thoughts recorded, that from the literary point of view the whole subject seems exhausted. As a necessary result, these different treatises are in many respects contradictory; and it would be quite impossible to construct anything like a consistent explanation and account of our subject, from the very voluminous writings of those who have treated it from the purely literary point of view, and whose ideas have been evolved from their inner consciousness, according to what seemed to them the fitness of things, and without questioning nature as to the truth of their conclusions. On the scientific side of the subject, however, the writings are not so voluminous, and additions to it are still required to enable us to determine which of the many conflicting opinions are correct. In ancient times it was thought that the moon and stars had an important influence on dew, probably because there is most dew on those nights when these orbs shine brightly on the earth; thus confusing two things which have a common cause, and making one the effect of the other. ArisToTLe placed the knowledge of this subject far in advance of his time. He defines dew to be humidity detached in minute particles from the clear chill atmosphere. The Romans, led by the writings of PLriyy, returned again to the primitive idea that dew fell from the heavens. This idea retained its position during the course of the Middle Ages. Then began an endless variety of theories, such as, that the air is condensed into water by the cold, that the moon’s rays caused it, and so on. In the beginning of the eighteenth century clearer ideas began to be formed, and a reformation took place, in which, as in most reformations, the swing of the pendulum went to the extreme on the opposite side. Dew was no longer believed to descend from the heavens, for GersTeNn advanced the idea that it rose from the earth; and in this opinion he was followed by M. Du Fay and Professor MusscHENBROEK, the latter, however, afterwards made some observa- tions which caused him to change his opinion. GrERsTEN was led to think that _ dew rose from the ground, because he often found grass and low shrubs moist VOL. XXXIII. PART I. B 10 MR JOHN AITKEN ON DEW. with it, while trees were dry. M. Du Fay followed up these observations with experiments, made by placing sheets of glass at different heights from the ground. He found that dew formed on the lowest pane first, and only appeared on the highest at a later hour; he also found that the lowest pane collected most moisture. Other observers gave somewhat different explanations of the pheno- mena connected with dew; but owing to a want of clearness, the subject did not advance much till the masterly Zssay on Dew by Dr WELLS made its appearance. Dr WELLS’ experiments were so simple, and his interpretation of the different phenomena connected with dew so clear, that he has been justly considered the ereat master of this subject. In his Essay he struck a medium between the two previous theories as to the source of the moisture that forms dew. He did not think with the ancients that it fell from heaven, nor with GERSTEN that it rose from the earth, but that it was simply condensed out of the air in contact with the surfaces of bodies cooled by radiation below the dew-point of the air at the place. ‘This opinion has, so far as I am aware, been generally received up to the present time. Some experiments I have recently made on this subject have caused me to differ entirely from Dr WELLS as to the source of the vapour that forms dew. As everything written by Dr WELLs is, so to speak, stereotyped and final, there seems to be the greater reason that any of his conclusions that seem doubtful should be carefully criticised and fully investigated; I shall therefore give an account of the experiments that have caused me to differ from so great an authority. Dr WELts thought that almost all the moisture deposited as dew at night was taken up by the air during the heat of the day; so that, according to his idea, vapour ascended from the earth during the day, and again descended and became condensed as dew on the surface of the earth at night. My observa- tions have led me to the very opposite conclusion. All my experiments indicate that dew, on bodies near the surface of the earth, is almost entirely formed from the vapour rising at the time from the ground; at least this would appear to be the case generally in this climate, to which my experiments have been confined. After GERSTEN gave his reasons for supposing that dew rose from the ground, and Du Fay extended the subject, Dr WELLS combated their conclu- sions, and successfully showed that their experiments did not prove that vapour rose from the ground, and that all the phenomena adduced in favour of their theory could be equally well explained according to his own. With regard to Du Fay’s reason for thinking that dew rises from the ground— namely, that it appears on bodies near the earth earlier than on those at a greater height—he says :* “But this fact readily admits of an explanation on other grounds, that have already been mentioned. 1. The lower air, on a *An Essay on Dew, by William Charles Wells, p. 109. MR JOHN AITKEN ON DEW. 11 clear and calm evening, is colder than the upper, and will, therefore, be sooner in a condition to deposit a part of its moisture. 2. It is less liable to agitation than the upper. 3. It contains more moisture than the upper, from receiving the last which has risen from the earth, in addition to what it had previously possessed in common with other parts of the atmosphere.” Then he goes on to give reasons why vapour cannot be rising out of the ground, but adds, that some of it must be from this source, as bodies near the surface of the ground get dewed sooner than those higher up, though equally cold with them, but says, “the quantity from this cause can never be great,” and proceeds to give his reasons, which are not altogether satisfactory, and need not be quoted here. He then sums up as follows :—‘‘ These considerations ..... warrant me to conclude that on nights favourable to the production of dew, only a very small part of what occurs is owing to vapour rising from the earth; though I am acquainted with no means of determining the proportion of this part to the whole.” I shall now proceed to detail the observations which have caused me to differ from the conclusion so distinctly set forth by Dr WELLs in the above quotations. I need not say that all my experiments only confirm the con- clusions of that observer as to the formation of dew—that is, as to the conditions most favourable for the deposition of moisture on the surfaces of bodies during dewy nights, while the earth is radiating heat into space. The point on which we differ is as to the source of the vapour that condenses on the radiating surfaces—a point which Dr WELLS admits there were no facts to determine, his own opinion being formed by experiments that did not bear directly on the subject. : When I began to doubt the truth of the generally received opinion as to the source of the vapour, I found a difficulty in beginning my investigation, as it was not easy to arrange experiments to give a direct answer to the question. My intention at first was to test, by means of a delicate hygrometer, the humidity of the air at different heights from the ground and under different conditions. This plan had, however, soon to be abandoned, owing to the impossibility of making anything like accurate observations with any instruments at present in use. For some time I have had in my possession a hair hygrometer constructed by CHEVALLIER of Paris. This form of instrument is perhaps one of the best for the purpose ; yet on making a few test experiments with it, for the special purpose under consideration, its indications were found to be nearly valueless. For instance, if the instrument was removed from saturated to drier air, and again replaced in the saturated, it was impossible to get the pointer back again to the same position on the scale ; and as the amount of dryness it would be required to measure was a very small degree removed from saturation, the error in the indications might be greater than the actual amount of dryness. 12 MR JOHN AITKEN ON DEW. Then again, all such hygrometers, as well as wet and dry bulb thermometers, will have their indications affected by radiation ; they will surround themselves with an envelope of cooled air, as there is but little wind during the time the observations require to be made. Their indications would therefore be of little value, and investigation by means of them had to be abandoned. What first caused me to doubt the present theory, and led me to suppose that dew is formed from vapour rising from the ground, was the result of some observations made in summer on the temperature of the soil at a small depth under the surface, and of the air over it, after sunset and at night. On all occasions in which these temperatures were taken, the ground a little below the surface was found to be warmer than the air over it. It is evident that, so long as these conditions exist, and provided the supply of heat is sufficient to keep the surface of the ground above the dew-point, there will be a tendency for vapour to rise and pass from the ground into the air, the moist air so formed will mingle with the air above it, and its moisture will be condensed, forming dew wherever it comes in contact with a surface cooled below its dew-point. These considerations suggested another method of experimenting than by the use of hygrometers. If vapour is really rising from the ground during night, it seemed possible that it might be trapped on its passage to the air, and that this might be accomplished by placing over the soil something that would check the passage of the vapour, while it allowed the heat to escape. To carry out this idea, I placed over the soil shallow boxes or trays, made of tinplate and painted. These trays were 3 inches (76 mm.) deep, and more than a foot (305 mm.) square in area; they were placed in an inverted position over the soil to be tested. The action of these trays will be somewhat as follows :—Supposing the roof of the small enclosure formed by the covering tray is not by the passing air or by radiation cooled below the temperature of the ground. Then evaporation will cease when the air between the tray and the ground is saturated, and no dew will collect on the inside of the enclosure. But if the tray is cooled below the temperature of the ground, vapour will condense on the inside, and more vapour will rise from the ground to supply its place, and this will go on so long as the ground is the warmer of the two. The effect of these trays will be very much the same as if there was no enclosure, and the air over the grass was nearly saturated, motionless, and of a lower temperature than the soil. But it is evident the trays will check the evaporation on most nights, on account of the slow circulation inside, and also on account of the air inside being always nearly saturated, which is not the case outside the enclosure, so that under most conditions it seems likely there will be less evaporation under the trays than outside them. This will be particularly the MR JOHN AITKEN ON DEW. 15 case on those nights when there is wind, and the air is not saturated, a con- dition which seems to be very frequent in our climate at ordinary elevations. We must remember that the air may not be saturated when dew is forming; and the dew-collecting surface requires to be cooled below the temperature of the air before it collects moisture. In experimenting with these trays different kinds of ground were selected, and the trays placed over them after sunset, that is, after the earth had ceased to receive heat, and the heat-tide had begun to ebb. They were generally examined between 10 and 11 p.m., and again in the morning. Dew on GRASS. Confining our attention to the trays placed over grass, the result of the experiments was that, on all occasions yet observed, there was—1. Always more moisture on the grass inside the trays than outside. 2. There was always a deposit of dew inside the trays. 3. There was often a deposit outside the trays, but the deposit on the outside was always less than on the inside, and sometimes there was no deposit outside when there was one inside. Now I think these facts prove that far more vapour rises out of the ground during the night than condenses as dew on the grass. This excess is evidenced by the greater amount of moisture on the grass inside the trays than outside, and by the amount of dew condensed inside the box. Under the ordinary con- ditions found in nature, this excess is carried away by the wind and mixed up with the air, while some of it is deposited on bodies further away from the ground. It should be noticed that the inside of the tray was more heavily dewed than the outside. This shows there was a higher vapour tension inside than outside the enclosure, which proved that the vapour rising from the ground outside the tray had got mixed up with drier air, as it did not form so heavy a coating of dew as the inside air, even though it had the advantage of a slightly lower temperature than the inside, on account of it being the side of the metal from which the heat was radiating. It may be as well to notice here some objections that may be made to this way of testing the point. It may be said, that though so much vapour does rise under these trays, yet if they were removed and the grass freely exposed, the vapour would not rise, and that the vapour rises because the tray keeps the ground under it warm. Observation certainly shows that the ground under the trays is kept slightly warmer than outside them. At night a ther- mometer is higher on the grass under the tray than on that outside, and next morning the ground at 3 inches below the surface is from 1 to 2 degrees warmer under the tray than outside its influence. This objection to the protecting influence of the trays has an appearance of reason about it; but if we examine the facts, I think it will be admitted that instead of being an argument 14 MR JOHN AITKEN ON DEW. against this method of experimenting it is rather a reason for it. We must remember the tray does not heat the ground; it does not add anything to its store of heat, and enable it to evaporate more moisture ; it simply prevents so much of its store of heat escaping. Now heat escapes from the ground at night in two ways—first, by radiation, and second, by absorption—to supply the latent heat of evaporation. From the area covered by these trays radiation goes on much as at other places ; the painted metal will radiate as much heat as the grass, but evaporation is checked, as there is but little circulation under the trays; and further, there is the heat recovered by the condensation inside the box. It would thus appear that the reduced evaporation and heat of condensation will be the principal causes of the higher temperature inside than outside; so that the trays, instead of increasing the evaporation, would rather seem to decrease it; and that the lower temperature outside is due to the greater evaporation there taking place, as both surfaces are exposed to the same loss by radiation. There is an objection that might be made to the whole theory that dew is formed from vapour rising from the ground. It might be urged that it is impossible for the vapour to rise from the ground, and that these trays interfere with the conditions existing in nature. On a cold clear night, for instance, when the grass gets cooled before the dew-point, it might be said that it is quite impossible for the vapour to rise up through it, as it would be all trapped on its passage to the surface by contact with the cold blades, and that the trays placed over the grass prevent this condensation by stopping the radiation from the grass, and thus they allow the vapour to come up. A. little explanation will, however, show this objection to be groundless, On a dewy night no doubt the top of the grass is at a temperature below the dew point, and if we may take the temperature of a thermometer placed on the grass to be the same as that of the grass, which we may do without sensible error, if we then remove the thermometer and place it among the stems of the grass, the thermometer will rise; and if we place the bulb among the stems close to, but not in the ground, we shall find it to be very much warmer than at the surface. On dewy nights I have frequently found it as much as 10 to 12 degrees warmer. From this we see that the warm air diffusing upwards with its burden of vapour only meets with a very small amount of surface cooled below the dew-point, so that the greater part of the vapour is free to escape into the air. Fairly considered, I think these trays more nearly represent natural conditions than might at first sight appear. Indeed, precisely similar results have been observed with natural conditions. If we examine plants with large blades, we shall often find, on dewy nights, that those leaves which are close to the ground have their under surfaces heavily dewed, while their upper surfaces MR JOHN AITKEN ON DEW. 15 are dry. The effect of the trays is very much the same as that of these large leaves on a perfectly calm night. The only difference is, the trays will lose more heat on account of their better conducting power, and more vapour will be condensed under them than under the bad conductor, while the temperature of the soil will be more nearly reduced to what it would have been if no large close surface prevented the free evaporation. The experiments described were made in August and September, when the ground was very dry, owing to the unusually small rainfall during the previous months. On all occasions the inside of the tray was dewed, however dry the soil, and the inside was always more moist than the outside. After these experiments were made, another method of testing the point under investigation suggested itself, and though, unfortunately, rather far on in the season for satisfactory work of this kind, I at once proceeded to carry it out, as it afforded a means of checking my previous experiments with the trays; but by this time October had arrived, and the conditions had very much changed. The temperature had fallen considerably, and the rainfall had greatly increased the humidity of the soil. . It is very evident that if vapour continues to rise from the ground during dewy nights, as well as during the day, the ground giving off vapour must lose weight. If this could be shown to be the case, it would prove in a more satisfactory manner than the previous experiments that vapour does rise from the ground during night, and that, therefore, dew on bodies near the surface of the ground is really formed from the vapour rising at the time, and not from the vapour that rose during the day. In the first week of October experiments were begun to test this point, by weighing a small area of the surface of the ground, before and after dew had formed, to see whether the ground continued to give off vapour or not while dew was forming. For this purpose a number of shallow pans 6 inches (152 mm.) square and + inch (6°3 mm.) deep were prepared. One of these pans was selected, and a piece of turf slightly smaller was cut from the lawn and placed in it. The pan with its turf was then carefully weighed with a balance sensitive enough to turn with + grain; but in experiments of this kind, which must be done quickly, accuracy of only one grain was aimed at, lest the time required for more accurate weighing might cause loss of weight by evapora- tion. To prevent loss from this cause, the weighing was done in an open shed. The turf was cut at sundown, and when dew began to form. The earth was removed from it till it weighed exactly 3500 grains (226-79 grammes). The pan with its turf was then rapidly restored to the lawn, and put in its place, where the turf had been cut out, and in as good contact with the ground as possible. The pan and turf were then brought back, the under side of the pan carefully cleaned and dried, and all weighed again to make sure nothing was 16 MR JOHN AITKEN ON DEW. lost in the manipulations ; after which it was again restored to its place in the lawn, and left exposed while dew was forming. A few experiments were made in this way, in all of which the ground was found to lose weight. For instance, on the 7th October, the small turf freely exposed to the sky at 5.15 p.m., when weighed again at 6.30 P.m., was found to have lost 54 grains (0°356 grammes), and by 10.15 p.m. it had lost 24 grains (1555 grammes). Fuller particulars of these experiments will be given further on. In making these experiments, the first thing done was to sink two thermo- meters in the ground, one to a depth of 3 inches (76 mm.), the other to a depth of 1 foot (805 mm.), and to place a third thermometer on the surface of the grass. Readings were taken when the experiment began, and again when the pans were removed for weighing. During the time the turfs were exposed, generally about 5 hours, the soil at 3 inches below the surface lost from 2 to 5 degrees, while at 12 inches the loss was small. No doubt part of the heat was lost by radiation, but in grass-land, where the surface of the soil is protected by a fairly good non-conductor, much of the heat will be spent in evaporating the moisture. These experiments prove clearly that under the conditions then existing, the soil loses weight, and that vapour really rises from the ground even while dew is forming; therefore the dew then found on the grass must have been formed out of the vapour rising from the ground at the time. The dew on the grass was, in fact, so much of the rising vapour trapped by the cold grass. The blades of grass acted as a kind of condenser, and held back some of the vapour which would have escaped into the air. It must not be supposed that these experiments in any way contradict the well-known observations of WELLs and others who have worked at the subject. It has long been the custom to expose different substances to radiation during the night, and to estimate the amount of dew on different nights by the increase of weight due to the moisture collected on them. It must be noticed that the conditions of the two sets of experiments are quite different. In those for estimating or measuring the amount of dew, the collecting body must not be in heat communication with the earth, an essential condition being that it shall receive no heat by conduction from surrounding bodies; whereas, in the experi- ments with the turf, the essential condition is that the body experimented on shall be in as good contact with the ground as possible. The result of these two conditions is, that in the former, the exposed surface loses heat by radiation into space, and soon gets cooled below the temperature of the air, and when cooled below the dew-point, dew collects upon it ; while in the latter case the exposed surface is in good heat communication with the ground, and tends to keep hotter than the other surface ; then being always moist it tends to give off vapour, which diffuses away from the hot ground and escapes into the air above, but in part is trapped by coming into contact with the cold grass. MR JOHN AITKEN ON DEW. als The experiments were generally stopped at night. It would be of no use to let them go on till morning, unless one were in attendance at sunrise; for the early morning heat radiated from the sun and sky would cause an increased evaporation, and make the loss appear too great. On one occasion, however, when the morning was dull, weighings were made, and the soil was then found to have lost weight during the late night and morning. The following simple observation is sufficient to convince us that, under the ordinary conditions of our climate, vapour is almost constantly escaping night and day from soil under grass. Go out any night, but it is best when terrestrial radiation is strong, place one thermometer on the grass, and push another under its surface, among the stems, but it need not be into the soil, and note the differ- ance in temperature. Asan example, I found, at 10.45 p.m. on the 10th October, this difference to be as much as 185 degrees. The thermometer on the surface of the grass was 24°, while the other, only about 14 inches underneath it, and not in the soil, was as high as 42°°5, the temperature of the air at the time being 32°°5. Of course, this difference varies, and is not always so great as on this occasion, when the sky was clear and the air still. An experiment of this kind causes us to doubt the value of the radiation observations made by com- paring the readings of a thermometer placed on the grass with the temperature of the air in the screen; because the temperature of the thermometer on the grass varies greatly according to its position. If its bulb is supported near the tips of the stems, the temperature is much lower than when it is allowed to press the grass close to the ground, because in the latter position it receives a good deal of heat from the earth. It might be objected that these experiments having been made late in the year, and when the soil was damp, they do not prove that evaporation would take place in summer when the soil was dry. Other considerations, however, lead us to suppose that this nightly evaporation does go on even after a con- tinuance of dry weather, though I have no direct experiments to prove it, other than those made with the inverted trays. But I find that soil, after it has been kept for some time in a house, and when it looks dry and incapable of support- ing vegetation, still gives off vapour, and saturates the air over it. This was shown by placing over some dry-looking soil a glass receiver, in which was hung the hair hygrometer. The instrument soon showed an incease of humidity inside the receiver, and after a time indicated saturation. To check the read- ing of the hygrometer, it was quickly removed and placed in saturated air, when it was not found to change its reading. Now as soil, even when it appears dry, tends to give off vapour, and saturate the air in contact with it, it is evident that under most conditions of our climate the vapour tension at the surface of the ground, amongst the stems of the grass, must, owing to the higher temperature, be very much greater than VOL. XXXII. PART I. c 18 MR JOHN AITKEN ON DEW. at the tops of the blades; and as the air and vapour are warmer, they tend to rise and diffuse themselves, and so come into contact with the colder blades at the surface, where the moisture gets deposited as dew. Having proved that, under the conditions existing during the experiments, the ground was giving off vapour during the night, I then proceeded to test the value of the observations previously described, and which were made by placing shallow trays over the grass, in order to see if those experiments were of any value. A small tray, similar to those used in the earlier experiments, was prepared. It was made to fit tightly into one of the shallow pans, in which, as before, was placed a small turf cut from the lawn. After the turf and its pan was weighed, the tray was placed over it, and the whole removed, and put in its place in the lawn. This was done at the same time as the other experiment previously described, in which the turf and pan, after being weighed, was freely exposed to radiation and evaporation. The result was that the tray was found to check the evaporation. The inside of the covering tray was dewed very much like another one placed over un- disturbed grass. The turf covered by the tray lost only 6 grains (0°388 grammes) during the five hours, or about + of the amount lost by the one freely exposed to the air. This shows that the trays check the evaporation; we may therefore conclude that the amount collected by them is less than would be given off by the exposed parts of the grass. There seems to be reason for supposing that the amount lost per unit of area in these experiments, with the freely exposed turfs, is too low an estimate for the loss of the lawn at the parts where it was undisturbed, because the under sides of the pans were not in good contact with the ground beneath them. The experimental turf would not therefore be so warm as the rest of the ground, and its evaporation would therefore be less. Most of the heat was conveyed upwards towards the experimental turfs by the rismg vapour, which condensed on the under sides of the pans on which the turfs rested, as they were always found to be dripping wet underneath when removed from the soil. The question now comes to be, Does this evaporation take place from grass- land on all nights and in all weathers? So far as my observations at present go, evaporation is constantly going on, however strong the radiation. On all nights on which the inverted trays have been exposed, dew has collected on their inner surfaces. There is, however, an indirect way of testing this point which may be noticed here, as it is specially applicable to observations on grass land. As soil capable of supporting vegetation tends to saturate the air in contact with it, it will be admitted that so long as the soil is hotter than the air in contact with the grass, vapour will tend to diffuse upwards. Now I find by placing a minimum registering thermometer on the grass, and another on the top of the soil among the stems of the grass, that there is always a difference MR JOHN AITKEN ON DEW. 19 between the minimum on and under the grass, often amounting to a considerable number of degrees, this difference being greatest on nights when radiation is strongest, and least when windy and cloudy. It is only as the day advances that the temperature on the grass approaches that wader it; this is caused by the upper thermometer being heated by solar radiation sooner than the lower ; but as the air is by this time drier, there is no tendency for it to lose moisture by contact with the colder soil, though some of the dew condensed on the grass will, after it evaporates, diffuse downwards, and condense on the soil. It may therefore be safely concluded that, on almost all nights in this climate, vapour does rise from grass-covered land, and it is this vapour that we see as dew on the exposed surfaces of the grass. DEw ON Soll. While the experiments previously described were being made on ground covered with grass, parallel ones were made on bare soil. The inverted trays placed over soil always showed a greater amount of condensed vapour inside them than those over grass. Sometimes there was a heavy deposit of dew inside, while there was none outside. This would be owing to the soil radiating directly to the trays, and to the amount of heat brought up and con- veyed to the trays by the vapour. The temperature of the trays was thus in some cases kept above the dew-point of the air outside. Experiments were also made by weighing a small area of the surface soil, to see if it also lost weight like the grass-land during dewy nights. One of the small pans was covered with a thin layer taken from the top of the soil. The pan and its soil was then weighed and put on the surface of the bare ground at the place where the soil had been taken out. It was left exposed the same time as the other trays with the turfs. On weighing, the soil was found to have lost 23 grains (1-490 grammes) in five hours, or nearly the same as the turf. Alongside this pan was placed another one of the same area, and with the same weight of soil, but covered with a small tray, to see whether the covering trays decreased the evaporation from soil as well as from grass. The result was the same as was found with the turf—a decrease in the evaporation. The protected soil lost only 8 grains (0°518 grammes). The following are the details of a few of the experiments on grass-land and on bare soil made on different evenings, and show the temperatures and the loss of moisture per 0°25 square foot, or 0:023225 square metre, during the experiments :*— * Throughout this investigation I have adhered to the Fahrenheit scale, as it is the one generally used for meteorological purposes in this country, and because it possesses what appears to me practical advantages over the Centigrade scale. The degrees are of a more suitable size, and combine ease in reading with accuracy. This scale also avoids a fruitful source of error, experienced by many, when taking readings above and below zero. 20 MR JOHN AITKEN ON DEW. OctosEer 7, 1885. 5.30 PM. Grass. Soil. j Temperature of soil, 3 inches below surface, 47°5 46° " " 12 uw " 47°5 44° AT 6.30 P.M. Grass exposed, lost 54 grains, or 0356 grammes. » under tray, u 3 " 0:194 Bare soil exposed, " 54 on 0:356 n undertray, 22 1 0178 At 10.30 P.M. Grass. Soil. Temperature on surface, 36°°5 40° " of soil, 3 inches below surface, 44°°5 AD aall W " a, " ! ATs 44° Grass exposed, lost 24 grains, or 1°555 grammes. n under tray, " 6 " 0388 Soil exposed, " 23 on 1:490 « under tray, " Suny 0518 OcToBER 12, 1885. 5.30 PM. Grass. Soil. Temperature of soil, 3 inches below surface, 44° 45° " " 12 " " 44°°5 43°5 AT 10.15 P.M. Grass. Soil. Temperature on surface, 31°°5 30°°5 " of soil, 3 inches below surface, 42° 40°2 " n- 12 " " 44° 43°°5 Grass exposed, lost 30 grains, or 1°944 grammes. n under tray " 6 4 0°388 Soil exposed, " 22 4 1:425 n under tray, " 64 " 0:421 There was a little wind on this occasion, and very little dew formed. During the night the min. on the grass was 28°'5, under it 41°; and it was not till 10 o’clock next morning that the thermometer on the grass was as high as the one wider it. The following reading were taken about the 20th October, the exact date unfortunately is omitted in note-book :— 5.15 P.M. Temperature of air—Dry bulb, 42°5; Wet bulb, 40. Grass. Temperature on surface, 39° " of soil, 3 inches below surface, 46° " " 13 " " 46°°3 Soil. 42° 48° 46°°1 MR JOHN AITKEN ON DEW. 20 AT 10.40 P.M. Temperature of air, Dry bulb, 38° Wet bulb, 36° at 4° 0” " n 4 334 " 38)" hear ground. a Grass. Soil. Temperature on surface, 31° 34° " of soil, 3 inches below surface, 44°:2 43° " " 12 u " 46° 46° Grass exposed, lost 9 grains, or 0583 grammes. » under tray, " 8 on 0°518 " Soil exposed, \" 163 0 1-069 " » under tray, " 9 nn 0583 " NEXT MORNING AT 9 A.M. Grass. Soil. Temperature on surface, 39°°5 39° " of soil, 3 inches under surface, 42°5 40°°5 " " 12 " " 45°°5 45°'5 Grass exposed, lost 19 grains, or 1:231 grammes. n under tray, " 3} " 0°842 " Soil exposed, " 30 4 1:944 " n under tray, " 18 " 1166 " These figures cannot be supposed to represent anything definite, they only indicate a condition of matters which has not been previously observed. They show that evaporation in our climate is going on night as well as day during dry weather, but the extent to which it takes place cannot be gathered from these observations, as they are far too few for the purpose—too few alike with regard to seasons, humidities, and exposures ; nor can the proportionate amount of evaporation from bare soil and from grass-land be arrived at from the weights given. These readings can only be considered true for the place and moisture at the time of the year when the experiments were made. For instance, the inverted trays over soil in my early experiments always indicated a larger evaporation from soil than from grass, while the later ones did not. But the early experiments were made over soil freely exposed to sunshine during the whole day, while the later ones were made at a place less freely exposed, on account of the situation where the first experiments were made being too far from the place of weighing. It is evident the amount of sunshine will be an important factor in this nightly evaporation, as it will greatly determine the amount of heat stored up during the day, and available for evaporation during the night. I extremely regret the season was so far advanced before these experiments were begun, as most of the weather suitable for the purpose was past. I have, however, endeavoured to check my results as well as possible. Still I feel that what has been done is only preliminary. Similar experiments would require to 22 MR JOHN AITKEN ON DEW. be made during the whole year, to determine whether this evaporation is constantly going on or not in fair weather, and to determine its amount under different conditions. The varieties of soils, of humidities, and exposures are so great that an enormous number of experiments would require to be made to determine with any degree of accuracy the amount of evaporation that takes place from any large tract of land. The temperatures of the soil and of the air during these experiments were not high, but we must remember they were taken in October. In summer we have to deal with much higher temperatures and greater vapour tensions, and therefore the possibilities of heavier dews. On the 18th August I find the temperature 4 inch under the surface of the soil at 4 P.M. was 82°, at 3 inches underneath it was 72°, the temperature of the air being 66°. At 9 pPM., at 3 inches deep, the temperature was 60° under grass and under bare soil. The temperature on the grass was 45°, while a thermometer placed on bare soil was 52°. Next morning the temperature at 3 inches under grass was 56’, and at the same depth under soil 52°. The soil at 3 inches down had thus lost 20 degrees during the night, and that nearer the surface would have lost a good deal more. Much of this loss would be spent in evaporating moisture. On this occasion it will be noticed that at night the difference between the temperature on the surface of the grass and on the bare soil was as much as 7 degrees, and this easily explains why the ground kept dry while the grass got wet. So far as my limited observations go, evaporation is constantly going on from soil under grass, but on a few occasions it was doubtful whether the reverse process had not taken place, and vapour got condensed on the surface of bare soil. On one or two occasions in autumn, I observed soil which had been dry the previous day to be damp in the morning. The soil had evidently received an increase of moisture. But the question still remains, Whence this moisture? Came it from the air, or from the soil underneath? The latter seems the more probable source, as the higher temperature below would determine a movement of the moisture upwards by the vapour diffusing; and the surface soil being cold, the vapour would be trapped by it before it escaped into the air, in the same way as it is trapped by grass on grass land. During summer it is difficult to trace the vapour condensed on the surface of the soil to its source, and to say definitely whether it came from the air or from the ground underneath, But on the morning of the 12th October I had an interesting opportunity of studying this question. During the night the radiation had been very powerful, the surface of the soil was greatly cooled, and a thin crust of frozen earth formed. After the sun had thawed the sur- face it was very wet. An examination of the soil before the sun had acted on it, showed that the vapour condensed near its surface had come from under- MR JOHN AITKEN ON DEW. 23 neath. On lifting the small clods on the surface, it was observed that their under surfaces and sides, when close to each other, were all thickly covered with hoar-frost so thickly as to be nearly white, while the upper surfaces exposed to the passing air had but little deposited on them,—the interpreta- tion of which seems to be, that the vapour rising from the hot soil underneath had got trapped in its passage through the cold clods. Its presence under- neath and on the sides of the clods was an evidence that the moisture was on its passage from the ground, when it met with the cold surface which im- prisoned it. This hoar-frost on the sides and under the clods could not be due to vapour condensed from the passing air, because the upper surfaces of the clods had scarcely any deposited on them, and that in spite of the fact that the upper surfaces would be the colder, as they were those from which the radiation was taking place. It seems probable that even the vapour condensed on the upper surfaces of the clods was part of the vapour escaping from the soil, and was not taken from the passing air. The occasions when the earth is most likely to receive vapour condensed upon it from the passing air, are not on clear nights when the radiation is strong, but rather when after strong radiation and cooling of the surface the weather changes, becomes cloudy, and a warm moist wind blows over the land. Occasions of this kind are seen most frequently after frosts, and undoubtedly much moisture is then condensed on the soil, but the moisture so condensed is not what we call dew. Dew on ROADS. There is considerable difference among works on dew as to the absence of dew on roads, but almost all agree in stating that it is never formed on roads ; and the presence of dew on grass, while none is visible on roads, is generally attributed to the greater radiating power of vegetation over that of the material of which our roads are composed. Now I find that this state- ment as to facts is wrong, and the explanation is also inaccurate. Dew really does form on roads in great abundance on dewy nights, and the material of the road is practically as good a radiator as the grass. The reason why it is generally said that dew is not seen on roads is owing, not to the less radiating power of the stones, but to the fact that dew has not been looked for at the proper place. The blades of grass are practi- cally non-conductors of heat, while stones conduct fairly well. The result of this is that we are not entitled to look for dew on the upper surfaces of stones, as on grass, but it must be sought for on their under sides, because the stones are good conductors, and the vapour tension under them is much higher than at their upper surfaces, owing to the higher temperature of the air laden with 24 MR JOHN AITKEN ON DEW. moisture rising from the ground. If we examine a gravel walk on a dewy evening, we shall find the under sides of the stones, especially those near the solid ground, to be dripping wet; and we may occasionally see isolated patches of stones wet on the upper surface, probably due to an openness in the ground at the place permitting a free escape of vapour. Another reason why the upper surface of the gravel does not get wet, is that it is in good heat communication with the ground; the stones are thus kept warm; and as a good deal of the vapour rising from the ground is trapped by the under surfaces of the stones, the vapour which escapes these surfaces is not enough to saturate the air at the temperature of the exposed surfaces of the gravel. The following temperature, taken at 10 p.M. on the 25th September, will give an idea of the difference in temperature on the surface of grass and on gravel, and show why no dew is formed on the top of the stones while it collects on the grass. A thermometer placed on the surface of the gravel was 34°, while one placed near it, but on grass, was 30°, or 4° lower. At the surface of the soil under the grass the temperature was 40°, and it was almost exactly the same temperature at the bottom of the gravel which was 14 inches deep. We see from the above that hot vapour, rising from the ground under grass, ascends till it comes into contact with the cold blades, and is condensed on their exposed surfaces; whereas on the gravel road the under sides of the stones are nearly as cold as their exposed surfaces, and much of the warm vapour gets condensed under them, while the vapour which escapes to the surfaces has its dew-point lowered by mixing with the surrounding air, and the upper surfaces of the stones being in good heat communication with the ground, are not cool enough to condense this vapour and form dew. A simple manner of studying the formation of dew on roads is to take, say, two slates, and place one of them on the gravel and one on a hard part of the road. If these slates are examined on a dewy night, their under sides will be found to be dripping wet, though their upper surfaces and the road all round them are quite dry. This experiment also shows us that under most conditions of our climate vapour does rise from hard dry-looking roads on dewy nights. In studying questions of this kind, and for showing the importance of the heat communicated by the earth to the radiating body, the following experiment may be useful. Place on the grass, soil, or road, a slate and a piece of iron, say an ordinary 7 lb. weight. Alongside of these place another slate and weight; but instead of the latter resting on the ground, elevate them a few inches on small wooden pegs driven into the earth. If we examine the surfaces of these bodies on dewy nights, the following will be the general result. While the grass all round is wet with dew, we shall find that the upper surfaces of the slate and the weight resting on the ground keep dry, and MR JOHN AITKEN ON DEW. 25 those of the elevated ones get wet like the grass. the reason for this is that the bodies on the ground as well as the elevated ones are constantly losing more heat by radiation than they receive by absorption; but those in contact with the ground have heat communicated to them by conduction and by the condensa- tion of vapour on their under surfaces; their temperature is thus prevented from falling as low as that of the elevated bodies, which only receive heat from the passing air; the latter are thus cooled more by radiation than those on the ground. Bodies out of heat communication with the ground thus tend to cool more than those in contact with it; and while the former get cooled below the dew-point and collect dew, the latter keep warmer than the dew- point, and thus tend to keep dry, or if wetted to become dry again. These considerations suggest a simple method of testing whether the surface of any particular part of the ground is giving off vapour or not. It is very evident that so long as the temperature of the surface of the soil is above the dew-point of the air, vapour will rise from the ground, and that if the surface is cooled by radiation below the dew-point, evaporation will cease, and vapour will condense upon it. In order to test this, all that is necessary is to place on the ground, and in good heat communication with it, some substance that is a good conductor, and shows dewing easily. A piece of metal covered with black varnish does well. It is painted black, not in order to radiate copiously, but because black shows any deposit of dew most quickly and easily. So long as this test surface keeps dry while in contact with the ground, the soil round it must be giving off vapour, because the temperature of its surface is higher than the dew-point. But if the temperature of the ground falls below the dew-pomt it will collect moisture, and this test surface will collect dew also, and will thus tell us that the surrounding soil is receiving moisture. In experiments such as these we are simply converting a small area of the earth’s surface into a condescending hygroscope, and our test surface tells us whether the earth’s surface at the place is cooled by radiation below the dew-point or not. So long as no dew forms on the test surface vapour is being given off. These test surfaces must not be large, at most only two or three centimetres, because if large they would check the free passage of the vapour to the air, and so prevent the soil under them from cooling to the same amount as the surrounding ground ; and further, it is difficult to get good contact with large surfaces, without which only a part of the test surface keeps clear, while the part not in contact gets dewed, even though the temperature of the surface of the ground is above the dew-point. This was confirmed by observations made on a frosty night. On lifting each test plate, it was observed that the soil was frozen to it under the clear parts, and no soil adhered under the parts that were dewed. In my experiments I have used small copper discs covered VOL. XXXII. PART I. D 26 MR JOHN AITKEN ON DEW. with black varnish, ordinary glass mirrors, and also small black mirrors, in order to get rid of the objection to ordinary silvered mirrors, namely, that they might not be good radiators. On no occasion up to the beginning of November have I yet seen dew on any of these at night, but it is difficult to say whether dew had not formed on them on some mornings, as the air was thick and misty, and the deposit then observed might have fallen as fine rain. The changes in temperature of the surface of the soil due to radiation, give rise to a downward movement of heat during the day, and to an upward move- ment of it during the night. These heat changes will be accompanied by corresponding movements of the moisture in the soil. During day, after the surface is heated, the vapour tension being higher above than below, a downward movement of moisture will take place; and at night this process will be reversed, the tension of the vapour at a depth being greater than near the surface, the vapour rises and condenses in the colder soil. Part of the latent heat so liberated by the rising vapour is spent in radiation from the surface, part in evaporating moisture, and a little in heating the air cooled by contact with cold grass, &e. We may conclude that, owing to the heat received during the day, and probably also to the internal heat of the earth, vapour continues to rise from the ground long after the sun has set, and in many conditions the vapour continues to rise the whole night; but under certain others it seems probable that the reverse will occasionally take place, and vapour condense on the ground. ‘This is most likely to take place soonest on bare soil, especially on those parts of it that are in bad heat communication with the ground under- neath. But over grass-land in most conditions of our climate, when dew is forming, the evaporation seldom seems to stop, but goes on night and day, on account of the surface of the soil being protected by the grass from losing its heat so quickly as the bare soil. The escaping vapour rises till it meets with some surface not in good heat communication with the ground, and which has been cooled by radiation, in the manner set forth by WELLS and others, These remarks refer to weather when dew is most abundant, as in spring, summer, and autumn, and do not apply to those conditions in which a warm vapour-laden air is brought over a cold ground. DEW AND WIND. It is well known that during windy nights no dew is formed. We pre- viously knew that wind acts in two ways to prevent the formation of dew; to these two ways we must now add a third. Wind prevents the formation of dew—(1) by mixing the hot air above the surface of the ground with the air cooled near its surface, this tends to prevent the air being cooled to the dew- MR JOHN AITKEN ON DEW. 27 point; (2) the wind by its passage over the surface of radiating bodies prevents these surfaces being cooled much below the temperature of the air; the wind thus tends to prevent the air in contact with these surfaces being cooled below the dew-point; and (8) wind blowing over the surface of the ground rapidly carries away the vapour rising from the soil, and mixes it up with a large quantity of drier air. The wind thus tends to prevent an accumulation of damp air near the ground. To illustrate this third effect of wind, let us use the observations made on the evening of October 12. The sky was clear, and there was a considerable amount of radiation, but a slight wind was blowing. The bare soil in the test- pan lost 22 grains and the corresponding turf lost 30 grains in about five hours. Almost no dew was formed on the grass, but trays placed over the bare soil and over grass had their inside surfaces covered with moisture, though not so heavily as was generally observed on dewy nights. The reason why so little dew formed on this occasion was, partly, that the wind prevented the tempera- ture of the air near the ground falling as much as it would have done if it had been calm. In the screen the temperature only fell to 40°. On the grass, however, it fell to 31°°5, and on the soil to 35°°5; but a good deal depended on the exposure of the thermometer to the wind. From the above we see that, though wind was blowing, the thermometer on the grass fell a good deal below the temperature of the air, and showed a considerable amount of radiation.. The wind apparently prevented the formation of dew on this occasion, principally by preventing an accumulation of moist air near the surface of the ground. The inverted trays showed that if the wind had fallen dew would have formed, because it formed in the still air under the trays. The deposit was not so heavy inside the trays on this occasion as was often seen in dewy nights, because the wind prevented the radiation cooling the top of the trays to the same extent as when it was calm, DEW AND VEGETATION. When I began to make observations on dew, one of the first things I did was to make a tour of the garden on a dewy night, and to examine the appearance of the plants. A very short survey was sufficient to show that something else was at work than radiation and condensation to produce the effects then seen. Let me briefly describe what I saw, and what at once struck me could not be explained by the ordinary laws of radiation and con- densation. Certain kinds of plants were found to be covered with moisture, while others were dry. Many plants of the Brassica family were heavily covered with glistening drops; while beans, peas, &c., growing alongside them, were quite dry. Again, in clusters of plants of the same kind some were wet, while others were not; and not only so, but some branches were wet, while 28 MR JOHN AITKEN ON DEW. others on the same plant were dry. These differences were noticed to be quite irrespective either of their exposure to the sky, or to the probable humidity of the air surrounding them. Tn illustration of this latter point, small clusters of dwarf French poppies may be mentioned. Most of the plants were quite dry, whilst others growing amongst them were dripping with moisture; and while some branches were dry, others on the same plant were studded with drops, and the general surface of the leaves in some cases wet. On examination of these plants next day, it was observed that those that were wet at night were all plants in vigorous growth, and the shoots that were dewed were those in which the vegetation seemed most active. It was also observed that it was always the same plants and branches that were dewed night after night during the short time the observations were made. A closer examination of the leaves of broccoli plants showed better than any others that the moisture collected on them was not deposited in the manner we should expect if it had been deposited as an effect and according to the laws of radiation ; nor was it deposited in accordance with the laws of condensation; indeed, every appearance was at variance with these laws. Examination showed that the moisture was collected in little drops placed at short distances apart, along the very edge of the leaf, while the rest of the leaf was often dry. Now, if the moisture had been condensed by cold produced by radiation, then it would have been most abundant on the upper surface of the leaf; but there would have been none on its windward edge. This is well seen when we expose a small glass plate on a dewy night; the windward edge is always dry, and the deposit is spread evenly over the rest of the plate up to the opposite margin, because the temperature of the air when it first strikes the plate is higher than the dew-point, and it has to travel over more or less of the surface of the glass before it is cooled enough to deposit its moisture. Again, if these drops on the edge of the leaf had been deposited according to the laws of condensation, then the moisture would have been deposited on the surface more in accordance with the distribution of temperature at the different points; the moisture would therefore have been more equally distributed, and not been in large isolated drops. On further examining these plants, I placed the lantern behind the blade, and then observed that the position of the beautiful sparkling diamond-like drops that fringed its edge had a definite relation to the structure of the leaf; they were all placed at the points where the nearly colourless and semi-transparent veins of the leaf came to the outer edge, at once suggesting that these veins were the channels from which the drops had been expelled. These isolated drops on the edges of the leaves were therefore evidently not dew, but an effect of the vitality of the plants. .An examination of grass MR JOHN AITKEN ON DEW. 29 blades showed that they also tend to have large drops attached to them, while the rest of the blade is dry, and these drops were always found to be situated at certain definite points; they were always near the tips of the blades. These large drops seen on plants at night are therefore not dew at all, but are watery juices exuded by the plants. Now this excretion of water by the leaves of growing plants is not a new discovery—it has been long well known. But what seems extremely curious is, that its relation to dew has never been recognised, at least so far as I am aware, and it must be admitted that it is one of considerable importance. It is well known that plants transpire from their leaves an immense amount of moisture, which passes off in an invisible form. Prof. J. Boussineautr found that mint transpired 82 grammes of water per square metre in sunshine, and 36 grammes in shade; but if the roots of the plants were removed, they only transpired 16 and 15 grammes respectively. This simple experiment proves that the root sends into the stem of the plant a supply of water, that it acts as a kind of force-pump, and keeps up a pressure inside the tissues of the plant. This supply sent in by the root is in most conditions removed by means of transpiration from the surface of the leaves. Now what will be the result if transpiration is checked, while the root con- tinues to send forward supplies? It will evidently depend on two things— first, the pressure the root is capable of exerting before its action is stopped; and second, the freeness with which the water can escape from the leaves. If the root pressure is small, it will cease with the transpiration; but if it is great, the sap will be forced into the plant, and if nature has provided any outlets it will escape at these openings. Dr J. W. Moou* has given great attention to the subject, and has experi- mented on a number of plants. The method he employed in his researches was to place the leaves under the most favourable conditions for the excretion of drops, by diminishing the transpiration as far as possible, and by supplying them with water. He substituted for root pressure, a pressure produced by a column of mercury. Out of 60 plants experimented on by Dr Moot, he found that the leaves of 29 excreted drops without being injected, 13 leaves became injected and excreted drops, and 18 became injected and did not excrete at all. He says that the excretion takes place by water-pores, and by ordinary stomata, while in some cases it occurs at surfaces possessing neither of these organs. I have recently made a few experiments on this subject in its relation to dew. As, however, the season was far advanced before the experiments were begun, but little could be accomplished, for the activity of the plants was nearly over, and grass was almost the only plant possessing sufficient vitality for * Nature, vol. xxii. p. 403. 30 MR JOHN AITKEN ON DEW. experimenting. I however removed a branch of the poppy, which, during summer, had shown such a tendency to exude moisture, and connected it by means of an india-rubber tube with a head of water of about one metre. After placing a glass receiver over it, so as to check evaporation, it was left for two or three hours, when it was found to have excreted water freely—some parts of the leaves being quite wet, while drops had collected at other places. The broccoli plants which had excited my interest in summer were also experimented with. A full-grown leaf was fitted into the apparatus, and the pressure applied. In a little over an hour it also exuded water, and soon got fringed with drops along its edge in exactly the same way that was observed on it in summer. Another leaf from the same plant, but much younger, being about one quarter grown, on being tested in the same way did not excrete at all, after the pressure had been applied for twenty-four hours. Here we have the same result as that noticed in summer—one leaf exudes, while another on the same plant does not. | If the water pressed into the leaf is coloured with aniline blue, the drops when they first appear are colourless, but before they grow to any size, the blue appears, showing that little water was held in the veins, but the whole leaf got coloured of a fine deep blue-green, like that seen when vegetation is very rank, showing that the injected liquid had penetrated through the whole leaf. Most of my experiments on this subject were made with grass. I find that even in the middle of October, after having been severely frosted two or three times, which had probably reduced its vitality, it still exuded so abun- dantly that drops collected in air which was not saturated. A turf placed in a cellar, dry enough to keep glass quite free from dewy deposit, soon collected drops. These drops always appear near the tips of the blades; they are not exuded from every blade, and sometimes from only one on each stalk, but generally from more ; and it is always from the blades that seem to have the greatest vitality, and are nearly, but not quite full grown. Sometimes it is the youngest blade that exudes, but if it is very small, it is the second youngest. As the blades grow old they cease to exude; but this seems to be due to some change in the blade at the point where it exuded, and not to a diminution of root pressure, as it exudes freely when the tip is cut off. The question might be here raised, Are these drops really exuded by the plant Are they not due to some condensing power possessed by the leaves, by the presence at these points of some substance possessing an affinity for water vapour, or some process by which they may extract moisture from the air?. To get an answer to this question, I selected a small turf, placed over it a glass receiver, and left it till drops were excreted. Removing the receiver, a blade having a drop attached to it was selected. After being MR JOHN AITKEN ON DEW. 31 carefully dried, the tip of the blade was placed in a small glass receiver, so as to isolate it from the damp air of the larger receiver. This small covering glass measured about 10 mm. in diameter by about 15 mm. in height. Its open end was closed by means of a very thin plate of metal cemented to it. In the centre of this plate was pierced a small opening, of the same size and shape as the selected blade of grass. The tip of the blade was entered about 5 mm. into this small receiver, and to prevent moisture entering and coming in contact with the tip of the blade, an air-tight joint between the blade and the metal was made with india-rubber solution. The tip of the blade was thus isolated inside the small receiver in which the air was dry. The large glass receiver was then placed over the turf to prevent evaporation from the lower part of the blade, or the experiment was made in a room where the air was not very dry. After a time, generally some hours, the turf was examined. A drop was always found to have formed on the tip of the blade inside the small receiver, and this drop was, as nearly as could be judged, always as large as the drops formed in the moist air under the large receiver. It would thus appear that these drops are really exuded by the plant, and not extracted from the air. These exuded drops seem to be almost entirely the result of root pressure, because if we cut off the roots, and place the stems in water, putting over all a glass receiver standing in water so as to saturate the air, and as a test that the conditions are favourable, placing a small turf alongside the cut grass under the receiver, we shall find that scarcely any drops make their appearance on the rootless stems, while those with roots have drops attached to them, Again, if we take one of these rootless stems, and attach it by means of the india-rubber tube to a head of water, it is found to exude drops at the tips of its blades in moist air in the same way as when it was attached to its roots. These excreted drops are formed on grass on other than dewy nights, After rain, if there has been no wind, and the air near the ground becomes saturated, a rearrangement of the drops takes place. Some time after the rain has ceased, most of the blades will be found to be tipped with a drop at the same point as the exuded drop appeared at night—a position which no falling rain drop could keep. This tendency of plants to exude moisture explains why the grass is almost always wet during autumn. At that season evapora- tion is slow, and as the plants are constantly pouring in supplies to the drops, it takes a long time for the slow evaporation to overcome the wetting effect and dry up the grass. The question as to what degree of humidity in the air is necessary before plants will exude drops, would seem to be greatly determined by the rate at which the supply is sent into the leaf. If the supply is greater than the 32 MR JOHN AITKEN ON DEW. evaporation from the whole surface of the leaf, the drop grows; but if the supply is less, it does not form, or if formed, it decreases in size. ‘The rate of the supply will evidently depend on the kind of plant and the amount of its vital activity at the time. The formation of drops on plants that exude moisture will therefore depend on the rate of supply, the humidity of the air, and the velocity of the wind. It is not easy to get a satisfactory experimental answer to this question, on account of the soil near the grass tending to moisten the air over it. A small turf placed in an elevated position in the centre of a room has been observed to have drops on it, when there was a difference of more than one degree between the wet and the dry bulb thermo- meter hung alongside. As the drops are exuded at the tips of the blades, it is probable the air in contact with them was not much moistened by the small area of soil underneath. These observations entirely do away with the explanation usually given of the tendency of grass to get wet early and heavily on dewy nights. It has generally been explained by saying that grass is a better radiator than most substances, and therefore cools more, and sooner, than other bodies. We now see that those drops that first make their appearance on grass are not drops of dew at all, and their appearance depends, not on the laws of dew, but on those of vegetation. Hence the varied distribution of moisture on plants and shrubs on dewy nights. We have seen that much of the moisture that collects on plants at night does not form like dew on dead matter. Dead matter gets equally wet where equally exposed, and the moisture does not collect on it in isolated drops, as it does on plants. Those drops which appear on grass on clear nights are not dew, and they make their appearance on surfaces that are not cooled to the dew-point. If the radiation effect continues after these drops have been forming for some time, true dew makes its appearance, and now the plants get wet all over their exposed surfaces in the same manner as dead matter. This latter form of wetting or true dew is of rarer occurrence than we might at first imagine. On many nights on which grass gets wet, no true dew is deposited on it; and on all nights, when vegetation is active, the exuded drops always make their appearance before the true dew; so that when we walk in early evening over the wet lawn, it is not dew that we brush off the grass with our feet, but the sap exuded by the plant itself. The difference between these exuded drops and true dew can be detected at a glance. The moisture exuded by grass is always excreted at a point situated near the tip of the blade, and forms a drop of some size, which may form while the rest of the blade is dry, but true dew collects evenly all over the blade. The exuded liquid forms a large glistening diamond-like drop, whereas dew coats the blade with a fine pearly lustre. MR JOHN AITKEN ON DEW. 33 I feel that the dissecting hand of science has here done an injury to our poetic feelings. Every poet who has sung of the beauties of nature has added his tribute to the sparkling dew-drop, and BALLANTINE in his widely-known song has taught a comforting lesson from the thought that “ilka blade o’ grass keps its ain drap 0’ dew.” No doubt the drop of dew to which the poets refer is the large sparkling diamond-like gem that tips the blades of grass, and which we now know is not dew at all. While, however, our interpretation of nature has changed, the teaching of the poet remains, and the sparkling dew-drop may still teach the same comforting lesson. We must, however, change our views regarding the source of the refreshing influence. We may no longer look upon it as showered down from without, but as welling up from within—no longer as taken by the chill hand of night and given to refresh and invigorate exhausted nature ; we must rather look upon it as suggesting that we are provided with an internal vitality more than sufficient to restore our exhausted powers, after the heat and toil of the day are past. RADIATION. I have said in a previous part of this paper that the surface of bare soil and of roads will radiate at night as much heat as grass. It may be thought I have said this simply because we do not now require that grass should be the more powerful radiator to enable us to explain its greater wetness on dewy nights. Though it is not now necessary to suppose that grass is a powerful radiator, yet there is nothing in the above experiments to prove it either a good or a bad one. It therefore seemed desirable that some definite experiments be made on this point, and also to determine the radiating powers of different substances at night, as this is always an interesting and important point in questions con- nected with the deposition of dew; and the radiating power of grass, though not the principal cause of its wetness at night, might be still considered to play a subordinate part. We have already a great number of experiments on the radiating powers of different substances. Unfortunately most of the accurate measurements of this kind are from laboratory experiments, and do not appear to bear very directly on our subject. FRANKLIN’S early experiments, made with different coloured cloths placed on snow, seem to have given our ideas an unfortunate bias on this subject. From observing the different depths to which cloths of dif- ferent colours sunk in snow, when exposed to solar radiation, he came to the conclusion that the dark colours absorb most heat, and this conclu- sion seems for long to have influenced our ideas. If the heat radiated and absorbed by a surface was composed entirely of visible rays, then no doubt the colour of a body would be an index of its radiating and absorbing powers. VOL. XXXII. PART I. E 34 MR JOHN AITKEN ON DEW. But as the eye gives us no information about the greater proportion of the radiant energy, its indications are of no value in determining the radiating and absorbing powers of different surfaces. Experiment shows that different surfaces have different absorbing powers for different rays. MELLOoNI, for instance, found that white lead absorbed only about half as much heat from a Locatelli lamp as lamp black did, while it absorbed as much as lamp black when the source of heat was copper at 100° C. It is evident from this, that we cannot take the result of experiments made in the laboratory, and apply them to surfaces exposed to the temperature of the sky on a clear night. It may be possible that the radiating and absorbing powers of different surfaces may bear the same proportion to each other when the temperature is 0°, and they radiate into space, as when their temperature is 100°, and they are exposed to surfaces at the ordinary tem- perature of the laboratory. This may be so, but till it is proved we can- not apply these laboratory experiments to the cooling effect of radiation at night. Some experiments on the radiating power of different substances exposed to a clear sky were made by Danrett. He used for his purpose two similar parabolic reflectors. In the focus of each was placed the bulb of a ther- mometer. In experimenting he turned the reflectors to the sky, and coated the bulbs of the thermometers with the substances to be tested. Comparing garden mould with black wool, his measurements show, from the average of three readings given by him, that while the black wool fell 9° below the temperature of the air, the mould fell only 6°. The difference between the radiating powers of chalk and black wool, as given by him, was not quite so ereat. There seems to be an objection to this method of experimenting. The different surfaces here lose more heat by radiation into space than they receive. To supply this loss, they receive more heat by radiation from the reflector than they give, and they also receive heat from the surrounding air, conveyed to them by connection currents. Now in the experiment as arranged by DANIELL, the two surfaces will not receive the same amount of heat from the latter source. The wool surface will not have such a free circulation of air over it as the other one; it will therefore not receive so much heat, and its temperature will thus tend to fall lower. It appeared that something more might be done in this direction, and on con- sideration it was thought that the radiation thermometers, described by me in a previous paper, might be suitable for the purpose. It may be remembered that the principle on which these radiation thermometers is constructed is, that a large surface is more highly heated than a small one by radiation during the day, on account of the absorbed heat being more slowly taken away by the passing air from the former than from the latter; and for a similar reason MR JOHN AITKEN ON DEW. 39 a large surface is colder at night than a small one, as the small surface receives more heat, per unit of area, from the air than the large one. The absorbing and radiating surface of these instruments is a large flat area, painted black, and its temperature is taken by means of a thermometer, with its bulb placed under the centre of the radiating surface.* The construction of these radiation instruments has been altered, and those used in this investigation were made of metal in place of wood, as described in the previous paper, the radiating surface being a thin plate of metal, 14 inches (355 mm.) square. A thin metal tube is fixed close to and parallel with the under surface of the plate. One end of the tube terminates at the centre of the plate, and the other at the edge. The thermometer is placed in this tube with its bulb under the centre of the plate, and to prevent heat escaping or being absorbed at the back, a considerable thickness of cotton wool is placed under it. The instrument is practically a shallow box, 14 inches square by 2 inches (51 mm.) deep, packed with cotton wool. One of the flat areas of the box is exposed to radiation, and its temperature is taken by means of a thermometer placed under its surface. In the following I shall refer to this instrument simply as the thermometer box. One of the advantages of this form of instrument for solar radiation experiments is, that the readings given by different instruments agree with each other, at least this is the case so far as my experience goes; and it is well known that the vacuum radiation thermometers are unsatisfactory in this respect, no two almost ever reading alike. For instance, the vacuum radiation thermometers used at the Indian Stations, when compared with another of the same pattern as standard, were in some cases found to differ as much as 15°, though they were exact copies of each other, and similarly exposed.t I find that when the different instruments of the kind used by me are compared they agree very well when of the same size. It is of course necessary that they be of the same size—this results from the principle of their construction. It seems possible that we might make boxes of different sizes, and from them determine the law of variation for size; so that, knowing the size of the surface used in any particular set of observations, we could determine what temperature its readings corresponded to in another instrument of a different size, or all readings might be reduced to a standard size, say the temperature of a very large surface. I may mention that the temperature given by an instrument of the size here described when placed in sunshine is a good deal above that indicated by a vacuum thermometer, which had been carefully prepared for me by CasELLA of London. Generally the readings were about 12 per cent. higher. * Thermometer Screens, Proceedings of the Royal Society, Edinburgh, No. 117, 1883-84. t Report of the Meteorology of India, 1879, by H. F. Blanford, F.R.S. 36 MR JOHN AITKEN ON DEW. One objection to these large-surface radiation thermometers is that they are more affected by wind than the vacuum ones. If it is a question of solar energy we are considering, this certainly is an objection, but if it is one of climate it will scarcely be so. I need not say that for questions of terrestrial radiation at night the vacuum thermometer is of no use. In using these thermometer boxes for determing the radiating powers of different surfaces at night the following method was employed :—Two precisely similar boxes were prepared, and their upper surfaces painted black. They were placed in an elevated position in the open air, commanding a clear view of the sky all round. They were first exposed without anything on their surfaces, to see if their readings were exactly alike. In constructing them care was taken to put the same amount of cotton wool in each, in order that their non-conducting powers and heat capacities might be the same, so that both might take the same amount of heat to warm them, and both lose the same amount of heat at the back. On trial both instruments were found to read alike when similarly exposed. As the sky radiation is a rather variable quantity, it would not do, on most nights, to leave one of these test surfaces bare, and use it as a standard with which to compare the other, over which we have put the substance to be tested, because the uncovered surface will follow the changes in the radiation more easily than the other, and will change more, and sooner, than the one covered with the substance to be tested, particularly if the substance is a bad conductor. The method generally adopted was to place both surfaces as nearly as possible under the same conditions. For instance, the first substances tested were black and white cloths of different materials; of each kind a black and a white was selected, each pair being as much alike as possible, of the same material, of the same weight, and of the same texture. 63 laboratory, where it was connected by means of an india-rubber tube with a head of water of about 1°5 metres, and surrounded with saturated air. After a time drops appeared at the tips of most of the leaves, and also at some other points on them; but these drops were quite unlike those on grass, broccoli, and other water-repelling plants ; they spread themselves on the leaves, and adhered to them, no reflection being given from the back of the flattened drop. It could, however, be easily seen, when the experiment was made in this way, that moisture is exuded from the plant, whereas at night no exuded moisture is perceptible. The reason for this is, that under the condition of the experiment, the exuded drop only spreads to a certain extent, and the outline of the wetted surface is defined, because the whole surface of the leaf is not wet ; but at night the surface of the leaf is wet with dew, and the exuded drop spreads and thins away by imperceptible degrees into the dewed surface. This was illustrated in the above experiment by breathing on the leaf, so as to bring it into the same condition it is on dewy nights, the drop was then seen to spread rapidly outwards. We see from the above that a plant may be exuding, and yet we may not be able to notice it. This is specially the case while dew is forming, that is under natural conditions ; for dew is very generally forming while plants are exuding, and it is difficult to tell from an examination made at night whether any plant whose leaves have an affinity for water is exuding or not. It is therefore much better to test the plants under artificial conditions, by placing them in saturated air, but where no dew can be formed on their surfaces. This can be done by placing them at night under hand-glasses, and well protected from radiation, or even during the day under metal boxes, and well shaded. In this way a few plants, whose leaves got wet with rain, were tested, and all were found to exude if the evaporation from the leaves was stopped long enough, and time given for the tissues to get filled with sap. In all cases the exuded moisture adhered to the leaf and formed a wet patch. The plants tested were helichrysum, stocks, asters, mignonette, foxglove, celery, lettuce, turnips. The plants were taken at hazard, and while some, such as mignonette and stocks, exuded little, the others discharged a good deal. The root pressure of a stock was measured, and found to be only about one-half that of the more freely exuding Helichrysum. The root pressure will, however, be only one factor in determining the amount exuded, as it is evident the rate of supply sent in by the root will be of as much importance ; but no measurements of quantity have been made by me. It may be as well to note here, that though the few plants, taken at hazard, all showed powers of exuding, yet we must not therefore conclude that all plants have this property. It is interesting to note the effects of these two ways in which the surface of leaves behave towards their exuded sap and water. Take the different kinds 64 MR JOHN AITKEN ON DEW. of turnips, for instance. The Swedish variety exudes freely, the liquid forming little drops fringing the leaves, while the moisture exuded by the other varieties spreads itself over the leaves. One result of this is, that after dewy nights the softer varieties dry sooner than the Swedish, because the exuded moisture, by spreading itself over the surface of the leaves, dries up much more quickly than the drops on the others. This seems to be the explanation of a fact fre- quently observed by sportsmen and others who have occasion to walk through turnip fields on autumn mornings, namely, that the softer varieties generally wet them much less than the swedes. Again, after rain the swedes take longer to dry than the others, because their surfaces do not get wet, but the water col- lects in drops, imperfectly attached to them, and also fills the hollows of their leaves ; whereas the other kinds get wet, and the water runs off them, leaving only a thin film on their surface, which dries up much more quickly than the drops on the others, Further, when we walk through turnips immediately after rain, our feet brush the drops from the swedes in showers, which rapidly wet us, While the water adheres to and does not so easily leave the surfaces of the others. This last part of the investigation takes us a step further, and shows us that not only is the dew-drop a. result of the vitality of those plants on which it forms, but that much of the wetness spread over the leaves of others on dewy nights is produced by moisture exuded by the plants. III.—On the Foundations of the Kinetic Theory of Guses. By Professor Tarr. (Revised May 14, 1886.) INDEX TO CONTENTS. PAGE - PAGE INTRODUCTORY, ; [00 Part VJ. On some Definite Integrals, Part I. One Setof Equal cee 1-5, . 67 SS20-2;, 84 » LI. Mean Free Path among Equal » WII, Mean Path in a nitro of Spheres, §§ 6-11, : (a two Systems, § 28, . . 86 », LILI. Number of Collisions per Particle » VIII. Pressure in a System of per Second, §§ 12-14, . iyato Colliding Particles, §§ 29, » LV. Clerk-Maxwell’s Theorem, §§15-22, 77 30, . 86 ,, . Rate of Equalisation of Average » IX. Effect of External Potential, Energy per Particle in two §$ 31, 32, , : + Ol Mixed Systems, §§ 23, 24, = 82 APPENDIX, ; : : » Oe The attempt to account for the behaviour of gases by attributing their apparently continuous pressure to exceedingly numerous, but nearly infinitesi- mal, impacts on the containing vessel is probably very old. It certainly occurs, with some little development, in Hooxe’s tract of 1676, Lectures de potentid resti- tutivd, or of Spring ; and, somewhat more fully developed, in the Hydrodynamica of D. Bernoutu, 1738. Traces of it are to be found in the writings of Lu SacE and PrEvost some 80 or 90 years ago. It was recalled to notice in 1847 by HeERApPATH in his Mathematical Physics, and applied, in 1848, by JouLe to the calculation of the average speed of the particles in a mass of hydrogen at various temperatures. JOULE expressly states* that his results are independent of the number of the particles, and of their directions of motion, as also of their mutual collisions. In and after 1857 Cuausius greatly improved the treatment of the problem by taking account not only of the mutual impacts of the particles but also of the rotations and internal vibrations which they communicate to one another, with the bearing of this on the values of the specific heats; at the same time intro- ducing (though only to a limited extent) the statistical method. In this series of papers we find the first hint of the length of the mean free path of a particle, and the explanation of the comparative slowness of the process of diffusion of one gas into another. But throughout it is assumed, so far as the calculations * The paper is reprinted Phil. Mag. 1857, II. See especially p. 215. VOL. XXXIII. PART I. I 66 PROFESSOR TAIT ON THE are concerned, that the particles of a gas are all moving with equal speeds. Of the Virial, which CLausius introduced in 1870, we shall have to speak later. In the Philosophical Magazine for 1860 CiERK-MaAxweELL published his papers on the “Collisions of Elastic Spheres,” which had been read to the British Association in the previous year. In this very remarkable investigation we have the first attempts at a numerical determination of the length of the mean free path. These are founded on the observed rate of diffusion of gases into one another; and on the viscosity of gases, which here first received a physical explanation. The statistical method is allowed free play, and conse- quently the law of distribution of speed among the impinging particles is investigated, whether these be all of one kind or a mixture of two or more kinds. One of his propositions (that relating to the ultimate partition of energy among two groups of colliding spheres), which is certainly fundamental, is proved in a manner open to very grave objections :—not only on account of the singular and unexpected ease with which the proof is arrived at, but also on account of the extraordinary rapidity with which (it seems to show) any forced deviation from its conclusions will be repaired by the natural operation of the collisions, especially if the mass of a particle be nearly the same in each system. As this proposition, in the extended form given to it by BoLTzMANN and others, seemed to render the kinetic theory incapable of explaining certain well-known experimental facts, I was induced to devote some time to a careful examination of MAxWELL’s proof (mainly because it appears to me to be the only one which does not seem to evade rather than boldly encounter the real difficulties of the question*), with the view of improving it, or of disproving the theorem, as the case might be. Hence the present investigation, which has incidentally branched off into a study of other but closely connected questions. The variety of the traps and pit-falls which are met with even in the elements of this subject, into some of which I have occasionally fallen, and into which I think others also have fallen, is so great that I have purposely gone into very minute detail in order that no step taken, however slight, might have the chance of escaping criticism, or might have the appearance of an attempt to gloss over a real difficulty. The greater part of the following investigation is concerned only with the most elementary parts of the kinetic theory of gases, where the particles are * Compare another investigation, also by CLerk-MaxweELu but based on Boxtzmann’s processes, which is given in Wature, viii. 537 (Oct. 23, 1873). Some remarks on this will be made at the end of the paper. Meanwhile it is sufficient to point out that this, like the (less elaborate) investigations of Meyer and Watson, merely attempts to show that a certain state, once attained, is permanent. It gives no indication of the rate at which it would be restored if disturbed. As will be seen later, I think that this “rate” is an element of very great importance on account of the reasons for confidence (in the general results of the investigation) which it so strikingly furnishes. FOUNDATIONS OF THE KINETIC THEORY OF GASES. 67 regarded as hard smooth spheres whose coefficient of restitution is unity. The influence of external forces, such as gravity, is neglected; and so is that of internal (molecular) forces. The number of spheres is regarded as extremely great (say of the order 10” per cubic inch): but the sum of their volumes is regarded as very small in comparison with the space through which they are free to move; as, for instance, of the order 10-* or 10-7‘. It will be seen that several of the fundamental assumptions, on which the whole investigation rests, are justified only by reference to numbers of such enormous magnitude, or such extreme minuteness, as the case may be. The walls of the containing vessel are supposed simply to reverse the normal velocity of every sphere impinging on them. I. One set of Equal Spheres. 1. Very slight consideration is required to convince us that, unless we suppose the spheres to collide with one another, it would be impossible to apply. any species of finite reasoning to the ascertaining of their distribution at each instant, or the distribution of velocity among those of them which are for the time in any particular region of the containing vessel. But, when the idea of mutual collisions is introduced, we have at once, in place of the hopelessly com- plex question of the behaviour of innumerable absolutely isolated individuals, the comparatively simple statistical question of the average behaviour of the various groups of a community. This distinction is forcibly impressed even on the non-mathematical, by the extraordinary steadiness with which the numbers of such totally unpredictable, though not uncommon, phenomena as suicides, twin or triple births, dead letters, &c., in any populous country, are maintained year after year. On those who are acquainted with the Fiption developments of the mathe- matical Theory of Probabilities the impression is still more forcible. Every one, therefore, who considers the subject from either of these points of view, must come to the conclusion that continued collisions among our set of elastic spheres will, provided they are all equal, produce a state of things in which the per- centage of the whole which have, at each moment, any distinctive property must (after many collisions) tend towards a definite numerical value; from which it will never afterwards markedly depart. This principle is of the utmost value, when legitimately applied; but the present investigation was undertaken in the belief that, occasionally at least, its powers have been to some extent abused. This appears to me to have arisen from the difficulty of deciding, in any one case, what amount of completeness or generality is secured when the process of averaging is applied in successive steps from the commencement to the end of an investigation, mstead of being reserved (as it ought to be) for a single comprehensive step at the very end. 68 PROFESSOR TAIT ON THE Some of the immediate consequences of this principle are obvious without calculation: such as (za) Even distribution, at any moment, of all the particles throughout the space in which they move. (b) Even distribution of direction of motion among all particles having any one speed, and therefore among all the particles. (c) Definite percentage of the whole for speed lying between definite limits. These apply, not only to the whole group of particles but, to those in any portion of space sufficiently large to contain a very great number of particles. (dz) When there are two or more sets of mutually colliding spheres, no one of which is overwhelmingly more numerous than another, nor in a hopeless minority as regards the sum of the others, similar assertions may be made as to each set separately. 2. But calculation is required in order to determine the law of grouping as to speeds, in (c) above. It is quite clear that the spheres, even if they once had equal speed, could not possibly maintain such a state. [I except, of course, such merely artificial distributions as those in which the spheres are supposed to move in groups in various non-intersecting sets of parallel lines, and to have none but direct impacts. For such distributions are thoroughly unstable; the very slightest transverse impact, on any one sphere, would at once upset the arrangement.| For, when equal smooth spheres impinge, they exchange their velocities along the line of centres at impact, the other components being unchanged ; so that, only when that line is equally inclined to their original directions of motion, do their speeds, if originally equal, remain equal after the completion of the impact. And, as an extreme case, when two spheres impinge so that the velocity of one is wholly in the line of centres at impact, and that of the other wholly perpendicular to it, the first is brought to rest and the second takes the whole kinetic energy of the pair. Still, what- ever be the final distribution of speeds, it is obvious that it must be in- dependent of any special system of axes which we may use for its computation. This consideration, taken along with (%) above, suffices to enable us to find this final distribution. 3. For we may imagine a space-diagram to be constructed, in which lines are laid off from an origin so as to represent the simultaneous velocities of all the spheres in a portion of space large enough to contain a very great number of them. Then () shows that these lines are to be drawn evenly in all direc- tions in space, and (¢) that their ends are evenly distributed throughout the space between any two nearly equal concentric spheres, whose centres are at the common origin. The density of distribution of the ends (2.¢., the number in unit volume of the space-diagram) is therefore a function of 7, that is, of 8 ————— FOUNDATIONS OF THE KINETIC THEORY OF GASES. 69 J/e2+y?2+z2. But the argument above shows, further, that this density must be expressible in the form LOFMFO whatever rectangular axes be chosen, passing through the origin. These joint conditions give only two admissible results: viz., either f(@)=A, orf (x) =Be™. The first is incompatible with the physical problem, as it would make the percentage of the whole particles, which have one definite speed, increase indefinitely with that speed. The same consideration shows @ fortiorz that, in the second form of solution, which is the only one left, C must be negative. Hence the density of the distribution of “ends” already spoken of is Big-2r? If n be the whole number of particles, 7.e., of “ends,” we must obviously have tobe 2dr = 1. 0 1 wd Tt . AN h>’ so that the number of spheres whose speed is between 7 and 7 +d7r is The value of the integral is h? Sy reared New srinsts tinea Cl af dr (1) This distribution will hereafter be spoken of as the “ special” state. The mean speed is therefore ioe (pa 2) 4 |= fre iy ae ae erin while the mean-square speed is h’ ys rt Ja — 3 tof POT =a This shows the meaning of the constant 2. [Several of the results we have just arrived at find full confirmation in the investigations (regarding mixed systems) which follow, if we only put in these P for Q passim :—i.e., pass back from the case of a mixture of spheres of two different groups to that of a single group. | 4. Meanwhile, we can trace the general nature of the process by which the “special” arrangement of speed expressed by (1) is brought about from any initial distribution of speed, however irregular. For impacts on the containing vessel do not alter 7, but merely shift the particular “end” in question to a 70 PROFESSOR TAIT ON THE different position on its spherical locus. Similarly, impact of equal particles does not alter the distribution of velocity along the line of centres, nor along any line perpendicular to it. But it does, in general, produce alterations in the distribution parallel to any line other than these. Hence impacts, in all of which the line of centres is parallel to one common line, produce no change in the arrangement of velocity-components along that line, nor along any line at right angles to it. But there will be, in general, changes along every other line. It is these which lead gradually (though very rapidly) to the final result, in which the distribution of velocity-components is the same for all directions. When this is arrived at, collisions will not, in the long run, tend to alter it. For then the uniformity of distribution of the spheres in space, and the symmetry of distribution of velocity among them, enable us (by the principle of averages) to dispense with the only limitation above imposed; viz., the parallelism of the lines of centres in the collisions considered. 5. In what precedes nothing whatever has been said as to the ratio of the diameter of one sphere to the average distance between two proximate spheres, except what is implied in the preliminary assumption that the sum of the volumes of the spheres is only a very small fraction of the space in which they are free to move. It is probable, though not (so far as I know) thoroughly proved, that if this fraction be exceedingly small the same results will ulti- mately obtain, but only after the lapse of a proportionately long time ; while, if it be infinitely small, there will be no law, as there will be practically no colli- sions. On the other hand, if the fraction be a large one (i.¢., as in the case of a highly compressed gas), it seems possible that these results may be true, at first, only as a very brief time-average of the condition of the spheres in any region large enough to contain a great number :—that, in fact, the distribution of particles and speeds in such a region will be for some time subject to con- siderable but extremely rapid fluctuations. Reasons for these opinions will be seen in the next section of the paper. But it must also be noticed that when the particles fill the greater part of the space in which they move, simultaneous impacts of three or more will no longer be of rare occurrence ; and thus a novel and difficult feature forces itself into the question. Of course with infinitely hard spheres the probability of such multiple collisions would be infinitely small. It must be remembered, however, that the investigation is meant to apply to physical particles, and not to mere mathe- matical fictions ; so that we must, in the case of a highly compressed gas, take account of the possibility of complex impacts, because the duration of an impact, though excessively short, is essentially finite. FOUNDATIONS OF THE KINETIC THEORY OF GASES. 71 Il. Mean Free Path among Equal Spheres. 6. Consider a layer, of thickness $x, in which quiescent spheres of diameter s are evenly distributed, at the rate of m, per unit volume. If the spheres were opaque, such a layer would allow to pass only the fraction 1—n,78"62/4 of light falling perpendicularly on it. But if, instead of light, we have a group of spheres, also of diameter s, falling perpendicularly on the layer, the fraction of these which (whatever their common speed) pass without collision will obviously be only 1 —ny7s6x ; for two spheres must collide if the least distance between their centres is not vreater than the sum of their radii. It is, of course, tacitly understood when we make such a statement that the spheres in the very thin layer are so scattered that no one prevents another from doing its full duty in arresting those which attempt to pass. Thus the fraction above written must be considered as differing very little from unity. In fact, if it differ much from unity, this consideration shows that the estimate of the number arrested will necessarily be exaggerated. Another consideration, which should also be taken into account is that, in consequence of the finite (though very small) diameter of the spheres, those whose centres are not in the layer, but within one diameter of it, act as if they were, in part, in the layer. But the corrections due to these considerations can be introduced at a later stage of the investigation. 7. If the spheres impinge obliquely on the layer, we must substitute for 6 the thickness of the layer in the direction of their motion. If the particles in the layer be all moving with a common velocity parallel to the layer, we must substitute for 6a the thickness of the layer in the direction of the relative velocity. If the particles in the -layer be moving with a common velocity inclined at an angle : — 6 to the plane of the layer, and the others impinge perpendicularly to the layer, the result will be the same as if the thickness of the layer were reduced in the ratio of sin@:1, and it were turned so as to make an angle 6 with the direction of motion of the impinging particles. 8. Now suppose the particles in the layer to be moving with common speed %, but in directions uniformly distributed in space. Those whose directions of motion are inclined at angles between 6 and B+df£ to that of the impinging particles are, in number, n, sin BdB/2 ; and, by what has just been said, if »v be the common speed of the impinging 72 PROFESSOR TAIT ON THE particles, the virtwal thickness of the layer (so far as these particles are con- cerned) is 62/2, where U= Jv? +v,2—2vv, cos B is the relative speed, a quantity to be treated as essentially positive. Thus the fraction of the impinging particles which traverses this set without collision is 1—n,7s*duv, sin B dB/2v. To find the fraction of the impinging particles which pass without collision through the layer, we must multiply together all such expressions (each, of course, infinitely nearly equal to unity) between the limits 0 and z of 8. The logarithm of the product is Ny Ts*6x [7 5 Daas Jv? +02 — 2vv, cos B. sin BdB. Making 2 the variable instead of 8, this becomes 2 _ ty s*de Pag 2vv, J”? U9: If v be greater than 2, the limits of integration are v—»v,, and v+%, and the expression becomes 9 vo? = mymstn( 1 + a) ; but, if v be less than ~,, the limits are v,—v and v, +2, and the value is Vv _ mymsed( x + 5) ‘ These give, as they should, the common value —4n,7s8°dx/3 when v=%. 9. Finally, suppose the particles in the layer to be in the “ special” state. If there be 7 in unit volume, we have for the number whose speed is between the limits v, and v,+ dv, hs —hv,2 a =Anv ido, / He ais: Hence the logarithm of the fraction of the whole number of impinging particles, whose speed is v and which traverse the layer without collision, is hs ” = ho,? v* “a2 (VV, , V,3 / 2 — Sn 1 2 hd! ie 9 1 1 1 —4ans Ne OA («, +3 )dut/s (FP +"")ao,) FOUNDATIONS OF THE KINETIC THEORY OF GASES. 73 The value of the factor in brackets is easily seen to be LUE a ae ~ tht Be ae (ata) 1 —hv? it i v where V= Mf en de, and thus it may readily be tabulated by the help of tables of the error-function. When v is very large, the ultimate value of the expression is iVB which shows that, in this case, the “special” state of the particles in the layer does not affect its permeability. 10. Write, for a moment, — Cox as the logarithm of the fraction of the particles with speed v which traverse the layer unchecked, Then it is clear that —exr é represents the fraction of the whole which penetrate unchecked to a distance «into a group in the “special” state. Hence the mean distance to which particles with speed v can penetrate without collision is co —e2 fe wd a} ari ye - This is, of course, a function of v; and the remarks above show that it increases continuously with v to the maximum value (when 2 is infinite) nis?’ i.¢., the mean path for a particle moving with infinite speed is the same as if the particles of the medium traversed had been at rest. 11. Hence, to find the Mean Free Path among a set of spheres all of which are in the special state, the natural course would appear to be to multiply the VOL. XXXIII. PART I. K 74 PROFESSOR TAIT ON THE average path for each speed by the probability of that speed, and™take the sum of the products. Since the probability of speed v to v+dv is 3 4 oe Leiden 7 the above definition gives for the length of the mean free path, TRY pe es 6 Ef Mon To or, by the expression for e above, 1 ays g~y2dy NITrs? v v ;) fe 18 v,3 —hv 32 (4, 2 1 ad —hv,2( 271 “Vad Cama v € =o v 0 J: ( : T 38 a v 3 Vv : This may without trouble (see § 9) be transformed into the simpler expression 1 i Aate-** da which admits of easy numerical approximation. The numerical work would be simplified by dividing above and below by «~”, but we prefer to keep the present form on account of its direct applicability to the case of mixed systems. And it is curious to note that 4¢~™ is the third differential coefficient of the denominator. The value of the definite integral (as will be shown by direct computation in an Appendix to the paper) is about 0°677 ; and this is the ratio in which the mean path is diminished in consequence of the motion of the particles of the medium. For it is obvious, from what precedes, that the mean path (at any speed) if the particles were quiescent would be NTIS” |The factor by which the mean path is reduced in consequence of the “special” state is usually given, after CLERK-MAXwWELL, as 1/,/2 or 0°707. But this appears to be based on an erroneous definition. For if 2, be the fraction of the whole particles which have speed 2, p, their free path; we have taken the mean free path as X(N), according to the usual definition of a “ mean.” FOUNDATIONS OF THE KINETIC THEORY OF GASES. ra) CLERK-MAXxWELL, however, takes it as (npr) X (1 0/Py) i.é., the quotient of the average speed by the average number of collisions per particle per second. But those who adopt this divergence from the ordinary usage must, I think, face the question “‘ Why not deviate in a different direction, and define the mean path as the product of the average speed into the average timé of describing a free path?” This would give the expression X(NyV) . U(Nypy/) . The latter factor involves a definite integral which differs from that above solely by the factor ,/h/x in the numerator, so that its numerical deternination is easy from the calculations already made. It appears thus that the reducing factor would be about 2 Nie z.é., considerably more in excess of the above value than is that of CLERK- Maxwe.. Until this comparatively grave point is settled, it would be idle to discuss the small effect, on the length of the mean free path, of the diameters of the impinging spheres. | x 0650, =0°754 nearly ; III. Number of Collisions per Particle per Second. 12. Here again we may have a diversity of definitions, leading of course to different numerical results. Thus, with the notation of § 11, we may give the mean number of collisions per particle per second as 2(Nyd/Pv). This is the definition given by CLerK-MaxweELL and adopted by Meyer ; and here the usual definition of a “mean” is employed. The numerical value, by what precedes, is 233 2 ne? enw? ve Te UV; ve 16ns*h WG vido ft (02-435, dn, + Je (G+ A; )av,). MEYER evaluates this by expanding in an infinite series, integrating, and sum- ming. But this circuitous process is unnecessary ; for it is obvious that the two parts of the expression must, /rom their meaning, be equal; while the second part is integrable directly. 13. On account of its bearing (though somewhat indirectly) upon the treat- 76 PROFESSOR TAIT ON THE ment of other expressions which will presently occur, it may be well to note that a mere inversion of the order of integration, in either part of the above double integral, changes it into the other part. Otherwise :—we may reduce the whole to an immediately integrable form by the use of polar co-ordinates ; putting v=7 0080.) %,=71s1n0, and noting that the limits of 7 are 0 to o in both parts, while those of fare 0 to 7/4 in the first part, and 7/4 to 7/2 in the second. [This trans- formation, however, is not well adapted to the integrals which follow, with reference to two sets of spheres, because / has not the same value in each set. | 14. Whatever method we adopt, the value of the expression is found to be Se mes see am ys Ns" = 2 ns ; and, as the mean speed is (§ 3) ue Jth’ we obtain CLERK-MAXwELL’s value of the mean path, above referred to, viz., lone nis” /2 But (in illustration of the remarks at the end of § 11) we might have defined the mean number of collisions per particle per second as (nw) ,; al Tape)" © Saepr]) ; &e., &e. The first, which expresses the ratio of the mean speed to the mean free path, gives Bie METS Jah 0677’ and the second, which is the reciprocal of the mean value of the time of describing a free path, gives 2 TNS 1 Jh 0°650° The three values which we have adduced as examples bear to one another the reciprocals of the ratios of the above-mentioned determinations of the mean free path. FOUNDATIONS OF THE KINETIC THEORY OF GASES. TT IV. Clerk-Maxwell’s Theorem. 15. In the ardour of his research of 1859,* MAXxweE tu here and there con- tented himself with very incomplete proofs (we can scarcely call them more than illustrations) of some of the most important of his results. This is specially the case with the investigation of the law of ultimate partition of energy in a mixture of smooth spherical particles of two different kinds. He obtained, in accordance with the so-called Law of Avogadro, the result that the average energy of translation is the same per particle in each system; and he extended this in a Corollary to a mixture of any number of different systems. This proposition, if true, is of fundamental importance. It was extended by MAaxweEL. himself to the case of rigid particles of any form, where rotations perforce come in. And it appears that in such a case the whole energy is ultimately divided equally among the various degrees of freedom. It has since been extended by BottTzmMann and others to cases in which the individual particles are no longer supposed to be rigid, but are regarded as complex systems having great numbers of degrees of freedom. And it is stated, as the result of a process which, from the number and variety of the assumptions made at almost every stage, is rather of the nature of playimg with symbols than of reasoning by consecutive steps, that in such groups of systems the ultimate state will be a partition of the whole energy in equal shares among the classes of degrees of freedom which the individual particle-systems possess. This, if accepted as true, at once raises a formidable objection to the kinetic theory. For there can be no doubt that each individual particle of a gas has a very ereat number of degrees of freedom besides the six which it would have if it were rigid :—the examination of its spectrum while incandescent proves this at once. But if all these degrees of freedom are to share the whole energy (on the average) equally among them, the results of theory will no longer be consistent with our experimental knowledge of the two specific heats of a gas, and the relations between them. 16. Hence it is desirable that CLERK-Maxwe v’s proof of his fundamental Theorem should be critically examined, and improved where it may be found defective. If it be shown in this process that certain preliminary conditions are absolutely necessary to the proof even of CLERK-MAXWELL’s Theorem, and if these cannot be granted in the more general case treated by BoLTzMann, it is clear that Bottzmann’s Theorem must be abandoned. 17. The chief feature in respect of which MaxweEtt’s investigation is to be commended is its courageous recognition of the difficulties of the question. In this respect it far transcends all other attempts which I have seen. Those * Phil. Mag., 1860. 78 PROFESSOR TAIT ON THE features, besides too great conciseness, in respect of which it seems objec- tionable, are :— (a) He assumes that the transference of energy from one system to the other can be calculated from the results of a single impact between particles, one from each system, each having the average translational energy of its system. Thus (so far as this step is concerned) the distribution of energy in each system may be any whatever. (b) In this typical impact the velocities of the impinging spheres are taken as at right angles to one another, so that the relative speed may be that of mean square as between the particles of the two systems. The result obtained is fallacious; because in general the directions of motion after impact are found not to be at right angles to one another, as they would certainly be (on account of the perfect reversibility of the motions) were this really a typical impact. (c) CLeRK-MAxXwWELL proceeds as if every particle of one system impinged upon one of the other system at each stage of the process—z.e., he calculates the transference of energy as if each pair of particles, one from each system, had simultaneously a typical impact. This neglect of the immensely greater number of particles which either had no impact, or impinged on others of their own group, makes the calculated rate of equalisation far too rapid. (d) Attention is not called to the fact that impacts between particles are numerous in proportion to their ve/ative speed, nor is this consideration intro- duced in the calculations. (2) Throughout the investigation each step of the process of averaging is performed (as a rule) before the expressions are ripe for it. 18. In seeking for a proof of MAxwELL’s Theorem it seems to be absolutely essential to the application of the statistical method to premise :— (A) That the particles of the two systems are thoroughly mixed. (B) That in any region containing a very large number of particles, the particles of each kind separately acquire and maintain the error-law distribu- tion of speeds—z.e., each set will ultimately be in the “special” state. The disturbances of this arrangement produced in either system by impacts on members of the other are regarded as being promptly repaired by means of the internal collisions in the system itself. This is the sole task assigned to these internal collisions. We assume that they accomplish it, so we need not further allude to them. [The warrant for these assuinptions is to be sought as in § 4; and in the fact that only a small fraction of the whole particles are at any instant in collision; 7.¢., that each particle advances, on the average, through a consider- able multiple of its diameter before it encounters another. | (C) That there is perfectly free access for collision between each pair of particles, whether of the same or of different systems ; and that, in the mixture, FOUNDATIONS OF THE KINETIC THEORY OF GASES. 79 the number of particles of one kind is not overwhelmingly greater than that of the other kind. [This is one of the essential points which seem to be wholly ignored by BoitzMANN and his commentators. There is no proof given by them that one system, while regulating by its internal collisions the distribution of energy among its own members, can also by impacts regulate the distribution of energy among the members of another system, when these are not free to collide with one another. In fact, if (to take an extreme case) the particles of one system were so small, in comparison with the average distance between any two contiguous ones, that they practically had no mutual collisions, they would behave towards the particles of another system much as LE SAGE supposed his ultra-mundane corpuscles to behave towards particles of gross matter. Thus they would merely alter the apparent amount of the molecular forces between the particles of a gas. And it is specially to be noted that this is a question of effective diameters merely, and not of masses :—so that those particles which are virtually free from the self-regulating power of mutual collisions, and therefore form a disturbing element, may be much more massive than the others. | 19. With these assumptions we may proceed as follows:—Let P and Q be the masses of particles from the two systems respectively ; and when they impinge, let u, v be their velocity-components measured towards the same parts along the line of centres at impact. If these velocities become, after impact, u’, v’ respectively, we have at once P(w’—u) = 25 (wv) = —Q(v’—¥): an immediate consequence of which is 4PQ Ge a amaye (Pur— Qv2—(P— Q)uv) = —Q(v?—Vv’). Hence, denoting by a bar the average value of a quantity, we see that trans- ference of energy between the systems must cease when Pr Oe-(PsOmv=0, 9.8 fo, 6. ly, and the question is reduced to finding these averages. [I thought at first that uv might be assumed to vanish, and that u’ and v’ might each be taken as one-third of the mean square speed in its system. This set of suppositions would lead to Maxwet.’s Theorem at once. But it is clear that, when two particles have each a given speed, they are more likely to collide when they are moving towards opposite parts than when towards the same parts. Hence uv must be an essentially negative quantity, and therefore Pu? necessarily less than Qv’, if P be greater than Q. Thus it 80 PROFESSOR TAIT ON THE seemed as if the greater masses would have on the average less energy than the smaller. These are two of the pitfalls to which I have alluded. Another will be met with presently. | 20. But these first impressions are entirely dissipated when we proceed to calculate the average values. For it is found that if we write (1) in the form Pu2—uv—Qv?—uv=0, the terms on the left are equal multiples of the average energy of a P and of a Q respectively. Thus Maxwet’s Theorem is rigorously true, though in a most unexpected manner. There must surely be some extremely simple and direct mode of showing that u’—uv is independent of the mean-square speed of the system of Qs. Meanwhile, in default of anything more simple, I give the investigation by which I arrived at the result just stated. 21. Suppose a particle to move, with constant speed v, among a system of other particles in the “special” state; the fraction of the whole of its encounters which takes place with particles, whose speed is from v, to v,+ dv, and whose directions of motion are inclined to its own at angles from B to B+dB, is (§ 8) proportional to ey 2dv,v, sin B dB, or as we may write it for brevity vv, sin BdG. This is easily seen by remarking that, by § 8, while the particle advances through a space 6z, it virtually passes through a layer of particles (such as those specified) of thickness v,da/v. Here (§ 3) 3/24 is the mean-square speed of the particles of the system. Let the impinging particle belong to another group, also in the special state. Then the number of particles of that group which have speeds between v and v+dv is proportional to a edu=y , as we will, for the present, write it. Now let V, V,, Vo, in the figure, be the projections of v, v,, v) on the unit sphere whose centre is O; C that of the line of centres at impact. Then VOV, = 8. Let: V,OV = a; ViOV; =a, V,OC = >) and VV,C = ¢. The limits of y are 0 and 7/2; those of ¢ are 0 and 27. Also the chance that C lies within the spherical surface-element sin ydydd, is proportional to the area of the projection of that element on a plane perpen- dicular to the direction of 2, 7.¢., it is proportional to cos y sin y dyd@. > FOUNDATIONS OF THE KINETIC THEORY OF GASES. 81 But by definition we have u=v cos VOC=7(cos a cos y+8in a sin y ©)s #). v=, cos V,OC =»v,(cos a, cos y+sin a, sin y cos ¢) ; and by the Kinematics of the question, as shown by the dotted triangle in the figure, we have vsina—v, Sin a,=0. Thus, as indeed is obvious from much simpler considerations, Uu—V=¥, cosy, so that ae Jor sin B dB u(u—v) cos ysin y dydp eS ——————— Jr sin 6 d6 cos y sin y dydo hy% sin 8 dG v (cos a cos y+58in a sin y COS P)v COS *y Sin y dyd¢ Jes sin 8 dB cosy siny dyd¢ where each of the integrals is quintuple. The term in cos¢ vanishes when we integrate with respect to ¢ :—and, when we further integrate with respect to y, we have for the value of the expression Z VV, Sin B dBvv, Cos a Jvvnysin B dB where the integrals are triple. Now 2uv, CcoSa=v?+0,27—,7, and vv, sin BAB=v,dv,, so that the expression becomes 27) VAY, i 2 7 fw V+0,2—0v2 Ve 1 W, ( 0 1) 277) 4 v,2dv pl W, It will be shown below (Part VI.), that we have, generally an i 2n-1 ju Fe dv, = Toni oe vz, (A+k) 2 om, n+l 4 (hy and that it is lawful to differentiate such expressions with regard to / or to 4. Hence ad a : W-(G-a Jk = 4 T,23 i’ Thus CLERK-MAXwWELL’s Theorem is proved. VOL. XXXIII. PART I. L uw—uv= 82 PROFESSOR TAIT ON THE 22. The investigation of the separate values of the parts of this expression is a little more troublesome, as the numerators now involve second partial differential coefficients of I, ; but it is easy to see that we have d da\ drut = a at) 1-2( 35, gp JS+TI5 h-+2h ~ 16— I,/3 ~ Ohh) ad a? ada ws _ , Gam) Agta asa ae LG 1,/3 ~~ FREE) and, from these, the above result again follows. [It is clear, from the investigation just given, that the expression for the value of u?—uv would be the same (to a numerical factor prés) whatever law we assumed for the probability of the line of centres having a definite position, and thus that Maxwe.i’s Theorem would be true, provided only that the law were a function of y alone, and not of ¢ (ae., that the possible positions of the line of centres were symmetrically distributed round the direction of relative motion of the impinging particles). In my first non-approximate investigation (read to the Society on Jan. 18, and of which an Abstract appeared in Nature, Jan. 21, 1886) I had inadvertently assumed that the possible positions of C were equally distributed over the surface of the hemisphere of which V, is the pole, instead of over the surface of its diametral plane. The forms, however, of u? and of uv separately, suffer more profound modifications when such assumptions are made. | V. Rate of Equalisation of Average Energy per particle in two Mixed Systems. 23. To obtain an idea of the rate at which a mixture of two systems approaches the MAxwe tt final condition, suppose the mixture to be complete, and the systems each in the special state, but the average energy per particle to be different in the two. As an exact solution is not sought, it will be sufficient to adopt, throughout, roughly approximate expressions for the various quantities involved. We shall choose such as lend themselves most readily to calculation. It is easy to see, by making the requisite slight modifications in the formula of § 12, that, if m be the number of Ps and m that of Qs in unit volume, the number of collisions per second between a P and a Q is o /rht+k) amns', | at where s now stands for the sum of the radii of a P and of a Q. For if, in the > FOUNDATIONS OF THE KINETIC THEORY OF GASES. &3 formula referred to, we put (A)* for h’, and also put & for / in the exponentials where the integration is with respect to v,, it becomes 8ns*(hk)?1,/3 , according to the notation of § 21. This is the average number of impacts per second which a P has with Qs. Hence, if w be the whole energy of the Ps, p that of the Qs, per unit volume, the equations of § 19 become ee Se) pea Shing arg, eal (ns — mp) = =A, from which we obtain, on the supposition (approximate enough for our purpose) that we may treat 1/h+1/% as constant, n@—mp=Ce- "7 , CN a) eee. where 1-3 P+ The quantity NB — mp = mini w/n— p/n) , is mn times the difference of the average energies of a P and a Q, and (since «°=100 nearly) we see that it is reduced to one per cent. of its amount in the time 2 =46t— 128 (P+O%,/_le Reverie. aun 24, For a mixture, in equal volumes, of two gases in which the masses of the particles are not very different, say oxygen and nitrogen, we may assume as near enough for the purposes of a rough approximation m=n=5X10", whence m+ (per cubic inch) is double of this, == -= (12 x 1600 inch sec.)? , Se) 52 MUS: aoe ne so that 13.8 x 101° x 4 3 =T6x9x3xX102x12x1600" 4,7 = S10 seconds, nearly ; and the difference has fallen to 1 per cent. of its original amount in this period, i.¢., after each P has had, on the average, about four collisions with Qs. This calculation has no pretensions to accuracy, but it is excessively useful as showing the nature of the warrant which we have for some of the necessary assump- 84 PROFESSOR TAIT ON THE tions made above. For if the rapidity of equalisation of average energy in two systems is of this extreme order of magnitude, we are entitled to suppose that the restoration of the special state in any one system is a phenomenon taking place at a rate of at least the same if not a higher order of magnitude. CLERK-MAXWELL’s result as regards the present question is that, at every typical impact between a P and a Q), the difference of their energies is reduced in the ratio (Gra): P+Q/’ so that, if the masses were equal, the equalisation would be instantaneous. VI. On some Definite Integrals. 25. It is clear that expressions of the forms ex x a) rr) je “wade fs = rapsdy and J: marda fe - kyypsdly ; 0 0 0 x where 7 and s are essentially positive integers, may lawfully be differentiated under the integral sign with regard to # or to &. In fact they, and their differ- ential coefficients, which are of the same form, are all essentially finite. As, in what immediately follows, we shall require to treat of the first of these forms only when 7 is odd and s even, and of the second only when r is even and s odd, it follows that their values can all be obtained by differentiation from one or other of the integrals Jemrie fordy= ? 0 0 4h JSh+k chase [5 dy = Jt 6 I Me nts Tay These values may be obtained at once by noticing that the second form is integrable directly ; while, by merely inverting the order of integration, it becomes the first with 4 and & interchanged. 26. In §§ 21, 22 we had to deal with a number of integrals, all of one form, of which we take as a simple example and Vo" 1 = Jvv,— uv I;/3 1, ° Ze : We ~ hay da js -kyy d Kory y haa /) a ff ky ay( ypu 3 y aa = *)) wo 0 0 a e FOUNDATIONS OF THE KINETIC THEORY OF GASES. 85 From the remarks above it is clear that this can be expressed as 2 T ( d2 d? il a2 a hs \ aie ahah ir) i Jeph (?aeah aie) igs a2 2) _ Jr (4317) + (8kh?+13) 4 Wk(h +h)! Ja (h+k) ~ 2 ey The peculiar feature here shown is the making up of the complete cube of & + hin the numerator by the supply of the jirst half of its terms from the first part of the integral, and of the remainder from the second.* On trial I found that the same thing holds for I; and I,, so that I was led to conjecture that, generally, as in § 21 2Qn-1 Is fo n41_ Bao ! (h+k) 2 ; Ors a” ik After the preliminary work we have just ’ given, it is easy to prove this as follows. We have always (e+ yy"tt— (ayy) (at y)?+ (ey) = (a+ty)rt3 —(a—y)rrs + (2-7 )(@ 4+y)n-1— (e—y)"-1) : Operate on this by ie hod Ve 2 —'Pydy ( ie and on the same expression, with 2 and y interchanged (when, of course, it remains true), by —hax2 — ky fereeis ferry and add the results. This gives at once ae a z (@ eee. —2( 5 +o Monit =Teers+ a7 aE) Ton-1 5 which is found on trial to be satisfied by the general value given above. * Prof. Carney has called my attention, in connection with this, to the following expression from a Trinity (Cambridge) Examination Paper :— (a+b) = (a+b) ("+b") + (a+b)"-' (nab +nab") nea feed pan, Wt 1 2pn + (a4) (iz arte MN Ly ec 2 . an +(a+b) nN non “ai 14 qn- 1) . 86 PROFESSOR TAIT ON THE 27. Partly as a matter of curiosity, but also because we shall require a case of it, it may be well to mention here that similar processes (in which it is no longer necessary to break the y integration into two parts) lead to the Ton _ Pw? — _m 1.3.5...Qn—1) (+h) _ Ae guts a Wee (hk) ? And we see, by WALLIS’ Theorem, that (when 2” is increased without limit) I,, is ultimately the geometric mean between I,,_, and I,,., VII. Mean Path in a Mixture of two Systems. 28. If we refer to § 10, we see that, instead of what was there written as —eéz, we must now write —(¢+e,)6z2; where e,, which is due to stoppage of a particle of the first system by particles of the second, differs from e in three respects only. Instead of the factor 4s’, which appears in e, we must now write (s+s,)’; where s, is the diameter of a particle of the second system. Instead of h and we must write /, and n, respectively. Hence the mean free path of a particle of the first system is s/f hs ee. — ne? e+ a° 0 which, when the values of ¢ and ¢@, are introduced, and a simplification analogous to those in §§ 9, 11, is applied, becomes 1 = Ae—artda =) re~P + (14 20%) ede a an: =) (a e+ (1422,2) /¢ (e***de) 0 0 nh, in which Hi =x, /7 ad ho Thus the values tabulated at the end of the paper for the case of a single system enable us to calculate the value of this expression also. VILL. Pressure in a System of Colliding Particles. 29. There are many ways in which we may obtain, by very elementary processes, the pressure in a system of colliding particles. > FOUNDATIONS OF THE KINETIC THEORY OF GASES. 87 (a) It is the rate at which momentum passes across a plane unit area; or the whole momentum which so passes per second. [It is to be noted that a loss of negative momentum by the matter at either side of the plane is to be treated as a gain of positive. | In this, and the other investigations which follow, we deal with planes sup- posed perpendicular to the axis of «; or with a thin layer bounded by two such planes. The average number of particles at every instant per square unit of a layer, whose thickness is 8x, is ndz. Of these the fraction 3 — 4 [os ra hv 2dy T have speeds from v to v+dv. And of these the fraction sin B dB /2 are moving in directions inclined from 6 to @+dB to the axis of x. Each of them, therefore, remains in the layer for a time 62/v cos B, and carries with it momentum Pvcos B parallel to «. Now from B=0 to B=5 we have positive momentum passing ee 7 towards x positive. From $= to B=7 we have an equal amount of negative momentum leaving 2 positive. Hence the whole momentum which passes per second through a plane unit perpendicular to « is 1 nial ioe) se 2x bn ww / cos? B sin Bd8 =3Pnv? 0 0 where the bar indicates mean value. That is 2 Pressure =p = 3 (Kinetic Energy in Unit Volume). (b) Or we might proceed as follows, taking account of the position of each particle when it was last in collision. Consider the particles whose speeds are from v to v+ dv, and which are con- tained in a layer of thickness 6z, at a distance x from the plane of yz. Each has (§ 10) on the average ev collisions per second. Thus, by the perfect re- versibility of the motions, from each unit area of the layer there start, per second, Nvevox such particles, which have just had a collision. These move in directions uniformly distributed in space ; so that sin 8 d3/2 88 PROFESSOR TAIT ON THE of them are moving in directions inclined 8 to 8+d@ to the axis of x Of these the fraction —ex sec € B (where « is to be regarded as signless) reach the plane of yz, and each brings momentum Pv cos 6 perpendicular to that plane. Hence the whole momentum which reaches unit area of the plane is 1 z i 2x iP é cos 8B sin B dB Came emacs ry cos” 8 sin BdB, the same expression as before. (c) CLaustius’ method of the virial, as usually applied, also gives the same result. 30. But this result is approximate only, for a reason pointed out in § 6 above. To obtain a more exact result, let us take the virial expression itself. It is, in this case, if N be the number of particles in volume V, 5 PN 2 pV +52(Rr), where R is the mutual action between two particles whose centres are 7° apart, and is positive when the action is a stress tending to bring them nearer to one another. Hence, omitting the last term, we have approximately which we may employ for the purpose of interpreting the value of the term omitted. [It is commonly stated (see, for instance CLERK-MAXwELL’s Lecture to the Chemical Society®) that, when the term 32(Rzv') is negative, the action between the particles is in the main repulsive :—‘‘a repulsion so great that no attainable force can reduce the distance of the particles to zero.” There are grave objections to the assumption of molecular repulsion ; and therefore it is well to inquire whether the mere impacts, which must exist if the kinetic theory be true, are not of themselves sufficient to explain the experimental results which have been attributed to such repulsion. The experiments of REGNAULT on hydrogen first showed a deviation from BoyLr’s Law in the direction of less * Chem. Soc. Jour, xiii. (1875), p. 493. SEE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 89 compression than that Law indicates. But ANDREWS showed that the same thing holds for all gases at temperatures and pressures over those correspond- ing to their critical points. And AmacatT has experimentally proved that in gaseous hydrogen, which has not as yet been found to exhibit any traces of molecular attraction between its particles, the graphic representation of pV in terms of p (at least for pressures above an atmosphere, and for common temperatures) consists of a series of parallel straight lines. If this can be accounted for, without the assumption of molecular repulsion but simply by the impacts of the particles, a real difficulty will be overcome. And it is certain that, at least in dealing with hard colliding spheres if not in all cases, we have no right to extract from the virial, as the pressure term, that part only which depends upon impacts on the containing vessel; while leaving unextracted the part depending on the mutual impacts of the particles. The investigation which follows shows (so far as its assumptions remain valid when the particles are not widely scattered) that no pressure, however great, can bring a group of colliding spheres to a volume less than four times the sum of their volumes. If they were motionless they could be packed into a space exceeding the sum of their volumes in the ratio 6: 7,/2, or about 1°35: 1, only. | In the case of hard spheres we have obviously 7=s; and, with the notation of § 19, remembering that Q=P, s=h, we have R=—P(u—v). Hence we must find, by the method of that section, the mean value of the latter expression. It is easily seen to be - plrro sin 8B dB cosy sin ydyd¢ _ 2P Jrvyvp2dv,/vv, Jv sin B dB cosy sin ydydp ~——— 3 fiyv,2dv,/vv, all 2P I,/4 wa aT, Tin 3 I,/3 a 2h F But, § 14, the average number of collisions, per particle per second, is 2N ae | oS ahV Hence, for any one particle, the sum of the values of R (distributed, on the average, uniformly over its surface) is, in one second, ZANE; ASN ays a rie sp Peas? = —pArs?. x(R)=— Thus it would appear that we may regard each particle as being subjected to the general pressure of the system; but as having its own diameter doubled. VOL. XXXII. PART I. M 90 PROFESSOR TAIT ON THE It is treated, in fact, just as it would then be if all the others were reduced to massive points. The value of the term in the virial is 5 nsX(R) because, though every particle suffers the above average number of collisions, it takes two particles to produce a collision. This is equal to —inprs* = —6p (sum of volumes of spheres) ; so that the virial equation becomes nPv?/2 =50(V-4 (sum of volumes of spheres) ); which, in form at least, agrees exactly with AMAGAT’S* experimental results for hydrogen. These results are closely represented at 18° C. by p(V —2°6)=2731 ; and at 100° C. by p(V —2°7)=3518. The quantity subtracted from the volume is sensibly the same at both temperatures. The right-hand members are nearly in proportion to the absolute temperatures. The pressure is measured in métres of mercury. Hence the volume of the gas, at 18° C. and one atmosphere, is (to the unit employed) 2°6 + 2731/0°76 = 3596 nearly. Thus, by the above interpretation of AMaGaAT’s results, we have at 18° C. ns = 3°9/3596 . CLERK-MAXWELL, in his Bradford Lecture,+ ranks the various numerical data as to gases according to “the. completeness of our knowledge of them.” The mean free path appears in the second rank only, the numbers in which are regarded as rough approximations. In the third rank we have two quantities involved in the expression for the mean free path, viz., the absolute diameter of a particle, and the number of particles per unit volume (s and n of the pre- ceding pages). To determine the values of s and 2 separately, a second condition is required. It has usually been assumed, for this purpose, that the volume of a gas, “ when reduced to the liquid form, is not much greater than the combined volume of the molecules.” MAxweE.u justifies this assumption by reference to the small compressibility of liquids. * Annales de Chimie, xxii. 1881. + Phil. Mag., 1873, ii. 453. See also Nature, viii. 298. FOUNDATIONS OF THE KINETIC THEORY OF GASES. 91 But, if the above argument be, even in part, admitted, we are not led to any such conclusion, and we can obtain ms° (as above) as a quantity of the second rank. We have already seen that ms?’ is inversely proportional to the mean free path, and is thus also of the second rank. From these data we may considerably improve our approximations to the values of 7 and of s. Taking MaxweEtt’s estimate of the mean free path in hydrogen, we have (to an inch as unit of length) 0677 a Ds 380.107°. From these values of ms? and ns? we have, approximately, for 0° C., and 1 atmosphere, n=16.10", s=6.10-°. The values usually given are WEBW 5) a= 23105". It must be recollected that the above estimate rests on two assumptions, neither of which is more than an approximation, (@) that the particles of hydrogen behave like hard spheres, (6) that they exert no mutual molecular forces. If there were molecular attraction the value of ns’ would be greater than that assumed above, while ms? would be unaltered. Thus the particles would be larger and less numerous than the estimate shows. [Of course, after what has been said, it is easy to see that V should be diminished further by a quantity proportional to the surface of the containing vessel and to the radius of a sphere. But though this correction will become of constantly greater importance as the bulk occupied by a given quantity of gas is made smaller, it is probably too minute to be detected by experiment. | IX. Hgfect of Kxternal Potential. (Added June 15, 1886.) 31. Another of MAxwELL’s most remarkable contributions to the Kinetic Theory consists in the Theorem that a vertical column of gas, when it is in equilibrium under gravity, has the same temperature throughout. He states, however, that an erroneous argument on the subject, when it occurred to him in 1866, “ nearly upset [his] belief in calculation.”* He has given various investiga- tions of the action of external forces on the distribution of colliding spheres, but * Nature, vii., May 29,1873. Maxwetw’s name does not occur in the Index to this volume, though he has made at least five contributions to it, most of which bear on the present subject :—viz. at pp. 85, 298, 361, 527, 537. ‘ 92 PROFESSOR TAIT ON THE all of them are complex. The process of BoLTzMANv, alluded to in a foot-note to the introduction (anté, p. 66), and which CLERK-MAxXwELL ultimately pre- ferred to his own methods, involves a step of the following nature. An expression, analogous to the fof § 3, but in which B and C are unde- termined functions of the coordinates x, y, z, of a point, is formed for the number of particles per unit volume, at that point, whose component speeds, parallel to the axes, lie between given narrow limits. I do not at present undertake to discuss the validity or the sufficient generality of the process by which this expression is obtained, though the same process is (substantially) adopted by Watson and others who have written on the subject. However obtained, the expression is correct. It can be established at once by reasoning such as that in §§ 2, 3,4. To determine the forms of the aforesaid functions, however, a most peculiar method is adopted by BottzmMann and MAxweELL. The number of the particles per unit volume at x, y, z whose corresponding “ends” occupy unit volume at w, v, w in the velocity space-diagram (§ 3), is expressed in terms of these functions, and of w’?+v’+w’*, The variation of the logarithm of this number of particles is then taken, on the assumption that oz=udt, &., du= any) ot, &e., dz where U is the external potential; and it is equated to zero, because the number of particles 1s unchangeable. As this equation must hold good for all values of w, v, w, it furnishes sufficient conditions for the determination of B and C. The reasons for this remarkable procedure are not explained, but they seem to be as below. The particles are, as it were, followed in thought into the new positions which they would have reached, and the new speeds they would have acquired, in the interval 8¢, had no two of them collided or had there been no others to collide with them. But this is not stated, much less justified, and I cannot regard the argument (in the form in which it is given) as other than an exceedingly dangerous one; almost certain to mislead a student. What seems to underlie the whole, though it is not enunciated, is a postu- late of some such form as this :— When a system of colliding particles has reached its final state, we may assume that (on the average) for every particle which enters, and undergoes collision in, a thin layer, another goes out from the other side of the layer precisely as the first would have done had it escaped collision. 32. If we make this assumption, which will probably be allowed, it is not difficult to obtain the results sought, without having recourse to a questionable > FOUNDATIONS OF THE KINETIC THEORY OF GASES. 93 process of variation. For this purpose we must calculate the changes which take place in the momentum, and in the number of particles, in a layer; or, rather, we must inquire into the nature of the processes which, by balancing one another’s effects, leave these quantities unchanged. Recur to § 29, and suppose the particles to be subject to a potential, U, which depends on « only. Then the whole momentum passing per unit of time perpendicularly across unit surface of any plane parallel to yz is 1 it glee abn [=a where 7 (the number of particles per cubic unit), and (which involves the mean-square speed), are functions of x. At a parallel plane, distant « from the first in the direction of « positive, the corresponding value is i d\n =p te (1 ss ) h But the difference must be sufficient to neutralise, in the layer between these planes, the momentum which is due to the external potential, 7.¢., Hence or aU_1 du_idh dx nde hdx ° 5 : ; ; (1). ) Again, the number of particles which, in unit of time, leave the plane unit towards the side x positive is L fon fcwasing dpa! af 5 nN A V f 4 7 , VV Hence those which leave the corresponding area at distance « are, in number, 1 d s a (tom afm): But, by our postulate of last section, they can also be numbered as inf» (1—@/v), 4 dU i Oe ¢ “dae where 94 PROFESSOR TAIT ON THE This expression is obtained by noting that none of those leaving the first plane can pass the second plane unless they have © costs Bae v? COS B>2az. All of the integrals contained in these expressions are exact, and can therefore give no trouble. The two reckonings of the number of particles, when com- pared, give Ate ih Vda Siodh From (1) and (2) together we find, first dh deme which is the condition of uniform temperature ; and again W=MNpe = **CO—Vo), which is the usual relation between density and potential. [In obtaining (2) above it was assumed that, with sufficient accuracy, eS — 1 fe, To justify this :—note that in oxygen, at ordinary temperatures and under gravity, ct = 15502 in foot-second units, 2h an = so that, even if a=1 inch, we have approximately : adU u io= 2h-Te = 30,0001 It is easy to see that exactly similar reasoning may be applied when U is a function of x, y, 2; so that we have, generally, i= —2A(U — N=Npé n(U Uo), where / is an absolute constant. And it is obvious that similar results may be obtained for each separate set of spheres in a mixture, with the additional proviso from MaxweE w’s Theorem (§§ 20, 21) that P// has the same value in each of the sets. FOUNDATIONS OF THE KINETIC THEORY OF GASES. ACP EEN DX. 95 The following little table has been calculated for the purposes of §§ 11, 28, by Mr J. Crark, Neil-Arnott Scholar in the University of Edinburgh, who used six-place logarithms :— x Lge Sata Cel coe MAL, gle? lane eee OCMTNAAHKwWNMOE OKOATA AK ow wp HL Se es S) nm ty bo ke 2°3 no ty cho wah ID S) &) 6 xX, ‘000099 001537 ‘007420 021814 048675 090418 147091 215978 ‘291870 ‘367879 ‘436590 ‘491380 ‘527004 541119 533081 ‘506619 ‘464174 ‘409127 352543 293040 ‘236390 185224 141065 104541 075390 052962 036242 024155 015700 ‘009963 Here X,=a*e-* and X,=«%¢-*, while X,=wxe-” + (2a? + » fran 0 X, ‘200665 ‘405312 ‘617838 841997 1:081321 1:339068 1618194 1921318 2°250723 2°608351 2°995825 3414479 3°865384 4:349386 4°867132 5419114 6005696 6°627149 7283658 7:975359 8°702340 9°464667 10:262360 11:095474 11:964016 12:867980 13°807388 14°782249 15°792549 16°838302 X,/X, 00049 + 00379 + 01198+ 02591 — 04501+ 06752+ ‘09089 11241 — 12968+ 14104 — 145724 14388 + 13633 + 12441 + 10962+ 09348 — ‘07729 — 06203 + 04840 — ‘036744 027154 01956 — 01373+ 00941 + ‘00630+ ‘00411 — 002624 001624 00099+ 00057 + X, X,/X_ 000990 00493 + ‘007686 01896 + 024676 03994 — 054537 ‘06477 + ‘097350 09003 — 150698 11254— ‘210130 129854 269973 140514 324301 "14409 — ‘367879 14104 — ‘396900 "13249 — ‘409409 11990+ ‘405388 10488 — ‘386514 ‘08887 — 355721 ‘07309 — 316637 ‘05843 — ‘273044 04546 + ‘228404 ‘03447 — 185549 02547 +4 146520 ‘018374 112567 ‘01294 — 084193 ‘00889 061333 00598 — 043559 00393 — 030156 00252+4 020370 001584 013423 ‘00097 + 008627 ‘00058 + ‘005414 000344 003321 00019 + az The sum of the numbers in the fourth column is 1°69268, so that the approximate value of the integral in § 11, which is 0°4 of this, is 0°67707. The sum of the numbers in the sixth column is 1°62601, so that the value of the integral in [the addition to] § 11 is about 0°6504. ene) IV.—The Eggs and Larve of Teleosteans. By J. T. Cunnincuam, B.A, (Plates I—-VIT.) (Read 5th July 1886.) The purpose of this memoir is (1) to make known a number of drawings and descriptions of the eggs, embryos, and larvee of the species of Teleosteans which I have been able to study at the Scottish Marine Station; (2) to review as comprehensively as possible what is known at the present time concerning the structure of the embryos and larvee of the species of Teleosteans, and to discover what features are common to each family or each order ; (3) to discuss the changes which take place in the protoplasm and nucleus of the mature ovum immedi- ately after it is shed, both when fertilised and when unfertilised. The ova of the following species were taken directly from the parent fish, and artificially fertilised. The necessary operations were carried out, in some cases by myself, on board fishing boats—usually steam trawlers from Granton. In many instances I did not myself go out in the boats, but the ova were obtained and brought to me at the laboratory by ALEXANDER TuRBYNE, keeper of the station. But in every case there is no uncertainty as to the species of the fish from which the ova were taken; if there was any doubt, specimens of the parent fish were brought with the ova. 1. Clupea harengus, Linn. (Herring) (PI. I. figs. 1-3). The development of the herring has been described by Prof. C. Kuprrrer* in an elaborate memoir, which is illustrated by microscopic photographs. Among these, one figure of the hatched larva is given, but this is on too small a scale to exhibit the structure clearly. It is nearly two years since I studied the ova of the herring, and some of the drawings which I then made have been used to illustrate papers on particular problems in Teleostean development.t But, as far as I am aware, no good figures of the larva of the herring have been published, and I therefore think that the figures on Pl. I. will not be super- fluous. Herring, as is well known, have two spawning seasons on the east coast of Britain—one in the spring, in February and March, and one in the * Ueber Laichen und Entwicklung des Ostsee-Herings, Berlin, 1878. + “On the Significance of Kupffer’s Vesicle,” &c., Quart. Jour. Mier. Sci., 1885; and “On Relations of Yolk to Gastrula,” &c., Ibid. VOL. XXXIII. PART I. N 98 MR J. T. CUNNINGHAM ON THE autumn, in August and September. The eggs which I studied were obtained in August off the Longstone Lighthouse, Fearn Islands. Embryonic Period and Temperature.—Hatching took place on the eighth and ninth days, the temperature varying from 11°5 to 14°°5 C. In the herring ovum the yolk consists of a number of nearly spherical translucent vitelline globules; there are no oil globules. The blastodisc is large in proportion to the yolk. Diagnosis of Larva.—The length of the newly hatched larva is 5:2 to 5°3 mm., according to Kuprrer. The mouth is open, the body is wholly transparent except the eyes, which are of a deep black, and perfectly opaque ; there are no red blood corpuscles; the notochord is unicolumnar ; the anus is at a distance from the yolk sac, being 1 mm. from the end of the tail; the pectoral fin is present as a simicircular fold of membrane; the pelvic fin is not developed ; compact chromatophores are present on the sides of the body and tail. The larve of the herring I have taken occasionally, but not often, in the tow- net. Two were obtained at 5 fathoms depth west of Inchkeith, Oct. 7, 1885 ; 15 at a depth of 3 feet off St Abb’s Head, Sept. 30, 1885; a few at 3 fathoms east of Inchkeith, May 14, 1885; and a few at 5 fathoms north-east of Inch- keith, April 15, 1885. 2. Salmo levenensis (Loch Leven Trout) (PI. I. fig. 4). This figure is taken from an alevin of the species obtained from Sir JAMES MaItTLANp’s hatchery at Howietoun. The larva was three days old; the per- manent anterior dorsal, caudal, and anal fins have begun to develop, but the median larval fold is present behind the anterior dorsal, behind the anal, and between the anus and the yolk. The pelvic fins have appeared; they are situated some distance in front of the anus, and they have no connection with the preanal larval fin, which extends between them up to the yolk sac. 3. Osmerus eperlanus, Lacép (Smelt or Sperling) (Pl. I. figs. 5, 6). The mature egg of Osmerus, when first shed, is yellow in colour, and but: slightly translucent; it is surrounded by a double zona radiata, the inner surface of which is, as in all Teleosteans, in immediate contact with the vitellus. When the eggs are allowed to fall on to stones or glass plates in water contain- ing milt, they become attached and fertilised simultaneously. The attachment is effected in the following manner :—The outer zona radiata ruptures at the region of the ovum which is opposite the micropyle, and peals off the inner zona, becoming of course inverted in the process. Over a circular area sur- rounding the micropyle, the two layers of the zona remain firmly united. The EGGS AND LARV.X OF TELEOSTEANS. 99 outer surface of the external layer or zona externa is adhesive, and the ruptured edge becomes attached, so that the ovum swings in the water from the flexible suspensory membrane thus formed. I have elsewhere * described the separation of the two layers of the zona resulting in the formation of the suspensory membrane, but the relation of the united parts of the two layers to the micropyle is now described for the first time, and is shown in fig. 6 as it appears in optical section. When fertilisation takes place, a large perivitelline space is formed by the elevation of the internal zona. Unfortunately, I was unable to obtain a sufficient number of healthy ova to study the development. Fig. 5 was taken from an ovum fertilised at Stirling on May 6; it was drawn on May 7, twenty-five hours after the egg was shed. It shows the character of the egg, and the rela- tion of the blastodisc to the yolk; but the blastodisc was not segmented, and it is possible that the ovum was not really fertilised, the formation of the blastodise occurring normally in ripe Teleostean ova without fertilisation. The ovum resembles somewhat that of the herring. It is a little more transparent than the herring ovum, and the structure of the yolk is different. In the egg of Osmerus there are a number of oil globules, varying much in size, while the yolk of the herring ovum has no oil globules. The diameter of the fertilised ovum is 1°3 mm. 4. Pleuronectes platessa, Linn. (Plaice) (Pl. II. figs. 1-3). The eges of the plaice were artificially fertilised on board a steam trawler outside the Isle of May, February 3, 1886. The egg is 1:95 mm. in diameter, and like the other eggs of the Pleuronectide which I have examined, has a perfectly homogeneous yolk. The perivitelline space is small. The larvae were not actually hatched, but one taken from the ovum, when almost ready to hatch, is shown in fig. 3. Its length is 4:1 mm. The eye is faintly pigmented. There are three rows of yellow dendritic pigment cells down each side, and black dendritic cells in the head. The anus is open, and situated immediately behind the yolk sac. The notochord is multicolumnar ; the pelvic fin not developed. 5. Pleuronectes flesus, Linn. (Common Flounder) (PI. IT. figs. 4-8). Eggs of this species were obtained on March 30, 1886, in the Firth of Forth, in Aberlady Bay. It is the only species which has been found in abundance, and in the spawning condition, so far up the Firth. The egg is similar in all respects to that of the plaice except in size. It is 1:03 mm, in diameter. The newly hatched larva is transparent, and 3°01 mm. in length. * Proc. Zool. Soc., London, 1886. 100 MR J. T. CUNNINGHAM ON THE The anus is open; notochord multicolumnar ;* small round pigment spots along the sides and head ; pelvic fin not developed. On March 5 of the current year I visited some fishing boats at Kincardine on the Forth. These boats were fishing with what are called bag-nets or stow- nets. A net of this kind is fashioned very much like a beam-trawl, and is fastened beneath the boat, so that its mouth faces the current of the tide; the fish are thus washed into the net. Among the fish taken on the occasion of my visit were a large number of Pleuronectes flesus, which are commonly called fresh-water flounders, or mud flounders. Nearly all of these fish had a number of small round white tumours on the fins and on the upper or dark side. The tumours are cutaneous, and have been described more than once (see M‘Intosu, Third Annual Report of Scottish Fishery Board). ‘The fishermen stated that these tumours were the eggs of the fish, that the mud flounder carried its eggs on its back. On another occasion a bottle was sent to me from Elie, said to contain flounder spawn; the contents when examined proved to be the greenish gelatinous egg-cases of some species of Cheetopod, perhaps Arenicola piscatorum, and within the cases were the trochospheres, whose green colour was the cause of the colour of the cases. 5. Pleuronectes limanda, Linn. (Salt-water Flounder) (PI. II. figs. 9-11; Pl. III. figs. 1-6). Ripe specimens of this species were obtained by me in considerable numbers on board a steam-trawler six or seven miles east-north-east of the Isle of May, on May 21 of the current year. A number of the eggs were squeezed out, artificially fertilised, and conveyed to the Marine Station. Living specimens were also successfully carried to the aquarium, and upon eggs taken from these I was able to study the condition of the ripe eggs immediately on their escape from the oviduct, and the earliest processes of fertilisation and development. The egg, after the formation of the perivitelline space, is ‘84 mm. in diameter; the appearance, magnified 33 times at the close of simple segmentation, is shown in Pl. II. fig. 9. Hatching took place on the third day; the temperature of the surface of the sea where the eggs were taken was 7°°5 C., and the temperature of the water containing the eggs varied from this to 10° C. The newly hatched larva was 2°66 mm. in length; the structure closely similar to that of other species of the genus; notochord multicolumnar ; mouth not open; small black pigment spots on sides of the body; anus close to the yolk, and not open. * The terms unicolumnar and multicolumnar applied to the notochord refer to the arrangement of the vacuoles, which are very conspicuous in newly hatched fish; in the herring and a few other cases these vacuoles are cubical, and form a single linear series ; in other cases there are several series, EGGS AND LARVA OF TELEOSTEANS. 101 The figures given of the first processes of fertilisation and development will be considered in a subsequent section. On Dec. 5, 1885, I trawled with a fine meshed shrimp trawl across the Drum Sands, which are situated between Queensferry Point and Cramond Island, and obtained a considerable number of young Pleuronectes limanda. ‘These were about 2 inches long, and could be identified from the semicircular curve in the lateral line above the pectoral fin. Larger, nearly full-grown, specimens were also taken, and kept for some time in the aquarium, where they lived healthily. In June of the current year, Mr RamAGE, who is at present studying at the station, pointed out to me that the sands to the west of the laboratory were swarming with young flounders. These were about + inch long, and had already reached the condition of the adult; they showed no trace of larval structures. But I was unable to identify these young fish, as the lateral line could not be clearly distinguished. It is of course probable that the young of many different species are present in such situations in the summer months. It is pretty certain that nearly all our valuable flat-fishes pass the early post- larval stages of their existence on littoral sand-flats. Mr Grorce Brook informs me that large numbers of young flat-fish are destroyed by shrimpers in such situations. With regard to this particular locality, I have never seen any shrimping carried on in the neighbourhood. 6. Pleuronectes cynoglossus, Linn. (Witch) (Pl, IIT. figs. 7-9; Pl. IV., Pl. V.). Of the developing eggs of this species I made a particularly careful study, with the intention of obtaining, if possible, greater certainty on the various points in dispute concerning the earliest changes that the mature ovum under- goes after being shed. A number of living specimens of the fish were trawled by the “ Medusa,” on 23rd and 24th June of the current year, at a place called Fairlie Patch, opposite the town of Fairlie, in the channel between the island of Cumbrae and the mainland. The fish were taken alive to the little labora- tory known as the “ Ark,” which was originally a floating structure, but is now firmly established on the beach at the east side of Millport Bay. I have given a large number of figures, illustrating the successive stages in the development of this species. After the formation of the perivitelline space they are 1:155 mm. in diameter, The yolk is perfectly transparent, but the zona radiata is thicker than in most of the other species of the genus. The perivitelline space is very small. During the time the eggs were under observation the weather was very fine, and the laboratory being fully exposed to the sunshine, became in the middle of the day very hot. I had no means of regulating the tempera- ture of the water containing the eggs, and on two occasions it rose to 20°5 C. 102 MR J. T. CUNNINGHAM ON THE This temperature was fatal to a large number of the developing eggs. The temperature of the water in which the eggs were first placed was 12°5 C. With these great variations in temperature, hatching took place on the sixth day. The larva is not different from that of the other species of Pleuronectes ; its length is 3°9 mm.; there is no pigment in the eye; a number of very minute pigment spots are scattered down the sides. The anus is not open, and the coalesced segmental ducts do not communicate with the rectum (see Pl. V. fig. 5). Pl. V. fig. 7, shows the condition of the larva a little more than forty-eight hours after hatching. The length is now increased to 5‘9 mm.—a very rapid rate of growth. The median fin-fold is much wider. The eye is slightly pigmented, and pigment is largely developed in the skin of the body. The cutaneous chromatophores form five well-marked transverse stripes, arranged in longitudinal series along the sides, three of them on the tail, one in the region of the rectum, and one about the pectoral fin. No trace of the pelvic fin is to-be seen. The operculum is present as a slight fold, and beneath it the first branchial cleft is widely open; behind this are four clefts indicated but not perforated. The mouth is also still wanting. 7. Pleuronectes microcephalus (Lemon Sole). I have not obtained fertilised ova of this species, but I was able to ascertain from examination of unfertilised mature examples that there are no oil globules, and that the diameter measures 1‘1 mm. The ripe females, from which the mature eggs came, were taken in the trawl east of May Island, May 22 of the current year. 8. Gadus ceglefinus, Linn. (Haddock) (Pl. VI. fig. 1). The ova of Gadus morrhua, G. cyglefinus, and G. merlangus, in various stages of development, have been previously figured by me.* The larve, after hatching, were not described in the paper I refer to. The newly hatched cod has been correctly figured by Joun Ryper.t For the sake of comparison, I give a figure of the newly hatched larva of the haddock. The eye is pigmented, and there is a single row of dendritic chromatophores along each side ven- trally. The anus is not open, nor the mouth; the pelvic fin is also wanting. In all respects, except in size, the larva of the haddock resembles that of the cod. The following species of ova and larvae were not obtained directly from the parent fish, but identified from other considerations. * “Relations of Yolk to Gastrula in Teleosteans,” Quart. Jour. Micr. Sci., 1885. + Report of American Fish Commission for 1882, Washington, 1884. EGGS AND LARVZ OF TELEOSTEANS. 103 9. Cottus scorpius, Linn. (Pl. VI. fig. 2). The eggs ascribed to this species were brought in to the station on February 14 of the present year. They formed large masses of dark red colour, and were attached to the rocks between tide marks. The ova are but slightly translucent ; the zona radiata is thick. The figure shows the appearance under a low power of the microscope. The yolk is homogeneous, except for the presence of scattered oil globules, irregular in number and size, and contains the pigment, which, yellowish-red as seen in each separate ovum, gives the whole mass a darker red colour. The diameter of the vitelline membrane is 2:03 mm., of the ovum 1°81. The identification is founded on some remarks of Professor M‘Inrosn, who observed the deposition of similar eggs in the aquarium of the Marine Laboratory at St Andrews (see Third Annual Report Scottish Fishery Board, 1885, App. F.). Acassiz* has stated that the eggs of Cottus graenlandicus, which is only a variety of Cottus scorpius, are pelagic. His conclusion rests apparently on the identification of the oldest stage of larvee from a certain kind of pelagic eggs with the adult Coltus, and this mode of identification is of course not abso- lutely certain. 10. Liparis Montagu, Cuv. (Pl. VI. figs. 3, 4). Small masses of adhesive eggs are frequently obtained attached to tufts of Hydrallmannia falcata, Hincks. I have obtained such specimens in the months of May and June, both from long lines laid outside the Isle of May and from the dredge in the upper parts of the Firth. By the fishermen the eggs in question are usually believed to come from the herring or the haddock, and even naturalists of some experience have confounded them with herring spawn, which also often adheres to specimens of Hydrallmannia. The mass from which fig. 3 was taken was attached to a piece of Hydrallmannia left by the tide on the beach near Cramond Island, and was obtained May 7, 1886. The longest diameter of the egg, including the vitelline membrane (zona radiata) | was 1:27 mm., the transverse diameter of the yolk sac ‘87 mm. The zona radiata is of considerable thickness, and shows a division into two layers. The yolk is homogeneous and transparent, and contains three or more oil globules of various sizes. The mass of eggs seen with the unaided eye was colourless and transparent. I have identified the ova as those of Liparis Montagut, from some remarks of Prof. M‘InrosH in Report on the St Andrews Laboratory, in the Third Annual Report of the Scottish Fishery Board, but the identification is * Proc. Amer. Acad. Arts and Sct., vol. xvii. ; and Memoirs of Mus. Comp. Zool., Harvard, vol. xiv. No. 1, pt. i. 104 MR J. T. CUNNINGHAM ON THE not certain. M‘InrosH says that the eggs of Liparis Montagui are found in shallow water, attached to such zoophytes as Hydrallmannia and Sertularia, and also to red Algve, and are of a pale straw colour. The eggs I have described were well advanced in development, so that the colour may have been present at an earlier stage, the colour of such eggs often disappearing as develop- ment proceeds. The eyes were considerably pigmented. Fig. 4 is a sketch of a fish hatched from some eggs exactly similar to those above described, which were taken in the trawl between Inchkeith and Burnt- island, April 29, 1884. The age of the young fish was two days after hatching. The eyes are deeply pigmented, the mouth completely developed, the pectoral fin is large, and covered with black pigment spots, and there is a row of similar spots along the ventral edge of the tail on each side. A small remnant of the yolk is still present, containing a single oil-globule. “11. Cyclopterus lumpus, Linn. (Lump-sucker) (Pl. VI. fig. 5). To amateur naturalists on the coasts of Scotland the large masses of yellow- ish spawn of this fish, watched by the male parent, the “rawn and cock paidle,” as they are called in the Scotch dialect, are a not unfamiliar sight. I regret to say I have not had an opportunity of personally observing the phenomenon in its natural state. But masses of the ova of Cyclopterus have been frequently brought into the station by boys; they are found attached to the rocks near the station, not far from low water mark. The colour of the eggs varies from red to pale yellow or nearly white. The yolk contains numerous oil globules of various sizes, arranged in a cluster at the ventral pole, but is otherwise homo- geneous. The perivitelline space is small. The eggs are but slightly trans- lucent. The diameter is 2°60 mm., inclusive of the vitelline membrane. The young Cyclopterus, when first hatched, is 4 mm. in length, but not so far advanced in development as the stage figured by Acassiz* of the same length. The anus is immediately behind the yolk sac, which forms such a contrast in size to the tail that the fish is tadpole-like in form. The body is quite opaque, and the blood red. The eyes are completely pigmented. The embryonic fin fold persists extending forwards dorsally a little beyond the anal region, but fin rays have appeared in the membrane. Both paired fins are well developed, the ventrals forming a median sucker, which differs only from that of the adult in exhibiting the fin rays in a more primitive condition. The skin contains numerous regularly distributed chromatophores. The young Cyclopterus, both immediately after hatching and in later stages, occur very plentifully among the Algee on the shore at Granton, and everywhere on the British coasts. They are also frequently taken in the tow-net at a * Young Stages, iii. EGGS AND LARVA OF TELEOSTEANS. 105 distance from the shore, but in this case are usually attached by the sucker to floating pieces of sea-weed. The eggs are deposited in January and February, and the young stages are to be found on the shore or in the tow-net through- out the summer. It was observed by Mr Jackson, in the Southport Aquarium, that the male parent, watching over the eggs, kept up a continual motion of his pectoral fins in close proximity to the eggs, and it appears that this is neces- sary to secure the sufficient oxygenation of the eggs, which are laid in such large masses that the central ones might easily in still water be asphyxiated. Young specimens of Cyclopterus were taken in the tow-net in the following localities :—Surface, 30 miles north-east of May Island, July 17, 1885; surface, near Inch Mickery, Aug. 26, 1885; surface, Firth of Forth, two occasions, 1884; surface, east of Craig Waugh, May 1884. I have never taken any large numbers either of these or any other fish larvee in the tow-nets. Species not identified. A certain number of well-marked species have been obtained by tow-net collecting, which I have not yet been able to identify. There are two possible methods of identifying an unknown species of pelagic ovum. One is to compare it, or the larva hatched from it, with figures and descriptions of ova or larvee already known; the other, to keep a number of specimens of the ovum in question alive until they hatch, and then to keep the larvee till they attain the specific characters of the adult fish. Both of these methods are liable to error. Species No. 12 (Pl. VII. fig. 2). This form is easily distinguished by one conspicuous characteristic, namely, that the perivitelline space is very wide. The yolk is perfectly homogeneous and transparent. The diameter of the vitelline membrane is 2:1 mm., of the ovum 12mm. The eggs were obtained in the latter end of March, both in 1885 and 1886, about 10 miles east of the Isle of May. Unfortunately, time could not be found to give sufficient attention to the form to isolate it and keep it alive till hatching took place. Thus the characters of the larva were not ascertained, and no egg at all similar has been taken directly from an adult fish. Species No. 13 (Pl. VII. figs. 3, 4). The eggs of this species were obtained in the tow-net, 16 miles beyond the Isle of May, on April 30, and off Gullane Ness, May 27, 1886. The diameter of the ovum, including the vitelline membrane, is ‘84 mm. The pert- vitelline space is small; there is a single oil globule situated beneath the VOL. XXXIII. PART I, g 106 MR J. T. CUNNINGHAM ON THE posterior end of the embryo. Some of the eggs were hatched, and fig. 4 shows the form of the larva immediately after hatching; the length is 2:1 mm. ; the notochord, as seen in fig. 4a, is multicolumnar ; there are black pigment spots on the body, but the eye is unpigmented; the pigment on the post-vitelline part of the body forms two black transverse bands. The intestine was, I believe, not open, but a solid extension of it extended to the ventral edge of the larval fin. In a great many respects the present species agrees with Motella mustela, Linn., as described by GEorcE Broox,* from eggs actually observed to be deposited by the parent. There are several minute points of difference. Broox’s measurement of the ovum is ‘655 to ‘731 mm. in largest diameter, while he gives the length of the newly hatched larva as 2°25 mm.; thus the diameter of the ovum given by Broox is slightly less than my measurement, while the length of the larva given by him is slightly greater than what I have stated. The position of the oil globule and rectum is also different in my figure from that in Broox’s. But the points of agreement are more numerous and important than the points of difference ; the arrangement of the pigment, for instance, is exactly the same in the two accounts. It is evident, therefore, that the species I have described is either Motella mustela, Linn., or some other of the four British species of Motella. Pelagic eggs closely similar to the species here described, and to those of Motella mustela as described by Brook, have been described by A. AGassiz and C. O. Wuitman,t and referred with some uncertainty to Motella argentea, Rhein. Two other species of pelagic eggs have also been provisionally ascribed vy those authors to the genus Motella. Species No. 14 (Pl. VII. figs. 5, 6). This form is well characterised ; it possesses one feature which, as far as extant observations show, is present in no other pelagic ovum, namely, that the yolk is divided into a number of polyhedral masses. This egg is the most perfectly pellucid of all I have observed, and the planes of division in the yolk appear in optical section as extremely fine lines. The egg is slightly oval in shape, ‘94 mm. by ‘97 mm. in diameter. The newly hatched larva is 3°63 mm. in length, the notochord is unicolumnar, and the anus is separated from the yolk by two-thirds of the length of the post-vitelline part of the body, as in the herring; the larva is absolutely without pigment. The eggs were obtained in 1884 and 1886, in the latter end of May and during June. In each season they were taken within the Firth of Forth, between Gullane Ness and the island of Inchkeith. This form, from its conspicuous characteristics, has * Linn. Soc. Jour., vol. xviii. { ‘Pelagic Stages of Young Fishes,” Memoirs of Mus. Comp. Zool. Harv., vol. xiv., No. 1. EGGS AND LARVA OF TELEOSTEANS. 107 long been known, but all attempts to trace it with certainty into connection with a particular species of fish have hitherto failed. The eggs and larvee were first described by A. AGassiz* in 1882, under the name Osmerus mordaa, Gill, figures being given of the newly hatched larva and some older stages. AGASssiz appears to have obtained his figures of the later stages from specimens taken in the tow-net, not from larve reared in captivity directly from the egg. He states that at first he supposed the larve to belong to some Clupeoid species, until he saw a paper by Mr H. J. Rice, on the development of Osmerus, when he became convinced that his specimens were really to be ascribed to Osmerus mordax. He points out that the oldest larva he figures has a striking resemblance to Scombresox and Belone. As a matter of fact this resemblance is not very exact, and as it is known that the eggs of all the Scombresocide are provided with filamentous processes of the vitelline membrane, it is certain that the ovum under consideration cannot belong to any member, of that family. Agassiz also remarks that the resemblance of the development of Osmerus to that of the herring as given by SUNDEVALLt is very close. Now SUNDEVALL gives a figure of the larva of Osmerus eperlanus, which shows an oil globule in the yolk sac, and I have shown that the ovum of O. eperlanus is adhesive. Thus it is impossible that AGassiz’ larva should be that of Osmerus mordax. Two species of the same genus could not differ so greatly in the structure of their ova and the conditions to which those ova are exposed, as do the pelagic ovum we have been considering, and the ovum of Osmerus eperlanus. Moreover, Osmerus mordax does not occur in the British seas. It is certain that the herring cannot be the parent of the ovum in question, in spite of the resem- blance between the larva derived from it and the herring larva, for the ova of the herring are well known, and are not pelagic. This same pelagic ovum and larva have been described by V. HENSEN,{ and that gentleman, courteously replying to inquiries of mine on the subject, said, in his opinion, the parent species was the sprat. But here we have the same difficulty as in the case of Osmerus. Can any species of Clupea have pelagic ova? No instance is yet known of a typically adhesive and a typically pelagic ovum occurring in the same genus. Nevertheless the segregation of the yolk in our pelagic ovum is not altogether incomparable with the condition of the yolk in the herring. It seems absolutely certain that the problematic ovum belongs to some physosto- mous fish, but hitherto no physostomous fish is known to have a pelagic ovum. It has struck me as possible that the parent we are seeking to discover is really the eel, Anguilla vulgaris. At all events the fertilised spawn of the eel has never been examined. * Young Stages, pt. ii. + Svensk. Vetensk. Akad., 1855. { Vierter Ber. Com. Unt. Deutsches Meere, Berlin, 1883. 108 MR J. T. CUNNINGHAM ON THE Species No. 15, Pl. VII. fig. 7. These ova formed a cylindrical rope-like mass, and were brought up on a tow-net line from a depth of about 30 fathoms in the Gulf of Guinea. They were obtained by Mr Joun Ratrray, on two occasions when he was on board the steamer ‘“ Buccaneer,” a telegraph steamer placed at the disposal of Mr J. Y. Bucnanan, for hydrographical investigations. ‘The first occasion was on March 12 of the current year, in lat. 1° 17’ N., long. 13° 56°6’ W.; the second occasion was soon after, not far from the same locality. The depth of the ocean at the place was 2725 fathoms. The felted filaments in the rope-like mass were internal, the eggs external. Each ovum was 1°5 to 1-6 mm. in diameter. They are, as far as I am aware, the first Teleostean ova which have been found to have a group of filamentous processes at each of two opposite poles of the vitelline membrane. In these ova one group of processes is rudimentary and functionless, but nevertheless the system of processes is closely similar to that which occurs in Myzxine (see my paper on “ Reproductive Elements of Myaine glutinosa,” Quart. Jour. Micr. Sci., 1886). Gobiidee, Blenniidee, Pomacentride, Atherinidee, Scombresocide are the only families known in which processes of the vitelline membrane occur, but it is impossible to say which of these families, if any, includes the parent of the ova described. Identification of Ova.—The identification of the numerous pelagic ova which are taken in the tow-net at the mouth of the Firth, at different times of the year, cannot at present be carried out with complete certainty. If the eggs and larvee of every species known to occur were adequately described and figured, the feat might be possible; but at present the identification of any egg taken from the sea must always be subject to a certain degree of scepticism. I have several times attempted to assign the eggs in a tow-net gathering each to its parent species, and have satisfied myself that I had separated the eggs of the Plaice, Cod, Haddock, Pl. flesus, and Trigla gurnardus. But there may be other species with closer resemblances to these than I am at present aware of. General Comparative Review of the Structure of the Ova and Larve of Teleosteans.* The method followed in the present section is to take the families of each Order successively, and inquire what is known concerning the characters of * The classification employed in the present section is— I. Physostomi. II. Physoclisti. 1. Anacanthini. 2. Acanthopterygi. 3. Acanthopt. Pharyngognathi, 4. Lophobranchii. 5. Plectognathi. EGGS AND LARVA OF TELEOSTEANS. 109 the eggs and larve, then to ascertain what features are common to all the families of the order, and finally to compare the characters which belong to the several orders. We shall take the families as defined by GUNTHER in the article “ Ichthyology” of the Encyclopedia Britannica. Fam. 1. SILURID. The female of Aspredo batrachus attaches the eggs to the skin of her own ventral surface, and carries them about there until they are hatched. The male Arius carries the eggs about in his pharynx. The male Callichthys makes a nest. An account of the breeding and development of Amiurus albidus (Lesueur), Gill, is given by Jonn A. Ryver in Bull. U.S. Fish. Com., vol. iii. The ovum is adhesive, and 4 inch in diameter after fertilisation; the vitellus was } inch in diameter. The female deposited the whole of her eggs at one time in a tank, in one mass, which was 6 inches in length by 4 in width, by 2? inch in thickness. The male watched over the mass with great assiduity till hatching occurred, and constantly fanned the eggs with his anal, ventral, and pectoral fins. The perivitelline space in the developing ovum was crowded with free refringent corpuscles, a fact not noted in any other Teleostean ovum. Hatch- ing took place on sixth to eighth day. The intestine in the larva ends not very far behind the yolk sac. Fam. 2. SCOPELID. » 3 CYPRINIDA. The carps are all fresh-water fishes. The eggs are in most cases adhesive, and attached to aquatic plants. The zona radiata is double. Carassius auratus, L., the gold-fish, and the variety known as the _ telescope-fish, attach their eggs to water plants (M. von. KowaLeswkl, Zeit. f. wiss. Zool., Bd. xliii.). The larve of Cyprinus (Leuciscus) rutilus and C. idus are figured by SUNDEVALL. These figures are curious. In the newly hatched larva they show the yolk apparently extending back to the anus; that is to say, although the anus is near the end of the tail, as in other physostomous larve, the yolk, instead of being ellipsoidal in shape, is elongated, and occupies, in addition to its usual space, the interval ordinarily taken up by the preanal median fin-fold. The latter structure is shown in a normal state of development in stages sub- sequent to the absorption of the yolk, and it is possible that the apparent anomaly in the earlier stages is due to want of definition in the drawings, as in SUNDEVALL’s figures generally the limit between intestine and yolk sac is 110 MR J. T. CUNNINGHAM ON THE not clearly shown. The eggs were hatched in May. The newly hatched larva of Leuciscus rutilus is 6°5 mm. long, of L. idus, 7°3 mm. Fam. 4. KNERIIDA. ,, ». CHARACINIDA. » 6. CYPRINODONTID. Many of the Cyprinodonts are viviparous. The males are always much smaller than the females. A. AGassiz has figured some late stages of one species, Fundulus nigrofasciatus, C. and V., but his youngest stage has the homocercal tail already complete, and does not allow one to judge of the characters of the newly hatched larva. Fam. 7. HETEROPYGII. » 8. UMBRIDA. », 9. SCOMBRESOCIDA. The position of this family is somewhat doubtful; it is placed by GUNTHER among the Physostomi, although the air-bladder has no duct. By Ciaus (Grundziige der Zoologie, 4th ed., 1882), the family is added to the Anacanthini. The peculiarities of the vitelline membrane in this family were first noticed and described by Harcxen (Muller’s Archiv, 1855). Prof. K6LLIkeEr, in the Verh. d. Physik u. Med. Ges. zu Wiirzburg, 1858, corrected and added to HAEcKEL’s observations. A clear and satisfactory description of the membrane, with its filamentous processes, is given by Joun A. RybER, in his paper on the Develop- ment of Belone longirostris = Belone truncata, Giinther (Lesueur) (Bull. U.S. Fish Commission, vol. i. 1881). From that paper we learn that the egg of Belone is much heavier than sea water, and sinks rapidly to the bottom when un- disturbed ; and also, that by means of the filaments, large numbers of the eggs spawned from the same female are fastened together, and the clusters usually become attached to foreign objects in the water, which objects may of course chance to be either fixed or in a state of free suspension. The vitellus is optically homogeneous, and the whole egg transparent, though, I infer, less so than pelagic ova. The larva, after hatching, is not figured, but as far as can be judged froin figures of the embryo within the vitelline membrane, the anus is in immediate proximity to the yolk. This point cannot be definitely decided. The egg of Belone truncata is rather large, measuring, according to RYDER, + inch diameter, or, as measured from the figure given by him, 3-49 mm. The vitelline membrane is provided with filaments similar to those of dselone in the genera Scombresox, Hemirhamphus, and Exocoetus (flying fish). Arrhamphus’ eggs have not been examined. The eggs of Belone vulgaris, EGGS AND LARVA OF TELEOSTEANS. 111 Fleming, have been examined by Mr Francis Day, and a short description of the filaments is given in his British Fishes. The species occurs on the British coasts. It is not uncommon on the south coast, and, according to PARNELL, enters the Firth of Forth in July. I have not obtained any specimens hitherto. It is worthy of note, that if I am right in judging from Ryber’s figure that the rectum in the larva of Belone is in contact with the yolk sac, this fact confirms the view of Cuiaus, that the Scombresocide do not belong to the Physostomi. Fam. 10. Esocip£. There is only one genus in this family, Esox, the pike. The eggs of Hsox lucius, Linn., have frequently formed the subject of embryological investigation, and were part of the material on which was based the classical memoir of LEREBOULLET, “ Recherches d Embryologie Comparée sur le Brochet, l’Ecrevisse et la Perche” (Ann. d. Sci. Nat., ser. iv. vol. 1. 1854). The eggs are small, and are deposited in February and March. They are adhesive and attached to aquatic plants in narrow creeks or ditches (Day). The larve of the pike at different stages are described and figured by SUNDEVALL (Svenska Vet. Akad. Hand., 1855). The youngest stage figured is two days old. The anus is nearer to the end of the tail than to the yolk sac ; the pectorals are developed, but not the ventrals; the eye is considerably pig- mented, and chromatophores are scattered all over the body; the length is 10 mm. ; the newly hatched larva is 9 mm. long. It is noteworthy that the pelvic fins have no relation in development to the ventral fin-fold ; the latter persists, extending between the pelvic fins and in front of them long after they have begun to appear. Fam. 11. GALAXIIDA, » 12. MormyrRipDs#. », 13. STERNOPTYCHIDA. Argyropelecus hemigymnus, Cocco, was dredged between the Shetland and Faroe Islands by the ‘‘ Porcupine” in 1869. Most of the species are pelagic, some abyssal. The eggs are large (Day, Brit. Fishes). Fam. 14. STOMIATIDA. », 15. SALMONIDA. The ova of Salmo are large, heavy, and non-adhesive. In the newly hatched larva of this genus, or alevin as it is commonly called, the anus is at 112 MR J. T. CUNNINGHAM ON THE a distance from the yolk, a preanal embryonic fin separating the two, as in the herring. The notochord, however, is multicolumnar. As has already been mentioned in the case of Esox, the preanal fin-fold extends between and in front of the pelvic fins in the alevin of Salmo. The larva of Coregonus oayrhynchus, Nilss., at different stages is figured by SUNDEVALL (Joc. cit.). The ova were deposited from 6th to 10th November, and hatched in the following February ; they fall loose and separate to the bottom of the water; the diameter measures 3 mm. The newly hatched embryo is 11 mm. long; the anus is near the end of the tail, far removed from the yolk sac. The pelvic fins develop at the sides of the preanal fin long before the latter disappears, and the position of the pelvic fins is behind the anterior end of the preanal fin, where it meets the yolk-sac. The ova of Thymallus (Grayling) are similar to those of Salmo, but smaller. The ova of Osmerus (eperlanus at all events) are adhesive, the external adhesive layer of the zona radiata peeling off from the inner, and forming a suspensory membrane. SuNDEVALL (Svensk. Akad., 1855) gives figures and description of the newly hatched larva of Osmerus eperlanus; its length is 55 mm., the anus is near the end of the tail, there is a single oil globule in the yolk, the eye is slightly pigmented ; the eggs were obtained May 2, hatched May 20, 1855. Agassiz and WuiTMaAn (Pelagic Stages, p. 38) remark that the development of the pelagic egg they believe to be Osmerus mordax closely resembles that of the herring as given by SUNDEVALL. They seem to have overlooked SuNDE- VALL’s figures of Osmerus. The presence of an oil-globule in the larva of the latter genus is sufficient to prevent its being confounded with the larvasup- posed by the American authors to belong to Osmerus mordazx. Fam. 16. PERCOPSID. Fam. 20. PANTODONTID. » 17. HAPLOCHITONIDA. », 21. OSTEOGLOSSID/. » 18. GONORHYNCHIDA. » 22, CLUPEIDA. » 19. Hyopontipsé, The ova of Clupea harengus, Linn., have been carefully studied. The ova are heavy and adhesive. The yolk is composed of a number of spherical or nearly spherical yolk spheres, with no oil-globules. The blastodisc is large, forming about one-fifth of the whole egg. The newly-hatched larva is pelagic and very transparent, the anus is far behind the yolk sac, the notochord unicolumnar, the eyes slightly pigmented, but no pigment in the rest of the body. The fertilised eggs and larvae of Clupea sprattus, Linn., have never been observed. Eggs, apparently mature, were pressed by Mr Duncan MATTHEWS from a few specimens of the fish which had well-developed ovaries. EGGS AND LARVA OF TELEOSTEANS. 113 The ova were apparently adhesive, similar to those of the herring, but considerably smaller (Report on the Sprat Fishing of 1883-84, Second Annual Report of the Scottish Fishery Board, 1884). The eges of Alosa sapidissima have received much attention from the United States Fish Commission. They differ from those of the herring in not being adhesive; they are deposited in fresh or brackish water, and are but slightly heavier than the water itself, so that they remain in a state of suspension near the bottom. It is a curious fact that, although the artificial cultivation of shad ova has been practised on such a large scale in America, no memoir on the development of the fish has appeared in the publications of the U.S. Commission. I have not been able to find any figure of the ova at any stage of development, but Mr Joun A. Ryper, in a paper on the absorption of the yolk in embryo fishes (Bulletin U. S. Fish Commission, vol. ii., 1882), gives a figure of the anterior region of a larval Alosa, some days after hatching. All that can be drawn from this figure is that the notochord is multicolumnar. Fam. 23. BATHYTHRISSIDA. Fam. 28. HoPLOPLEURID&. » 24, CHIROCENTRID. » 29. GYMNOTIDA. , 20. ALEPOCEPHALIDA. ,», 930, SYMBRANCHIDA. » 26, NOTOPTERIDA. » ol, MURAENIDA. » 2/¢. HALOSAURIDA. A great deal has been written about the reproductive organs of Anguilla and Conger. The fertilised ova have never been seen, but young eels about 24 inches long are common enough in canals and rivers in spring. Specimens from the Forth and Clyde Canal were brought to me in April 1886. For an account of the investigations which have been made into the reproduction of the eel, see G. Brown Goope, Bull. U. S. Fish. Commission, vol. i., 1881. From the above survey it is seen that no physostomous fish is known at present. to have pelagic ova. In the newly-hatched larve, at present known, the anus is separated by a considerable interval from the yolk sac. In the Clupeidee the notochord is unicolumnar, but this is not a feature common to the order, as that organ is multicolumnar in the newly-hatched Salmo. It seems pretty certain that the problematic ovum, which AGassiz and WHITMAN identified as belonging to Osmerus mordaz, is derived from some physostomous fish. V. HENSEN suggested, in a letter to me, that it was the ovum of the sprat, but without evidence this is improbable, and it is not supported by the account of the sprat’s ovum given by Duncan Marruews. As no one has seen the embryo of the eel, it possibly belongs to Anguilla, but in that case one would expect to find the ova more plentiful. VOL. XXXIII. PART I. P * 114 MR J. T. CUNNINGHAM ON THE Orprer II, ANACANTHINI. Fam. 1. Lycopip&. T am not aware that the development of any species of this family has been studied. Fam. 2. GADID. Gadus—A large number of the species of this genus have been studied— Gadus morrhua, merlangus, and ceglefinus by myself, G. morrhua-by Joun A. Ryper. The eggs are, of course, closely similar except in size. The largest of the three species above menticned are those of G. wglefinus. The eggs are pelagic, the yolk is optically homogeneous, and destitute of oil globules. In the newly hatched larva the anus is not open, the rectum is in immediate proximity to the yolk sac, the notochord is multicolumnar, the pelvic fins not developed, and the mouth not open. In the newly-hatched haddock the eyes are considerably pigmented ; there are stellate chromatophores scattered over the sides of the trunk, and a single row of them along the ventral edge of each side of the tail. Motella—The development of Motella mustela, Linn., the five-bearded rockling, has been studied by GrorcE Broox (Jour. Linn. Soc., 1884, vol. xviii.). The eggs were deposited in his aquarium, under observation. The eggs are pelagic, and have usually one large oil globule, exceptionally more than one. (The buoyancy of the egg is in the paper attributed to the oil globule, an error which has been repeatedly made; there are many pelagic ova which have no oil globule.) The eggs are somewhat oval in shape and slightly variable in size. Length of longer axis, ‘655 to ‘731 mm.; of shorter, ‘640 to-716 mm. Hatching took place in 53 to 6 days, at a temperature of 51° to 62° F. In the newly hatched larva the rectum is immediately behind yolk, but not open, and not extending to the edge of the fin-fold. The eyes are slightly pigmented, and there are two small patches of pigment on the tail. The anus was not open seven days after hatching; the mouth not open at hatching. Spawning took place in May and June. Motella argentea, Rhein—The young in various stages were identified and described by A. Acassiz, July 1882, in Young Stages, pt. iii. (Proc. Amer. Acad. Arts and Sei., vol. xvii.). In the youngest stage, 4 mm. in length, the embryonic fin-fold is continuous, notochord multicolumnar (a point not ascer- tainable from Broor’s figures), pelvic fins palmate and large. In oldest stage, 3°4 cm. in length, two dorsal and one anal fin all distinct ; pelvic fins very long and narrow. There is some uncertainty about the identity of the specimens ; they may belong not to Motella argentea, but to some species of Onus. EGGS AND LARVA OF TELEOSTEANS. 115 Eggs taken by the tow-net at Newport, identified as belonging to Motella argentea, are described by Acassiz and WHITMAN in Pelagic Stages of Young Fishes. The identification is based on the character and distribution of the pigment in the larva hatched from the eggs, and is to some extent doubtful. The average size of the eggs is ‘78 mm. There is a single oil globule (in one case two, which coalesced) which is large and colourless, and measures ‘15 to ‘16 mm. in diameter. The figure given of the newly-hatched larva agrees closely with Broox’s figure of Motella mustela. The embryonic period varied with the temperature from three to six days. The eggs were taken from May to July. Motella cimbria, Nilsson (Linn.).—The four-bearded rockling. PARNELL’s example, captured in June, had the ova almost mature. Three specimens were taken by me, in the trawl, off Fast Castle Point, Haddingtonshire, March 12, 1886. In these the reproductive organs were very small. The largest specimen was ‘26 m. long. A species allied to Motella, probably actually a species of that genus, is figured in Pelagic Stages, pl. xii. The ovum is ‘70 mm. in diameter, and has a single oil-globule. The newly hatched larva agrees with that of Motella mustela, in that the rectum terminates, apparently blindly, immediately behind the yolk, and does not extend to the edge of the ventral fin-fold. The eggs were obtained in March and April. Another species allied to Motella is figured in Pelagic Stages, plate ii. figs. 1 to 3. The ovum of Merlucius is mentioned and figured by KinesLey and Conn,* but the size is not stated. Like that of Motella, it has a single large oil globule at the vitelline pole. The eggs and larve of Lota vulgaris, the burbot, have been described by Cart J. SUNDEVALL in Svenska Vetensk. Akad. Hand., 1855. The species is entirely confined to fresh water, and is thus unique among the Gadide, all the rest of which are marine and produce pelagic ova. The ova of Lota are shed separate and loose at the bottom of the water; some ova are opaque, some transparent. According to SUNDEVALL, they are small, but measurements are not given. Figures of the newly hatched larva and somewhat later stages are given; the drawings are not quite adequate, but show some essential points. In the newly hatched larva there is a single oil globule in the yolk, and there- fore probably in the ovum; the anus is close behind the yolk, but not in contact with it; the larva is 3 mm. in length. This larva bears a close resemblance to that of Motella, as figured by Broox. There are two differences in Lota; the oil globule is not so far back, and the two transverse stripes of pigment in the tail of Motella are wanting. The pigment in an eight days old larva of Lota * Memoirs of Boston Society of Natural History, vol. iii. No. 6, 1883. 116 MR J. T. CUNNINGHAM ON THE formed a series of spots along the dorsal edge of the side of the body and tail. This case is interesting, as showing how little modification is necessary to adapt the ova of two allied fishes to such apparently different environments as the surface of the sea and the bottom of a river or stream. The ova of the cod sink in fresh water, but they probably would not develop in that condition. The ova of Trigla gurnardus sank in the water of the Scottish Marine Station, but they invariably died in that condition after some days. The conditions in which the ova undergo development are not constant in a given family, but the structure of the ovum is more so, and the structure of the larva is always characteristic of families, and even to some extent of whole orders. Fam. 3. OPHIDIIDA. The eggs of Fierasfer (acus and dentatus) have been described by EMEry in his Naples Station monograph on the genus. The ovum has a single large oil globule ; it is small, ‘8 mm. in diameter. The ova when deposited are united together in masses, each mass containing many thousand eggs in a thick gela- tinous envelope. The masses are pelagic, floating at the surface of the sea. In the newly hatched larva the anus is in immediate proximity to the yolk, which still contains its oil globule situated at the anterior end. A great deal of pigment along the sides of the trunk, and a single row of chromatophores on each side at the ventral edge of the tail. A little in front of the level of the anus a median dorsal papilla interrupts the continuity of the fin-fold. This papilla grows rapidly, and ultimately forms a long filament supported on a short upright stalk. The filament bears a number of leaf-like appendages, and is called the vexillum. No stages of embryonic development are figured, and in the figures of the larva the internal structure is not shown. The structure of the notochord cannot be seen. A very lucid and complete account is given of the ovarian development of the ovum. The vitelline nucleus is described, and shown to be merely the starting point of the development of the vitelline spheres, which by their coalescence form the yolk in the mature ovum. The oil globule similarly arises from the coalescence of a number of small ones. The differences in the structure of mature ova are thus explained, and no support is given to the ideas recently advanced concerning the origin of the yolk from follicular cells, or of the latter from the germinal vesicle. Fam. 4, MaAcrurip&. Development not yet studied. EGGS AND LARVA OF TELEOSTEANS. a7 ANACANTHINI PLEURONECTOIDEI. Fam. PLEURONECTID. The development of a great number of species belonging to this family has been studied. In the preceding section of this memoir, ova of four species of Pleuronectes are described. Mention of the study of several species has been made by M‘Inroso. In Appendix F. of the Third Annual Report of the Scottish Fishery Board, he states that the ova of the cod, haddock, whiting, grey gurnard, common flounder, turbot, sole, lemon dab, common dab, and long rough dab had been examined in the Marine Labo- ratory at St Andrews. E. E. Prince describes the ova of Pleuronectes platessa, P. fiesus, P. limanda, as well as those of Gadus ceglefinus, G. morrhua, G. merlangus, and Trigla gurnardus, in Ann. and Mag. Nat. Hist., May 1886, but gives no figures. The young of Pleuronectes Americanus, Walb., are described and figured by AGassiz in Young Stages, plate ii., from a stage at which the larva is 4mm. in length. The eggs and newly hatched larva of this species are figured in Pelagic Stages, plate xvi. In the larva the rectum is, as far as can be judged from AGassiz’s figure, a little distance behind the yolk, and the notochord seems to be unicolumnar ; but on neither of these points is the figure very distinct. In all the species of Pleuronectes which I have figured the rectum is in contact with the yolk, and the notochord multicolumnar. Pseudorhombus.—The eggs and larvee of Pseudorhombus oblongus, Storer, the Sienna flounder, are figured by Acassiz in Young Stages, ii. plate ix. figs. 1-3, and in Pelagic Stages, plates xiv., xv., figs. 1-14. There is one oil globule, which in the newly hatched larva is at the posterior end of the yolk ; at the same stage the rectum is in contact with the yolk. The character of the notochord is not shown in the figures. The egg of the transparent flounder, Pseudorhombus oblongus, Stein, has no oil globule, and no pigment on the yolk. The figures 1-4 on plate vi. of Young Stages, ii., given under the name of P. melanogaster, Stein, really belong to Tautoga onitis. Rhombus maculatus, Mitch.—Some advanced larve of this species are figured in Young Stages, ii., but the eggs and newly hatched larva are not given. Hippoglossoides limandoides.—I have been unable to cbtain eggs of this species. Many specimens were obtained in the months of May and June, which were spent; occasionally a ripe male was obtained, but never a ripe female. It probably spawns in the neighbourhood of the Firth of Forth in April. M‘IntosH states that he obtained the ova of this species before 1st June 1884, but he does not describe them. (Second Annual Report, Silo.) 118 MR J. T. CUNNINGHAM ON THE Arnoglossus megastoma, Donovan.—A specimen of this species, taken 16 miles E. by N. of May Island, April 30, 1886, in the trawl, was brought to the station. The ova were quite immature. THompson at Belfast, according to Day, ascertained that it spawned in October. Plagusia.—The young, about 1 inch long, is figured by Acassiz in Young Stages, ii., but not the eggs or larvee. Thus all the Anacanthini, as far as at present known, except Lota, have pelagic ova, and in all the rectum at the time of hatching is in contact with the yolk. In Gudus and Motella the anus is not open, and does not extend to the margin of the ventral fin-fold. OrpER IIT. ACANTHOPTERYGII. Div. I. PERCIFORMES. Fam. 1. PERcIDz#. The young of Labrax lineatus, Bl. and Schn., are figured by A. Acassiz in Young Stages, iii. In the youngest stage figured, 3-5 mm. in length, the yolk sac is already absorbed. ‘The larvee were taken at the surface of the sea with the tow-net, but the eggs were not found. The ova of Perca fluviatilis are adhesive, and attached to fresh-water plants. An account of them is given by SUNDEVALL. Hatching occurred fourteen days after fertilisation. The newly hatched larva was 5 mm. long; there was a single oil globule in the yolk sac, ‘and the anus was slightly separated from the latter. Spawning took place in May. The ova of Serranus cabrilla are stated by HorrMann to be pelagic. Fam. 2. SQUAMIPENNES. Fam. 5. CirRHITID-. » oo MULLIDA. » 6. SCORPANIDA, » 4 SPARIDA. This family consists exclusively of marine fishes, and all the species that have hitherto been studied from the embryological point of view have pelagic ova. An account of the ovum of Scorpena is given in HorrMann’s memoir, published in the 7ransactions of the Amsterdam Academy, 1881. The species observed were S. porcus and S. scrofa. The ripe ovum before fertilisation consists of a perfectly homogeneous glassy yolk surrounded by a thin envelope EGGS AND LARV OF TELEOSTEANS. 119 of protoplasm, which has a faint reddish tinge, and is as usual principally accumulated at the micropylar pole. The ovum before fertilisation has a slightly oval form, ‘95 mm. by ‘84 mm. in diameter. After fertilisation the perivitelline space is very small. HorrMann deals only with fertilisation and segmentation, and gives no figures or descriptions of embryos or larve. His examination of the ova of Scorpena was made at Naples. The ova of Scorpeena are deposited in masses, each mass consisting of a large number of ova enveloped by a slimy substance. Horrmann believes that the slimy sub- stance is not a product of the egg-membrane, but probably the peculiarly modified connective tissue of the theca folliculi. This conclusion seems extremely unlikely, but no investigator has yet inquired into the origin of the gelatinous envelopes which contain the ova of Scorpwna or of Fierasfer. The newly hatched larva of Scorpena is 2:07 mm. in total length, and the anus is almost in contact with the yolk sac, ‘07 mm., according to Horrmany, being the distance between the two. Hemitripterus americanus, C. and V. (H. acadianus, Storer.)—The ova and larve of this species are described by A. Agassiz in Pelagic Stages. The ovum is pelagic, and possesses a single oil globule, which in the early stages of development is at the pole of the yolk opposite the centre of the blastoderm. Diameter of ovum, 1:02 to1:10 mm. The developing embryo is distinguished by the large number of brownish-yellow chromatophores which, interspersed with a few black ones, are present on the sides of the body of the embryo, and over the whole surface of the yolk. In the newly hatched larva the rectum is separated by a very slight interval from the yolk. The fin-fold is very wide, and in it are three pigment patches—two dorsal and one ventral. The anus is apparently not perforated. The structure of the notochord is not shown. The identification of these ova and embryos seems to be based on the characters of the older stages of the larvee. Fam. 7. NANDID. 8. POLYCENTRID&: 9. TEUTHIDIDA. I} 2) Div. IJ. ACANTH. BERYCIFORMES. Fam. 1. BERYcIDA. Div. II]. Acantu. KuRTIFORMES. Fam. 1. Kurtip&. 120 MR J. T. CUNNINGHAM ON THE Div. IV. AcAnTH. POLYNEMIFORMES. Fam. PoLYNEMID®. Div. V. ACANTH. SCIANIFORMES., Fam, 1. ScL©nIp&. Div. VI. ACANTH. XIPHIIFORMES. Fam. 1. XIPHIID. Div. VII. AcANTH. TRICHIURIFORMES, Fam. 1. TRICHIURID. Div. VIII. AcANTH. CoTTO-SCOMBRIFORMES. Fam. 1. ACRONURID&. » 2 CARANGIDA. Capros, according to Day, was observed to shed pelagic ova by Mr Dunn at Megavissey, July 20, 1882. Temnodon saltator, Linn. (Pomatomus saltatriz, Gill), is called the Blue- fish on the Atlantic coast of the United States. Pelagic ova, believed to belong to this species, are described by Acassiz in Pelagic Stages, and a long larva, 9 mm. in length, identified as Temnodon, is figured in Young Stages, pt. iii. pl. ii. The ova to which I have referred are, in one respect, unique among all the kinds of pelagic ova hitherto described. In AGassrz’s own words, the egg exhibits a partial segmentation of the yolk—that is to say, at the stage when the embryonic ring has just been formed, there is a ring of definitely limited large cells round the edge of the blastoderm. After the blastoderm has enclosed the yolk, the large cells seem to form a complete envelope round the yolk beneath the blastoderm. To judge from the figures given by AGAssiz and WHITMAN, I should have concluded that in this species of ovum the periblast, instead of being a syncytium, was divided into cells, and should have been ready to agree with the view expressed by those authors in their “Preliminary Notice” (Proc. Amer. Acad. Arts and Sci., vol. xx.), namely, that the actual cleavage of the yolk in this instance was positive proof that the nucleated periblast in all cases, and the yolk, are “integrant portions of the ovum.” But in Pelagic Stages it is stated that closer examination has shown EGGS AND LARVA OF TELEOSTEANS. 121 that the large cells are situated beneath the periblast, and belong to the yolk ; that they are not protoplasmic elements, but vitelline, although they have an epibolic growth, and extend round the unsegmented yolk as this becomes enclosed by the blastoderm and periblast. It is pointed out that the change in relations of these superficial yolk segments shows that a transposition occurs in the Teleostean ovum among the yolk elements closely analogous to the in- vaginatory movement of the yolk in holoblastic ova. The diameter of the ovum is ‘70 to °75 mm. The ova occur at Newport from the middle of June to middle of August. At the yolk pole there isa single large oil globule. The newly hatched larva is 2:15 mm. in diameter; the rectum is separated by a distance of ‘275 mm. from the yolk sac; pigment is scanty; a series of black chromatophores along the dorsal edge of the tail, and a few brownish-yellow ones along the body and rectum. The structure of the notochord is not shown. The development of the young fish was traced till a stage at which it measured 9 mm. in length. The presence of an oil globule, the externally segmented yolk, and the slight separation of the rectum from the yolk sac, are the diagnostic features in Temnodon, but how far these are characteristic of the family is not known. Fam. 3. CytTTip&. , 4. STROMATEIDA. This is a small family of marine fishes, containing only two genera. Figures of Stromateus triacanthus, Peck, from a length of 7 mm. upwards, are given by AGassiz in Young Stages, pt. 11. The notochord is apparently multicolumnar, but no other embryonic or larval characters are to be discovered from the figures. The species is called Butter-fish in America, and the young at the length of 10-20 mm. are in the habit of sheltering themselves beneath the umbrella of Dactylometra, one of the Scyphomeduse. Fam. 5. CorYPHANID. » 6. NoMEIDz. » 4 SCOMBRIDA. According to Day, the eggs of Scomber scomber, the common mackerel, are shed in May and June, and in the Brighton Aquarium have been observed to be of the pelagic kind. The development of Cybiwm maculatum, the Spanish mackerel, has been described by Joun A. Ryper (Bull. U.S. Fish Commission, vol. i., 1881). The investigation was carried out in July 1880 at Mobjack Bay, Virginia, and in 1881 at Cherrystone Harbour, Va. The eggs hatched twenty- four hours after fertilisation, but the temperature to which they were exposed is not stated. Evidence was obtained that spawning naturally takes place at VOL. XXXIII. PART I. Q 122 MR J. T. CUNNINGHAM ON THE night. The ovum measures ;!; to »5 inch in diameter, or, as measured from the figures given, ‘856 to 1:06 mm. It is pelagic; there is a single large oil globule, otherwise the yolk is homogeneous ; the perivitelline space is small. The newly hatched larva is 2°52 mm. long as measured from the figure; the notochord is multicolumnar ; the anus immediately behind the yolk, and open ; pigments spots are present on the body and round the oil globule, and also form one conspicuous transverse stripe in the middle of the tail. The oil globule in the hatched larva is situated on the ventral side of the yolk, a little posteriorly. The mouth opens twenty-one hours after hatching. Fam. 8. TRACHINID. The development of Trachinus vipera has been described by Gro. Brook (Lin. Soc. Jour., vol. xvii, 1884). The eggs were shed in that author’s aquarium. Spawning takes place at night, and is continued through the months of May, June, and July. The ovum is pelagic, 1°32 mm. in diameter, and contains from 20 to 30 small oil globules. The oil globules are external to the vitellus, and contained in depressions of its surface. It is probable that this is often the case; it certainly is m Trigla gurnardus, but whether the oil globules are always external is doubtful. The perivitelline space is small. Hatching took place on ninth, tenth, and eleventh days, at a temperature of 54° to 60° Fahr. In the newly hatched larva the rectum is immediately behind the yolk sac, the notochord multicolumnar. The eyes are pigmented; black pig- ment cells are scattered over the body and the surface of the yolk sac, and aggregated in a transverse stripe at the middle of the tail. The ventral fins are well developed at the time of hatching. The length of the newly hatched larva is 355 mm. The yolk sac is absorbed, and the mouth well developed twenty- four hours after hatching. Fam. 9. BATRACHIDA. The young Batrachus tau, Lin., 2 mm. in length, has been figured by Storer (Mem. Amer. Acad., v. pl. xix.). AGAssiz figures a specimen 6 mm. in length in Young Stages, pt. iii., but this shows only traces of the larval characters. The anal is still continuous with the caudal fin, and the “ ganoid” lobe of the tail is well marked. Fam. 10. PEDICULATI. The eggs of Lophius piscatorius, Lin., are described in Young Stages, pl. iii The eggs are held together by gelatinous mucus in a single flat layer which floats horizontally in the sea, forming a large sheet 3 feet broad and 25 to 30 feet long. The spawn is shed on the American coast from June to EGGS AND LARV OF TELEOSTEANS. 123 August. It has also been observed on the British coast, but I have not myself met with it. A more complete description, with better figures, is given by Acassiz and Wuitman in Pelagic Stages. The egg is large, 1°75 mm. in diameter, and has a single immense oil globule ‘4 mm. in diameter, of a transparent copper colour. Black chromatophores are developed very early, and are aggregated chiefly about the ventral side of the embryo, present in less abundance on the surface of the yolk sac, round the oil globule, and on the tail. In the newly hatched larva the yolk sac is globular, and very large in compari- son with the body; the oil globule is ventral and posterior; the rectum is immediately behind the yolk ; the eyes are deeply pigmented; the notochord multicolumnar ; the pelvic fins not developed. The successive forms of the larva, which is up to a late stage pelagic, are described and figured in Young Stages, iii. AGAssiz points out the resemblance, both in the character of the spawn and the structure and development of the larva, between Zophius and Fierasfer, comparing the long anterior dorsal spine in the former, which is a permanent organ, but develops at a very early stage, to the vexillum of Mierasfer, which is a temporary appendage disappearing completely in the adult. It seems probable that in the Merasfer larva the vexillum is morpho- logically derived from a fin ray, as are the appendages in Lophius. The male of Antennarius, another species of this family, a pelagic fish, makes, according to GUNTHER, a nest, and guards the eggs deposited in it. We have thus in this family a series of steps in the transition from ordinary littoral adhesive ova to typical pelagic ova. The ova of Antennarius are probably adhesive, and are deposited in a pelagic nest. The ova of Lophius are also adhesive, but float as a detached mass unprotected by an apparatus formed from pelagic alge. If the ova of Lophius were separate, instead of adhering together in a mass, they would be typical pelagic ova. Fam. 11. CoTtip&. The question of the ova of Cottus has been discussed in a previous section. The pelagic ova of rigla gurnardus have been described.* In this family we have a greater difference between the ova of closely allied genera than in the preceding, for the eggs of Cottus are typical examples of littoral adhesive ova, while those of Trigla are typically pelagic. SuUNDEVALL (loc. cit.) gives an account of the development of Cottus gobio and Cottus quadricornis. The eggs of the former species are deposited in May. The larva twenty-four hours after hatching was 8 mm. long; there was a single oil globule in the yolk, and the rectum was in contact with the yolk sac. The larva of Cottus guadricornis is * “Yolk and Gastrula,” J. T. Cunningham, Quart. Jou. Mier. Se7., 1885. 124 MR J. T. CUNNINGHAM ON THE very similar; its length three days after hatching was 115 mm. The eggs of both species are adhesive, and form masses sticking to objects on the shore. Fam, 12. CATAPHRACTI. The fertilised ova of Aygonus cataphractus have never been described; but Prof. M‘Inrosu, in 37d Ann. Rep. S. F. B., says he found nearly mature ova in a specimen trawled near St Andrews on March 12. The ova had a pale salmon colour, were 1°3 mm. in diameter, and probably adhesive. Fam. 13. PEGASIDA. Div. IX, ACANTH. GOBIIFORMES. Fam. 1. DIScoLALI. The ova of Cyclopterus lumpus have been mentioned in the previous section. Liparis is the only other genus, and what is known of the spawn of Liparis Montagui has also been stated. Fam. 2. GoBlip&. Gobius Ruthensparri is stated by Day to have been bred in confinement by Mr Roserts of the Scarborough Museum. The ova were adhesive, and were deposited within the shell of a barnacle. The male watched over the mass of eggs, and fanned them with his fins. Horrmann (loc. cit., p. 19) gives a description and figure of the ovum of Gobius minutus. The ovum has a peculiar elongated pyriform shape, with a very large perivitelline space, and at the narrow end are a number of filaments, in the centre of which is the micropyle. The eggs are attached by the filaments. The ova of Callionymus lyra are pelagic, and have been described by M‘Intosu, in Ann. and Mag. Nat. Hist., Dec. 1885. -On the 8th August a female specimen was obtained at St Andrews, from which ripe ova could be pressed out. The ova are pelagic, transparent and buoyant, small in size, being of about the same diameter as the ova of Pleuronectes flesus. The exterior surface of the vitelline membrane or zona radiata exhibits a reticulum of slightly elevated ridges, the meshes of the reticulum being hexagonal; from this characteristic the ova can be easily identified. At Millport, in June of the present year, I obtained a pelagic ovum from the tow-net which agreed exactly with Prof. M‘Inrosn’s description. M‘Inrosu adds, that Zrophon, hermit crabs, and bivalve mollusca were found in the stomach of Callionymus. THe gives no figures or description of any embryonic or larval stages of the species. EGGS AND LARVA OF TELEOSTEANS. 125 There is here another example of a family in which some genera produce adhesive, others pelagic, ova. Div. X. ACANTH. BLENNIIFORMES. Fam, 1. CEPOLIDs, » 2. HETEROLEPIDOTIDA, » & BLENNIIDZ, The ova of Anarrhichas lupus have been discovered by Prof. M‘IntosuH to be deposited in February ; they are large, heavy, and non-adhesive, and the larve, when hatched, are well advanced in development (see Nature, June 17, 1886). The ova of Blennius galerita, according to Day, are adhesive, and attached to the under surface of stones. Blennius pholis also deposits adhesive ova, which are attached to small caverns in the rocks of the sea-shore. The ova of Blennius are stated by HorrMaNnn to possess processes extend- ing from the zona radiata. The ova of Centronotus gunnellus, according to W. ANDERSON SMITH, are deposited from February to April. The ova are adhesive, and form a spherical mass about the size of a walnut; this ball is quite free, and both parents lie coiled round it. In Zoarces viviparus the ova are retained during development within the cavity formed by the coalesced ovaries. Breeding takes place in the winter months, chiefly in December, January, and February. Specimens in which the young were ready to be born were obtained on the shore at Granton in February and March. The young at parturition are about 14 inches long, and in all respects, except size, similar to the parents. I met with several specimens in which the young in the ovary had been killed by some cause or another, and when the cavity was cut into, their bodies were discovered in a shrunken state, but not decomposed. Fam. 4. MASTACEMBELID. Div. XI. ACANTHOPTERYGII MUGILIFORMES. Fam. 1. SPHYRENID&. ,» 2 ATHERINIDA. Several stages of Atherinichthys notata, Giinther (Chirostoma notata, Gill), are figured by Acassiz in Young Stages, pt. iii. In the youngest the embryonic fin-fold is still unaltered, but the yolk sac is absorbed, and the mouth open. 126 MR J. T. CUNNINGHAM ON THE The ovum of this species is stated by Rypger to possess four filamentous processes connected with the vitelline membrane (“Development of Belone longirostris,’ Bull. U.S. Fish Commission, vol. i.). The threads or filaments are more completely described by Ryprr in vol. ii. of the same bulletin, the fish being there called Menidia, which is asynonym. The threads are in length about eight times the diameter of the ovum, and when the latter is first emitted the threads lie coiled spirally round it. There can be little doubt that the four threads are merely the outer layer of the zona radiata in a specialised form, and are homologous with the suspensory membrane in Osmerus. Fam. 38. MuGILip®. Diy. XII. ACANTH. GASTROSTEIFORMES, Fam. 1. GASTROSTEID. The ova of G'astrosteus are adhesive, and deposited in nests made with water plants, and guarded by the male. Spinachia vulgaris makes nests of seaweeds, Fucus, &¢c., on the sea-shore; its ova are similar to those of G‘astrosteus, but larger. It has been shown by Prof. Karu Mostus of Kiel, that the white fila- ments, by which the nest of Spinachia is held together, are spun by the male fish, and that they are formed from a substance resembling mucin which is pro- duced in the kidneys (see Schr. Naturwiss. Vereins fiir Schleswig Holstein, Ba. vi., 1885; translated in Ann. and Mag. Nat. Hist., Aug. 1885: also E. E. Prince, Ann. and Mag., Dec. 1885). The ova of Spinachia, according to PRINCE, are ‘085 inch in diameter. A large mass of pale yellow oil globules are aggregated at the yolk pole. At temperature 41° to 51° Fahr., in June the ova hatched in twenty-five to forty days. No figures of the development are given by PRINCE. Fam. 2. FISTULARIIDA. Div. XIII. AcAntH. CENTRISCIFORMES. Fam. 1. CENTRISCIDA. Div. XIV. AcANTH. GOBIESOCIFORMES, Fam. 1. GoBIESOCID®. Lepadogaster Decandolit.—Some observations on the development of this species are described by W. ANvERSON SmitH in Proc. Roy. Phy. Soc. Edin., 1886. The ova are adhesive, attached to stones or shells, and watched over EGGS AND LARVA OF TELEOSTEANS. 127 by both parents. Spawning takes place in June and July on the west coast of Scotland. The ovum has a single oil globule, and is hatched twenty-eight days after fertilisation. The ovum of Z. bimaculatus are always found adhering to the inner surface of shells of Pecten operculatus; they are deposited likewise in June and July, and are guarded by at least one of the parents. Div. XV. ACANTH. CHANNIFORMES. Fam. 1. OPHIOCEPHALIDA. These are fresh-water fishes of the Indian region. The male Ophiocephalus is stated by GUNTHER to make a nest and guard the presumably adhesive ova. Div. X VI. AcANTH. LABYRINTHIBRANCHII. Polyacanthus viridiauratus, Gimther, the Macropus viridi-auratus of Lacépede, commonly called the Paradise-fish, is a native of the East Indian Archipelago, but is commonly kept in aquaria in Europe, and breeds freely in confinement. Some account of the ova is given by Dr Miecz. von KowALEwskt in Zeit. f. wiss. Zool., Bd. xliii. The perivitelline space is small; the yolk apparently broken up into small masses, and large oil globules are pee but the appearance of the living ovum is not described. OrDER II. ACANTH. PHARYNGOGNATHI. Fam. 1. PoMACENTRID. The ovum of Heliasis chromis is described by Horrmann (loc. cit., p. 19). The name he uses seems to be slightly erroneous. It is the WHeliastes chromis of Giinther’s British Museum catalogue. The ovum forms a some- what long ellipsoid with blunt ends, and at one of the poles is a group of eight or nine long straight filaments attached at their basis to the vitelline membrane. The micropyle is situated in the centre of the group of filaments. HOFFMANN remarks that the filaments in this ovum, and in Belone, Blennius, Gobius, represent the external zona in adhesive ova such as Leuciscus and Perca, in which the zona radiata is differentiated into two layers. On the other hand, in pelagic ova, such as those of Scorpena, or in heavy non-adhesive ova, such as those of Sa/mo, no division into two layers can be discovered in the zona radiata. We may add to this comparison of HorrmMann’s that it is pro- bable from what RypER observes concerning the development of the filaments in Belone, that these processes are actually formed by a splitting up and unequal development of the external zona; and thus there is no fundamental difference between the origin of the suspensory membrane in Osmerus eperlanus and the filamentous processes in Belone, Atherinichthys, Heliastes, &c. The 128 MR J. T. CUNNINGHAM ON THE yolk of Heliastes resembles in structure that of the herring, being composed of a number of ellipsoidal vitelline discs; but there is also present a large oil globule at the vitelline pole. The protoplasm in the mature unfertilised ovum forms as usual an envelope round the vitellus which is thickest beneath the micropyle, and thins away all round that point. The blastodise and blastoderm during simple segmentation is large in proportion to the yolk. The perivitelline space is considerable. HorrMANN gives no figures of the embryonic or larval stages. Fam. 2. LABRID. The development of a large number of the wrasses has been studied. Tautoga onitis, Linn.—The pelagic ova of this species are described and figured in Pelagic Stages. The diameter of the ovum measures ‘90 to ‘95 mm. The yolk is homogeneous, and there is no oil globule; the perivitelline space is of moderate dimensions. ‘The newly hatched larva is 3:05 mm. in length; the rectum is not in contact with the yolk sac, but at a distance of +55 mm, from it (not nearly so far back as in Clupea). The anus is open, the notochord multicolumnar; the eye is scarcely pigmented, but there are small compact pigment spots along the dorsal region of the sides of both body and tail; the pectorals are scarcely developed, the ventrals not at all. The eggs of Tautoga were artificially fertilised, so that the identity of the ova and newly hatched larvee is certain. But the authors point out that the ova of Ctenolabrus, Ps. melanogaster, and Tautoga are so similar, both in structure and size, that it is scarcely possible to distinguish them with certainty in the produce of the tow-net. The authors state that figs. 1, 2, 3, and probably fig. 4 (in my opinion fig. 4 also, certainly) in plate vi. of Young Stages, part ii., belong to Tautoga, and not to Pseudorhombus melanogaster. Thus the position of the rectum with respect to the yolk sac in the newly hatched larve is shown to be a constant family character ; and there is no exception to the statement that in Pleuronectide the rectum at that stage is in contact with the yolk sac. Ctenolabrus adspersus, Walb. (C. caruleus, Storer)—The ova of this species have been described by AGassiz and WuiTMan in Pelagic Stages, and by Kinestey and Conn. The ovum is ‘85 to ‘92 mm. in diameter. Before fertilisation the peripheral layer of protoplasm is densely filled with refractive granules, which render the ovum opaque; but after fertilisation the granules disappear, and the egg becomes perfectly transparent. In the newly-hatched larva the rectum is separated by ‘25 mm. from the yolk sac, the total length of the larva being 2°30 mm. There are black dendritic chromatophores along the sides of the body and tail. The time required for hatching varies from two to six days. The eggs are shed at Newport in the months of May and June. The ova of Julis vulgaris are described by HorrMann (loc. cit., p. 43). The diameter of the ovum measures ‘75 mm. The yolk is homogeneous, but con- EGGS AND LARVA OF TELEOSTEANS. 129 tains an oil globule ‘15 mm. in diameter. The ova are suspended separately in the water not united in masses. The dimensions of the newly-hatched larva are 1:77 mm. in total length, ‘15 mm. from the yolk sac to the anus. The ova of Crenilabras, of which genus HorrmMANN examined four species, are not pelagic, but adhesive. The zona radiata shows the division into two layers, which occurs in most adhesive ova. The diameter of the ovum is ‘7 to 75 mm. The yolk is not homogeneous, but contains a number of vitelline globules; there seem to be no oil globules. The newly-hatched larva is 3°6 mm. long, and the anus is ‘6 mm. from the yolk sac. Thus we see that considerable variations occur in the family of Labride in the character of the ova. Most of the genera produce pelagic ova, but the ova of Crenilabrus are adhesive. As in the Gadide, there is either a single oil globule in the yolk or none at all. Two characters seem constant throughout the family—(1) that the notochord is multicolumnar, (2) that the anus is at some little distance from the yolk sac, though not nearly so far back as on the Physostomi. The separation of rectum and yolk sac occurs also in the Carangidee (Temnodon),and among the Physoclisti seems to be confined to these two families. Fam. 3. EMBIOTOCID-. Fishes of the North Pacific, most abundant on the American coast. All viviparous. Fam. 4. CHROMIDES. OrvER V. LOPHOBRANCHII. Fam. 1. SOLENOSTOMID. According to GunruEr, the female bears the eggs attached to filaments developed on the ventral fins, the inner edges of which are united to the skin of the body. Fam. 2. SYNGNATHIDA. In Siphonostoma typhle, which is common on the British coasts, the ova are carried till the time of hatching by the male in a pouch formed by longitudinal folds of the skin behind the anus. | In Syngnathus there is a similar pouch in the male. According to RYDER, yolk contains numerous oil globules. In Nerophis the ova are attached to the abdomen of the male by a viscid secretion in front of the anus. JN. lwinbriciformis and N. aquoreus are not un- common on the east coast of Scotland, but I have not had an opportunity of examining the ova of either. Some account of the development of Hippocampus is given by JOHN A. Ryper in Bull. U.S. Fish. Commission, Bd. 1. In the embryonic Hippo- cumpus the fin-fold is wanting, in Syngnathus it is but slightly developed. VOL. XXXIII. PART I. R 1380 MR J. T. CUNNINGHAM ON THE OrvDER VI, PLECTOGNATHI. Fam. 1. SCLERODERMI. » 2. GYMNODONTES. The development of these has not been studied. The Maturation and Fertilisation of the Teleostean Ovum. In considering the subject of the phenomena which take place in the ripe Teleostean ovum immediately after its separation from the parent, two ques- tions chiefly excited my curiosity, neither of which have I yet solved to my complete satisfaction. These questions refer to the account of the phenomena which has been given by Professor C. K. Horrmann.* The first is, Is there any foundation for Horrmann’s statement that the first segmentation spindle is directed radially, and divides into a superficial nucleus which belongs to the archiblast, and a deeper one which belongs to the periblast? The second is, Can we trace in the fish ovum the transformations of the nucleus which accompany the expulsion of the polar bodies, and compare these transforma- tions with those which E. vAN BENEDEN t has described in Ascaris megalocephala. The subject of the last question will be considered first. The investigation of the matter is one of considerable difficulty. It is necessary, in the first place, to have a plentiful supply of healthy living specimens of some species with pelagic ova; and in the second place, to have at command the most approved appliances and reagents for their microscopic examination. The first opportunity I had of making the attempt was in May of the present year, when I had a number of ripe Pleuronectes limanda alive in the aquarium of the Station. In the ripe ovum of P. imanda, immediately on its escape from the ovary, the zona radiata is in immediate contact with the ovum. The condition of the ovum is shown in Plate III. fig. 1. There is no doubt that the eggs of all the species of Pleuronectes and Gadus are closely similar except in size, and RyDER is in error when he indicates a perivitelline space in his figure of the ripe newly shed ovum of the cod. There is a layer of protoplasm round the ovum in the neighbourhood of the micropyle, which thins out at the pole opposite the micropyle. In the living egg, within half an hour after it is shed, whether milt be added to the water in which it is contained or not, the expulsion of a transparent spherical polar body through the micropyle can be readily observed. Its appearance is shown in Plate II. fig. 10, taken from an unfertilised ovum, and Plate IIT. fig. 4, three hours after fertilisation. At this latter stage the perivitelline space has begun to appear ; it develops first in a ring round the micropyle as a centre. The protoplasm, immediately after the ovum is shed, begins to collect at the micropylar pole of the ovum, and this process begins * See Verhandelingen der konink. Akad. der Wetenschappen, Th. xxi., Amsterdam, 1881. } La Maturation, Fécondation, etc., et la Division Cellulaire, Paris et Gand, 1883, EGGS AND LARVA OF TELEOSTEANS. 151 the rhythm of segmentation. At the stage shown in Plate III. fig. 3, one and a half hours after fertilisation, the central part of the protaplasmic disc is much the thickest, forming a somewhat conical protuberance downwards iuto the yolk. The protuberance afterwards disappears, and at the end of three hours the blastodisc has the shape shown in Plate III. fig. 4, the lower surface being uniform. Then a second aggregation of the protoplasm begins, but this time towards two points, as shown in Plate III. fig. 5, producing the first division of the blastodisc. Thus the aggregation of the protoplasm towards the micro- pylar pole may be regarded as a contraction towards one central point, and the first division as due to a contraction towards each of two separate points. The expulsion of a polar body through the micropyle I observed repeatedly in the ova of Pleuronectes cynoglossus studied at Millport last June. It took place in both fertilised and unfertilised ova. But I was unable to discover either in the living eggs, or in the fresh eggs treated with reagents on the slide, any nuclear spindle either before or during the expulsion of the polar body. In one or two instances I noticed a minute pyriform projection of protoplasm on the surface of the blastodisc after the latter had withdrawn itself from the vitelline membrane (Plate III. fig. 7) ‘This might be either a second polar body, or simply the proximal part of the first drawn away with the receding blastodisc, from the inner end of the micropyle. Acassiz and WaurTMAN, in Pelagic Stages, p. 19, mention the formation of two polar bodies in Teleostean ova, and under- take to describe them in a subsequent memoir. My results, as far as they go, concerning the unfertilised ova, are in agreement with those of Horrmann. In the unfertilised ovum, as a rule, the expulsion of the polar globule through the micropyle and the concentration of the protoplasm take place just as in the fertilised ovum, with the exception that the latter process goes on much more slowly m the unfertilised ovum. I have never seen any traces of segmentation in the unfertilised ovum. The small proto- plasmic body on the blastodise seen two hours after fertilisation, and shown in Pl. III. fig. 7, was seen also at the same stage in the unfertilised ovum. The aggregation of the protoplasm in the fertilised ovum is finished about three hours after fertilisation, and the stage shown in PI. IIL. fig. 4, is reached. At this stage the unfertilised ovum stops unchanged, being perfectly incapable of segmenting. PI. III. fig. 9 shows an unfertilised ovum of Pl. cynoglossus six hours after shedding, at which time the fertilised ova were in the eight-cell stage, and the cells again dividing to form the sixteen-cell stage. The unfertilised ova were in the condition shown in PI. III. fig. 9, twenty-four hours after being shed, and remained unchanged till they died. I was not able to determine with absolute certainty at what stage the spermatozoon entered the fertilised ovum. This occurs, as is evident from Pl. IV. fig. 2, during the first half hour, and I am inclined to believe that it takes place immediately the ripe 132 MR J. T. CUNNINGHAM ON THE ovum is exposed to the milt, so that the spermatozoon remains within the protoplasm while the polar globule or globules are being expelled. To take up now the first of my two questions. The changes which occur in the blastodise after the ovum has been exposed to the influence of milt are shown in the figures of the ova of Pl. imanda and Pl. cynoglossus. The separa- tion of the vitelline membrane from the blastodisc occurs almost immediately after the ovum has been placed in sea water containing milt. The protoplasm ageregates at the micropylar pole, and half an hour after fertilisation it projects at the pole considerably into the yolk (Pl. ILI. figs. 3 and 8). At this stage, in ova treated with acetic acid and methyl green, I was able with a high power to see distinctly the male and female pronuclei in close proximity to one another (PI. IV. fig. 2). I was not able to discover the spindle produced from the union of these two bodies. The first segmentation of the blastodisc takes place gradually by the aggregation of the protoplasm round two centres, as in Pl. III. fig. 5, and Pl. IV. fig. 1. The protoplasm towards each side of the blastodise projects downwards into the yolk, so that there are now two of these projections instead of one, with a deep broad furrow in the under surface of the blastodise between them. No furrow on the upper surface of the blastodisc is at first visible. After treatment of the ovum with acetic acid and methyl green at this stage, a nucleus can be made out very distinctly in the two halves of the blastodisc, and these are the only two nucleiin the ovum. ‘The line joiming the nuclei is a chord of the sphere of the ovum, and not a radius, as stated by Horr- MANN. ‘The nuclei are best seen when the ovum is placed on the slide with the blastodisc downwards, so that the blastoderm is seen through the transparent yolk, the stained ova being mounted in glycerine. It was from an ovum in these conditions that Pl. IV. fig. 3, was taken, the ovum having been killed with acetic acid half an hour after fertilisation. The outline of the blastodisc on the surface of the ovum is not circular, but elliptical, and the plane of division passes through the short axis of the ellipse. This plane of division contains the principal axis of the ovum, by which I mean the axis passing through the centre of the blastodisc and the centre of the ovum. HOFFMANN states that the plane of the first division is perpendicular to the principal axis of the ovum (/oc. cit., p. 105). It seems to me possible that HorrmMann may have been led into this error by the relative positions in which the two nuclei are seen when the ovum is in a certain position with respect to the axis of the microscope. To explain this I must refer to the diagrams shown in figs. 1 and 2, Fig. 1 represents a section of the ovum passing through the principal axis and per- pendicular to the plane of the first division of the blastodisc. Now, if the axis of the microscope occupies, with respect to the ovum, the position wy, and the plane which is in focus, perpendicular of course to that axis, occupies the posi- tion shown by the line a 4, then the appearance of the section of the ovum seen EGGS AND LARVA OF TELEOSTEANS. 133 in this plane will be that represented in fig. 2. The two nuclei will be seen projected on to the focus-plane as at n'n’, fig. 2; while the under surface of that part of the blastodisc, which is nearer to the observer, will be projected on to the focus-plane as a curved line, apparently dividing the blastodisc into two portions, one internal and one external, each containing one of the nuclei. In Pl. III. fig. 8, the blastodisc is seen thus apparently divided before the first division has taken place, but the dividing line is nothing but the under surface of the near part of the blastoderm projected on to the focus-plane. A com- parison between the diagram in fig. 2 and figs. 4 and 12, pl. iii. of Horrmann’s Fig. 1. Fig. 2, memoir, will show how completely his views are explained by my supposition. At the same time all his figures cannot be so explained. Figs. 2 and 4 of his pl. v. are in direct opposition to my results. It is a remarkable fact that, as will be evident from a reference to Pl. IV. fig. 3, the nuclei of the two-cell stage are not at first in the thickest part of their respective cells. The centre of aggregation of the protoplasm lies nearer the edge of the blastoderm than the nucleus, and it would seem as if the protoplasm were active in the division and the nucleus passive, a hypothesis quite contrary to current conceptions. I have been unable to find any evidence of the exist- ence of periblast up to the eight-cell stage. Pl. IV. fig. 5, shows an optical section of the four-cell stage, in which it is evident there is no separate sub- blastodermic layer. 134 MR J. T. CUNNINGHAM ON THE | Spawning Periods of some of the Fishes of the Firth of Forth. Oct <= | Nov. <= | Dee Clupea harengus, L., Osmerus eperlanus, Lac., Gadus morrhua, L., : x x G. eglefinus, Z., . ; : x X G. merlangus, L., . : ; xX Xx Pleuronectes platessa, L., : x X Pl. flesus, Z., : : 5 x X Pl limanda: 2, : : x X Pl. cynoglossus, L., ; : x X Pl. microcephalus, L., Trigla gurnardus, Z., : : x Xx Zoarces viviparus, L., . . |X X|X X Spinachia vulgaris, 4 4 x xX Callionymus lyra, . 5 : xX X xX X x X x x x x BIBLIOGRAPHY. The following list contains the titles of the memoirs and books used in the preparation of the preceding paper. It comprises all the works I have been able to discover which give speciegraphical details concerning ova or larvee:— 1855. Cart SunpevaLtt. Om Fisk Utveck, Svensk. Vetensk. Akad. Hand., vol. i., New Series. 1867. A. W. Matm. Pleuronektidermas Utveck, Jdid., vol. vii. 1877. A. Acassiz. Young Stages of Osseous Fishes—I. Devel. of Tail, Proc. Amer. Acad., vol. xiii. 1878. A. Acassiz. Young Stages, &c.—II. Devel. of Flounders, Jb7d., vol. xiv. 1882. A. Acassiz, Young Stages, &c.—III., Lbid., vol. xvii. 1884. A, Acassiz and C. O. Wuirman. Devel. of some Pelagic Fish Eggs, Prelim. Notice, Jd:d., vol. xx. 1885, A. Acassiz and C. O. Wu1tman. Devel. of Osseous Fishes—I. Pelagic Stages of Young Fishes, Mem. Mus. Comp. Zool. Harv., vol. xiv., No. 1, pt. i. 1881. Joun A. Ryper. Devel. of Silver Gar (Belone longirostris), Bull. U. S. Fish. Com., vol. i. 1881. Joun A. Ryppr. Dev. of Spanish Mackerel (Cybiwm maculatum), Ibid., vol. i. 1881. Joun A. Ryper. Devel. of Lophobranchiates (Hippocampus antiquarum), Ibid., vol. i. 1882. Joun A. Ryper. Obs. on Embryo Fishes (Alosa supidissima, §c.), Ibid., vol. ii. 1883. Joun A. Ryper. Devel. of Amiurus albidus, Ibid., vol. iii. 1884. Joun A. Ryper. Embryography of Osseous Fishes (Cod), Rep. U. S. Fish. Com. for 1882. 1878. C. Kuprrer. Entwickl. des Ostsee-herings, Aus, dem Jaresb. der Com. zur Unt. der Deutschen Meere, Berlin. 1884, C. Kuprrer. Die Gastrulation au den meroblast. Eiern der Wirbelthiere, &c., Teleostei., Arch. f. Anat. und Phys., His and Du Bois Reymond. 1881. C. K. Horrmann. Ontogenie der Knochenfische, Verh. Konink Akad. van Wetens, Amsterdam, Deel xxi. 1883. V. Hensen. Vorkommen und Menge der Eier einiger Ostseefische, 4" Bericht der Com. Unt. Deutschen Meere, II. Abt. years 77-81. 1884, Geo. Brook. Prelim. Acct. of Devel. of Lesser Weever-fish, Trachinus vipera, Linn. Soc. Jour., vol. xviii. 1885, Gro. Brook. Devel. of Motella mustela, Linn. Soc. Jour., vol. xviii. 1885. J. T. Cunnrycnam. Significance of Kupffer’s Vesicle, &c., Quart. Jour. Micr. Set. 1885. J. T. Cunnrncuam, Relations of Yolk to Gastrula in Teleosteans, Jdid, EGGS AND LARVA OF TELEOSTEANS. 135 1885, Mrecz v. Kowatewsx1, Furch. und Keimblatteranlage der Teleostier, S. B. der physik. medicin. Societat, Erlangen, 15th Dee. 1886, Mincz vy. Kowatewsxi. Ueber die ersten Entwickl. processe der Knochenfische, Z. jf. wiss. Zool., Bd. xliti. 1886. Mrscz v. Kowatewskr. Gastrulation und Allantois bei den Teleostiern, S. B. d. phys. med. Societat, Erlangen, June 7. 1880, Carto Emery. Fauna and Flora des Golfes von Neapel.—II”, Monographie, Fierasfer. 1880. A. C. Ginruer. Art, “Ichthyology,” Hncy. Britannica, 1880. A. C. Giéxtuer. Introduction to Study of Fishes, Edinburgh, 1880-84, Francis Day, British Fishes, London and Edinburgh. Fig. Fig. Fig. Fig. Fig. Fig. Fig. e Fig. DESCRIPTION OF PLATES. Puate I. . 1, Egg of herring towards the close of period of simple segmentation, 11 hours after fertilisation. Aug. 26, 1884. Zeiss A, Oc 3. 2. Anterior end of herring embryo, 6 days after fertilisation, 1 day before hatching. Sept. 1, 1884. Zeiss A, Oc 3. . 3. Herring larva nearly 24 hours after hatching. Aug. 22, 1884. Zeiss A, Oc 2. . 4. Outline of alevin of Salmo levenensis, 3 days after hatching. Magnified about 4 times. . 5. Ovum of Osmerus eperlanus, Lacép, 254 hours after fertilisation. May 7, 1886. Mag. 33 times, . 6. Ovum of Os. eperlanus: optical section through suspensory membrane, internal zona, and micropyle, Zeiss CC, Oc 2. Puate II. . 1. Embryo of Plewronectes platessa, Linn., in ovo, 23 days 5 hours after fertilisation. Mag. 33 times. Feb. 26, 1886. 2. Sculpturing of surface of vitelline membrane of ovum of Pl. platessa, L. Mag. 50 times, . 3. Larva of Pl. platessa, taken artificially from ovum on point of hatching. 27 days after fertili- sation. Mag. 33 times. .4, Ovum of Pi. flesus, 2 days 24 hours after fertilisation. Mag. 33 times. The development of this egg was retarded: end of the period of simple segmentation. . 5. Pl. flesus, 22 hours after fertilisation, stage just before appearance of segmentation cavity. Mag. 33 times. March 31, 1886. Pi. flesus, 2 days 24 hours, First appearance of segmentation cavity. Mag. 33 times. . Pl. flesus, newly hatched. 7 days. April6, 1886. Pigment black, anus open. Mag. 33 times. . Ovum of Pleuronectes limanda, L., 20} hours after fertilisation. Stage preceding formation of segmentation cavity. May 22, 1886. Mag. 33 times. 6. . 7. Pl. flesus, 2 days 22 hours. Mag. 33 times. 8 9 . 10. Blastoderm of unfertilised ovum of P/. imanda, showing expulsion of polar globule. g. 11, Spermatozoon of Pl. limanda. Zeiss DD, Oc 4. Puate IIL. 1. Newly shed unimpregnated ovum of Pl. limanda, optical section showing micropyle and rela- tions of protoplasmic layer. Zeiss A, Oc 3, Abbé’s camera. Mag. 70 times. 2. Pl. limanda, 4 hour after fertilisation. Zeiss A, Oc 3, camera. Mag. 70 times. 3. Pl. limanda, 14 hours after fertilisation. Mag. 70 times. 4, Pl. limanda, 3 hours after fertilisation, Mag. 70 times. 5. Pl. limanda, little more than 3 hours: process of first division. Mag. 70 times. 6. Pl. limanda, larva newly hatched. May 28, 1886. Mag. 33 times, 6a. Notochord of same. Zeiss CC, Oc 3. 7. Blastodise of Pl. cynoglossus, unfertilised 2 hours after shedding, shows what may be the second polar body. Mag. 70 times. Fig. Fig. Fig. Fig. Fig. sills Best oo: . 10. Pl. cynoglossus, 1 day 234 hours. Mag. 33 times. . 11. Pl. cynoglossus, 2 days 4 hours. Mag. 33 times. . 12. Pl. cynoglossus, 2 days 19 hours. Mag. 33 times. A Is Ys ss 4, . 4a. Notochord of Pl. cynoglossus. : OF 50; ann ae a) J 2. 3. 4, MR J. T. CUNNINGHAM ON THE EGGS AND LARVA, ETC. . Ovum of Pl. cynoglossus, 2 hours after fertilisation. Mag. 33 times. . Pl. cynoglossus, unfertilised, 6 hours after shedding. Mag. 33 times. Puatst IV. . Blastoderm of Pl. cynoglossus in process of first division, 34 hours after fertilisation. Zeiss CC, Oc 3. . Pl. cynoglossus, 4 hour after fertilisation, ovum treated with acetic acid and methyl green, and examined entire in glycerine ; optical section of blastodise ; shows male and female pronuclei. Zeiss CC, Oc 3, camera. . Blastoderm of P/. eynoglossus, immediately after first division: shows nucleus in each of the two cells. Acetic acid and methyl green. Zeiss Cc, Ce 2, without camera. . Pl. cynoglossus, 4-cell stage, 4 hours. Mag. 33 times, . Optical section of 4-cell blastoderm, acetic acid and methyl green: shows absence of periblast beneath blastoderm. Zeiss CC, Oc 2. . Pl. cynoglossus 8-cell stage dividing ; acetic acid only without glycerine, 6 hours after fertilisation. Mag. 33 times. Pl. cynoglossus, blastoderm commencing to spread, 24 hours. Mag. 33 times. Pl. cynoglossus, segmentation cavity, 1 day 5 hours. Mag. 33 times. Pt. cynoglossus, 1 day 8 hours. Mag. 33 times. Puate V. Pl. cynoglossus, 2 days 19 hours. Mag. 33 times. Pl. cynoglossus, 3 days 14 hour. Mag. 33 times. Pl. cynoglossus, 4 days. Mag. 33 times. Newly hatched larva of Pl. cynoglossus, 5 days. Mag. 33 times. Condition of rectum, 7, and coalesced ends of segmental ducts, s.d., in newly-hatched larva. Zeiss CC, Oc 2. Condition of same parts, 30 hours after hatching. Larva of Pl. cynoglossus, 2 days after hatching. Puate VI. Gadus eglefinus, L., newly hatched. Mag, 33 times. Adhesive ovum from shore near station, probably Cottus scorpint. Mag. 33 times. Adhesive ovum attached to Hydrallmannia falcata, perhaps Liparis Montagui. Mag. 33 times. Young fish, 2 days after hatching. Hatched from adhesive ova taken in trawl April 29, 1884. Liparis Montagui. Ovum of Cyclopterus lumpus. Mag. 33 times. Prats VII. . Newly hatched larva of Cyclopterus lumpus, ventral surface, from specimen preserved in spirit. Mag. 33 times. Pelagic ovum taken 10 miles F.S.E. of May Island, March 23, 1886. Mag. 33 times. Pelagic ovum taken in Firth of Forth, off Gullane Ness, May 27, 1886. Mag. 33 times. Larva newly hatched from ovum shown in previous figure. Mag. 33 times. 4a. Notochord of same. Zeiss CC, Oc 3. 5. Pelagic ovum taken off Gullane Ness, May 27, 1886. Mag. 33 times. Fig. 6. Larva newly hatched from same. Mag. 33 times. Fig. 7. Teleostean ova from Gulf of Guinea, Mag. 18 times. Species unknown. Roy. sec, Edin’ . VOL. XXX. PL. 4 FIGS. 1-3, CLUPEA HARENGUS, L. FIG. 4, SALMO LEVENENSIS. FIGS. 5, 6, OSMERUS EPERLANUS, LACEP. Th Lagan Cumming amp. ) Soc. Kdin* VOL. Sek. PLT: cm. The numbers in their Table (p. 456), Column V., are Green, P : ; : 4 . 8°41 d= pis ’ : : ; : 8:93 Yellow-green, . , : : : 9°64 so that the agreement is pretty good. I would remark in passing that the diagram does not recognise a yellow-green of this order; but the appearance of such may perhaps be explained by contrast. § 13. The series of colours complementary to those of Table III. are found by subtraction of the numbers there given from those representative of white, viz., 3°97, 6°52, 6°46, respectively. The resulting numbers are exhibited in Table IV., in which the first entry for zero retardation corresponds to the full white. * “On the Electrical Resistance of Thin Liquid Films, with a Revision of Newton’s Table of — Colours,” Phil. Trans., 1881. + In comparing with Table III., it should be remembered that the numbers there given under the head of V=0 are relative only, the true values being infinitely small. VOL. XXXIII. PART I. Z 170 LORD RAYLEIGH ON THE COLOURS OF THIN PLATES. TABLE IV.—Second (Complementary) Series. V. (24). (44). (68). Vv. (24), (44). (68). 0 3:97 6-52 6-46 5200 1-28 1:80 5-04 10065 15 06 59 5300 ‘91 2-29 451 1300 22 1-45 3°67 5400 64 2-74 3°78 1500 96 3:32 5-64 5600 46 3:77 2:31 1604 1:46 4-29 6-19 5800 77 451 1-43 1688 1:93 5-03 6-28 6000 1:44 475 153 1755 2-30 5-51 6-29 6200 2:23 4-41 2-49 1846 2°77 5-99 5°77 6400 2:89 3°70 3°72 1951 3-22 6-26 4-83 6600 3-22 2:96 461 2013 3-48 6-26 4-21 6700 3:23 2-65 4-75 2154 3:84 5:85 2°65 6800 3:14 2-40 4-71 2328 3:88 4-70 1-02 6900 2:97 2-26 447 2630 2:98 2:08 59 7000 2°72 2-20 £05 2927 1:38 ‘BT 3-09 7100 9-49 2-25 3:55 3100 68 75 4:75 7200 211 2°39 3-05 3300 - 19 1:84 587 7400 1°52 2°83 2-29 3400 16 2:48 5:88 7600 114 3:36 1:98 3500 23 3:35 5°53 7800 1:04 3°76 2-44 3600 47 4-12 4:87 8000 1-25 3:96 3-22 3800 1:29 5-26 3-04 8200 1°63 3:90 4-01 4000 2:30 5-59 1:38 8400 2-08 3-67 4-18 4200 3:19 5-04 "75 8600 9-45 3:35 3-94 4400 3°68 3:84 1:44 8800 2-65 3-05 3:48 4600 3°62 2:56 3-03 9000 2-63 2:87 2°77 4800 3-06 161 4-60 9200 2-44 2-79 2-49 4900 2-64 1:39 515 9400 215 2:85 2-62 5000 2:18 1:35 5:43 The curve representative of this series of colours on NEwTon’s diagram is given by the dotted line in the Plate, so far. as the tabulated numbers permit. It starts from the point White, and passes rapidly through a whitish-yellow to a very dark red and purple at V=1006'5. This part of the curve can not be drawn from the tabulated data,—a defect of no great consequence, for the quantity of light being so insignificant, its quality is of little interest. From V =1300 onwards the curve is pretty well determined. It will be seen that the two series of colours are of pretty much the same general character. The green at 5800 in the second series compares favour- ably with the greens of the third and fourth orders in the first series. G8 en 80= 72 \ \ 16 fa \, / NN / Violet nN \ 66 . 5r00 5300" 5600 \ 6000 46 Green 46: AB ; Fa Ad. Archibald 4 Peck Engravers Edin foubz i ) VII.—On the Electrical Properties of Hydrogenised Palladium. By Carcit G. Knort, D.Sc. (Edin.), F.R.S.E., Professor of Physics, Imperial University, Tokayo, Japan. (Plate XI.) (Despatched to Royal Society of Edinburgh, May 25, 1886. Read 19th July 1886.) In the following paper I desire to place on record the results of certain | experiments which have lately engaged my attention. The facts established | are, so far as I am aware, novel and in themselves interesting. Many of the physical properties of hydrogenised palladium or hydrogenium have been carefully studied by GRAHAM, DEwar, and others ; but no one seems | to have called attention to its thermoelectric peculiarities, or to have made a | special study of its electrical resistance. These two inquiries form the subject of this paper. Throughout I shall use, for brevity’s sake, the name Hydrogenium, | which was applied by Granam to the fully-saturated form. Here, however, it | is applied generally to any alloy of the two substances, without any regard to a | possible chemical compound of definite molecular constitution. The paper | naturally divides itself into two sections—the first part relating to the electrical resistance, the second part to the thermoelectric properties. ELECTRICAL RESISTANCE OF HyYDROGENIUM. The steady increase of the resistance of hydrogenium with the charge of | hydrogen was noticed by DEwar ;* and further details were given by myself in a short paper published a few years ago.t There I obtained a resistance of | 1518 for fully-saturated hydrogenium, of which the originally pure palladium | wire had a resistance of unity. In the present inquiry I have easily obtained a | much greater increase of resistance—such as 1°634, 1°7775, and even as much | as 1°83. Whether this may be a result of impurities being present in the acid which was used as the electrolyte, I cannot say. It may be noted, however, that the palladium wire itself was obtained in Paris, and was guaranteed to be very pure indeed. The main purpose of the present investigation was to study the temperature | characteristics of the resistance of hydrogen-charged palladium. Throughout | each series of experiments the same palladium wire was used, the hydrogen * Trans. Roy. Soc. Edin., vol. xxvii. + Proc. Roy. Soc. Edin., 1882-83. VOL, XXXII. PART I. 2A 172 DR CARGILL G. KNOTT ON THE being added in small successive doses. Not till the maximum saturation was reached was the wire subjected to any excessive heating, such as is generally supposed to be necessary to drive the hydrogen out. The temperature was regulated by means of an oil-bath, into which the wire, firmly bound to the ends of stout copper rods, dipped along with the thermometer which measured the temperature. The heating was applied gently by means of a spirit-lamp. In the first series of experiments the temperature was raised gradually to about 300° C. ; and in these experiments the loss of hydrogen was beautifully shown in the manner by which the resistance began to decrease at a temperature of about 260° C. = £ D7 i a. VE SIAGF: Li) LESS Db = . = ky ; ———— Bs Ge A} ; Set z Zt. Z = Seay Se : fs, Wi See 7 oe Ife yf 87 WE rs st QE Ne bm va Wp iit; = is if, = ae : Z ns. Roy. Soc. Edin? homson del. ns. Roy. Soc. Edin’ Vol. XXXIIL, Pl. XVIE Qe epr mera 5 “ey iO: F. Huth, Lith* Edin* ( 247 ) XI.—The Reproductive Organs of Bdellostoma, and a Teleostean Ovum Jrom the West Coast of Africa. By J. T. CunnincHam, B.A. (Read 5th July 1886.) During a short visit I paid to Oxford in the month of June last I had the opportunity of examining, by the kind permission of Professor MosE.ey, a number of specimens of Bdellostoma Forsteri, which were some of a large number brought from the Cape by Mr Apam Sepewick of Trinity College, Cambridge. This examination showed what, from the close affinity of the two forms, was naturally to be expected, namely, that the structure of the repro- ductive system and the development of the reproductive elements in Bdel- lostoma were very closely similar to the structure and development of the corresponding parts in Myxine. A short time ago I described before the Society some ovarian eggs of Myxine, obtained at the beginning of the present year, which were approaching maturity. In these eggs there were slight pro- jections at the poles, and on the surface of the projecting parts a number of papille. The projections were caused by the growth of a number of threads from the vitelline membrane within the ovarian capsule, and the papillee were the separate elevations produced by the threads. In one of the specimens of Bdellostoma which I examined at Oxford there were a number of ovarian eggs in an exactly similar condition. These eggs of Bdellostoma are of course much larger than those of Myxine; the eggs of the latter, in the condition I refer to, were 2°1 cm., those of the former are 35 cm. No one has seen the perfectly ripe eggs of Bdellostoma after their escape from the ovary, but the specimens I have described prove conclusively that the eggs of this species when shed are provided with a number of polar threads, which are processes of the vitelline membrane, exactly as in Myxine. I have not yet made a microscopic examination of the reproductive organs in Bdellostoma, but from what I could see by ordinary dissection, it is evident that all the peculiarities which exist in the reproductive system in Myxine occur also in Bdellostoma. A number of specimens possessed sexual organs, in the anterior part of which were minute ova, while the posterior part was evidently testicular tissue; and in one or two other specimens the whole organ seemed to be testicular. The small quantity of testicular tissue in a given specimen was also noticeable, as in Myxine. I found no specimens which showed indications of having recently discharged their eggs. I have ascertained from Mr Sepewick that his specimens were collected in August and September, and this fact shows that the breeding period VOL. XXXIII. PART I. 2N 248 MR J. T. CUNNINGHAM ON THE REPRODUCTIVE ORGANS of Bdellostoma agrees with that of Myxine in falling within the coldest season of the year. Mywine glutinosa, in the North Sea, deposits its eggs in December, January, and February, and the two latter months agree in meteorological conditions with the months of August and September in the latitude of Cape Town. The egg of Bdellostoma at the stage under consideration has a thicker and stronger vitelline membrane than the egg of Myxine. I found it impossible to strip off from preserved specimens of the latter the connective tissue and follicular epithelium without rupturing the vitelline membrane. In the eggs of Bdellostoma this could be accomplished with ease. The membrane, when exposed, was seen to be yellowish-brown in colour, and translucent. Round the micropylar end of the capsule formed by the membrane is seen a distinct thin line, forming a complete ring, and it is evident that the micropylar end forms an operculum which separates from the rest of the capsule along this le, STEENsTRUP has figured a detached operculum in the figure he gives of the ova of Myxine, but in the latter form I have not yet detected indications of the structure. There can be no doubt, from the appear- ance seen in the Bdellostoma ovum, that the escape of the embryo in the Myxinoids is effected by the removal of an operculum specially adapted for that purpose. The Teleostean ova I have next to describe resemble in the character of the vitelline membrane the ova of the Myxinoids. Each ovum is spherical in shape, 1°5 to 16 mm. in diameter, and about one pole of the sphere is pro- vided with a number of long thin flexible filaments springing from the vitelline membrane. Each filament commences at the attached base with a conical papilla, which is thicker than the filament itself. By the interlacing of the filaments a large number, many thousands, of eggs are connected together to form a cylindrical mass about an inch wide, and a foot or more in length. The felted filaments form a rope-like core to the cylinder, the eggs forming an external layer. Besides the long filaments, each egg shows a similar number of short filaments springing from the opposite pole. These are very slender, and only from 2 mm. to 1°5 cm. in length. In other respects they resemble the long filaments, of which they are evidently rudimentary repre- sentatives. They seem to have no function, being too small to afford any assistance in the process of attachment. It is probable, though I have not been yet able to demonstrate the fact, that the micropyle is situated in the centre of the region whence the long filaments arise. If this were so, the rela- tions of the filaments and vitelline membrane in this Teleostean egg would be exactly similar to those which obtain in the ovum of the Myxinoids. And whatever be the position of the micropyle, it is interesting to note that the occurrence of a group of filamentous processes of the vitelline membrane at each OF BDELLOSTOMA AND A TELEOSTEAN OVUM. 249 of the two opposite poles of the ovum is not peculiar to the Myxinoids. It is as certain as an inference from the unfertilised ovum can be, that the segmentation of the egg of the Myxinoids is meroblastic, as in Teleosteans, and thus in two points the Myxinoid ovum agrees with the Teleostean, and. differs from that of Petromyzon, while in respect of the mass of the yolk the Myxinoids agree more with Elasmobranchs. I have not succeeded in identifying the species of fish to which belong the eggs above described. The eggs of several species are known to be provided with filamentous processes. In the Scombresocide the filaments are equal in length to the diameter of the ovum, and are uniformly distributed over the surface of the membrane. The filaments in this family were first described by Professor HAECKEL.* JoHN A. RYDER gives a very clear and complete account of them in the Bulletin of the U. S. Fish Commission, 1881, vol. i., as studied in Belone longirostris. In Chirostoma, one of the Atherinide, Ryprr found there were only four filaments attached at one pole of the egg close together. In this latter case the filaments were during development closely wound round the vitellme membrane in one equator of the sphere, so that the method of their formation differs from that of the Myxinoid filaments, which are perpendicular to the surface of the membrane throughout their growth in the follicle. Filamentous processes of the vitelline membrane occur also in the family | Pomacentride ; they have been described by HorrMann in Heliastes chromis of the Mediterranean (see Konink. Akad. d. Vetensk. Amst., vol. xxi.) Here | they occur at one end only of the ellipsoidal ovum. They occur also in Gobius and Bleunius, but in neither of these cases are two sets of processes present, situated at opposite poles of the ovum. It is thus impossible to say ‘whether the ova described in this paper belong to a fish of the family | Scombresocidee among the Physostomi, of the family Pomacentride, or coral- ‘fishes among the Pharyngognathi, of the family Gobiide, Blennide or | Atherinide, or to a species of some other family whose eggs are alto- _ gether unknown. The ova were obtained on two occasions, each time a | single cylindrical “rope,” by Mr Joun Rarrray, F.R.S.E., in the Gulf of 'Guinea. Mr Rarrray was on board a steamer called the “ Buccaneer” last | winter, in the capacity of naturalist, having been invited to accompany Mr J. |Y. Bucuanan, who was carrying out some hydrographical investigations off the coast of Africa. The eggs were obtained in the following manner :—A small / conical buoy was attached at the end of a rope, and along the rope were fastened two or three muslin tow-nets. The whole was then thrown overboard |in such a way that the mouth of the tow-net faced whatever current was flow- ing. The eggs were found entangled on the line when the apparatus was * Muuuer’s Archiv, 1850. 250 THE REPRODUCTIVE ORGANS OF BDELLOSTOMA, ETC. recovered. On the first occasion, March 12th of the current year, the position was lat. 1° 17’ N., long. 13° 566 W. The depth at which the ova were caught by the line was 30 fathoms. The total depth of the ocean at the spot was 2725 fathoms. The other mass of eggs was taken in a similar way, not far off the locality just defined. Thus these eggs were in a pelagic condition, suspended in the water, and freely obeying the ocean current. Mr Rarrray states they were very transparent. ‘ P.S.—Since the above was written I have found that the meroblastice nature of the ovum of Myxine has been actually proved. Fertilised eggs, 1 in which the blastoderm had already begun to spread over the yolk, were examined and described by W. Mitver several years ago (Jenaische Zeitschrift, Bd. IX.). These eggs were from the collection of the Gdteborg Museum, and were obtained at Lysekil in Bohuslain in 1854. W. MULLER, however, did not give a correet account of the development of the vitellme membrane and polar threads. 26 JUL. 1887 “ Ann. : y ene We iee, ay 5 ns Rog The Transactions of the Roya Sociery or EpinsurGs will in future be Sold at the following reduced Prices :— ie - ri Price t ri rice t ihe Vol. area Fallow Vol. ae a vase | ; L IL Tt, | Out of Print. XXIL Part 1.| 42 50 “| £y-a) Tye. 180) .9 Ge |. 20) voae . Pentidel< Ot ee 07 & |. oO ute 09 0 ” Part 3.| 1 5 0 eee wl | Oe cs, 0 9! 6 VexxIM Part 1.|> Oabhe ol 6 Vit. | 0" fa 0 015 0 Pant 2.1 + (one O° sata Vii | Buy <0 014 0 8 Bart |e oh eee 1-10 Om 2 ee) eo 017 0: J) SeRnve Peed ao Tee 1 0mm eee aa 016 0 . Part2| 1.8.0 7) on be t-te 012 0 J a-Pant 9.) Uieton @ 15 0 > cae ee 012 0 XXV. Pati!" e948 9 O13. ey XT | 80 015 0 Paes | 1 Sea ree aa xiv, | & 5.10 110 | XXViPatL| 1 Qo | COUR SG ae ee 1 6:0 pata | or ee 10,0 XVI. » Part 3. 016 0 "0 12 6 ee Part Lit Oe oe Sa ce * Pat 4, | 6-12 0 oem Danae |". 0°18 20 014 0 |) XXVil,Bart 1) 0.16 0 9! J6 72) .0mN Bas,” ||. 00) 10. 10 01 FB Part | 68° 0 4 te Peat & |, 0. 5 68 6 4 0 Beta. | Ale See 016 0 Beeb: | 0.7 0 0:5 6 eet a |e 0a 016 0 KVIL »| Out Print, WX VET Part dy 1d ee ‘ae: AVGE ee avo 1 1) . peggel | 1 y Oaaee an XIX. Y Pee. | Oe 013 6 Pat 1} Ae ae Pd 0. Sat, Pach | 11 Oo ete Part we |. Ode 10 015 0 Part 2. | 016 6 012 0 XX. XXX, Patt 1.| 1 12.0 i. 68 Patt} ee ee oh ie » Part?) $016 0 | 042 a Part 2. 010 O 0-7 6 oy artis 0 5 0 0 4.0 Parts; | 010 0 O72 6 (Part 4, | Oneal 058 Panta 4| Sono 0 0.706 I me a “a XXL XXXIL Port ly] 1 0 0 016 0 ment OD eae -"S Paré 2). 0 1800 013 6 Part 2.-| 0.10 0 07 6 2S set's.) aT 117 6 Bark | See Fe 6 05 3 te Part 4k © 400 0 4 0mm Boa ied 013: aft SER Baar: ait 016 0 Ja art 4. ee ; PRINTED BY NEILL AND COMPANY, EDINBURGH. wo 1 1888 TRANSACTIONS OF THE ‘OYAL SOCIETY OF EDINBURGH. VOL. XXXIII. PART+II.—FOR THE SESSION 1886-87. XX. XXI. XXII. XXIII. XXIV. XXV. CONTENTS. . On the Foundations of the Kinetic Theory of Gases. Il. By Professor Tarr, . Tables for Facilitating the Computation of Differential Refraction in Position Angle and Distance. By the Hon, Lorp M‘Laren, . On a Class of Alternating Functions. By Tuomas Muir, LL.D., . Expansion of Functions in terms of Linear, Cylindric, Spherical, und Allied Functions. By P. Auexanpur, M.A. Communicated by Dr T. Murr, . On Cases of Instability in Open Structures. By E. Sane, LL.D., . On the Fossil Flora of the Radstock Series of the Somerset und Bristol Coal Field (Upper Coul Measures). Parts I., 11. By Roperr Kinston, F.R.S.E., F.G.8, (Plates XVIII-XXVIIL.), . A Diatomaceous Deposit from North Tolsta, Lewis. By Joun Rarrray, M.A., B.Se., of H.M. “Challenger” Commission, Edinburgh. (Plate XXIX.), . On the Minute Structure of the Eye in certain Cymothoide. By Franx E. Brpparp, M.A., F.R.S.E., F.Z.S., Prosector to the Zoological Society, and Lecturer on Biology at Guy’s Hospital. (Plate XXX.), Report on the Pennatulida dredged by H.M.LS. “ Porcupine.’ By A. Mitnes Marsuaut, M.D., D.Sc., M.A., F.R.S., Beyer Professor of Zoology in the Owens College ; and G. H. Fowrer, B.A., Ph.D., Berkeley Fellow of the Owens College, Manchester. Communicated by Joun Murray, Esq. (Plates XXXI., XXXIT.), On the Determination of the Curve, on one of the coordinate planes, which forns the Outer Limit of the Positions of the point of contact of an Ellipsoid which always touches the three planes of reference. By G. Puarr, Docteur és- sciences. Communicated by Professor Tarr, On the Partition of Energy between the Translatory and Rotational Motions of a Set of Non-Homogeneous Elastic Spheres. By Professor W. Burwnstpz, A Contribution to owr Knowledge of the Physical Properties of Methyl-Alcohol. By W. Dirrmar, F.R.SS. Lond. & Edin., and CHarues A. Fawsir. (Plate XX XIII), : . , On the Thermal Conductivity of Iron, Copper, and German Silver. By A. Cricuton Mircustn, Esq. Communicated, with an Introduction, by Pro- fessor Tait. (Plates XXXIV., XXXV.), Critical Experiments on the Chloroplatinate Method for the Determination of Potassium, Rubidium, and Ammonium; and a Redetermination of the Atomic Weight of Platinum. By W. Dittmar and Jonn M‘Artuur, [LIsswed April 13, 1888.1 335 419 443 453 465 501 509 535 561 XII.—On the Foundations of the Kinetic Theory of Gases. II. By Professor Tarr. (Read December 6, 1886, and January 7, 1887. Revised April 4, 1887.) INDEX TO CONTENTS. PAGE PAGE INTRODUCTORY AND PRELIMINARY, . . 251 Part XI. Pressure in a Mixture of Two Ne Sets of Spheres, § 35, . . 258 Part X. On the Definite Integrals, sft ye" ,, XIL Viscosity, §§ 36, 37, : _ 959 ae Warnes ,, XIII. Thermal Conductivity, sg 38-44, 261 and 4 oe §§ 33, 34, . 256 ,, XIV. Diffusion, §§45-56, . . 266 er ris AppENDIX. ‘Table of Quadratures, . 207 (Zrratum in Part I., anté, p. 65. For 1676, read 1678, as the date of Hooxr’s Pamphlet. | In the present communication I have applied the results of my first paper to the question of the transference of momentum, of energy, and of matter, in a gas or gaseous mixture; still, however, on the hypothesis of hard spherical particles, exerting no mutual forces except those of impact. The conclusions of §§ 23, 24 of that paper form the indispensable preliminary to the majority of the following investigations. For, except in extreme cases, in which the causes tending to disturb the “special” state are at least nearly as rapid and persistent in their action as is the process of recovery, we are entitled to assume, from the result of § 24, that in every part of a gas or gaseous mixture a local special state is maintained. And it is to be observed that this may be accompanied by a common translatory motion of the particles (or of each separate class of particles) in that region ; a motion which, at each instant, may vary continuously in rate and direction from region to region; and which, in any one region, may vary continuously with time. This is a sort of generalisation of the special state, and all that follows is based on the assumption that such is the most general kind of motion which the parts of the system can have, at least in any of the questions here treated. Of course this translational speed is not the same for all particles in any small part of the system. It is merely an average, which is maintained in the same roughly approximate manner as is the VOL. XXXIII. PART II. 20 252 PROFESSOR TAIT ON THE “special state,” and can like it be assumed to hold with sufficient accuracy to be made the basis of calculation. The mere fact that a “steady” state, say of diffusion, can be realized experimentally is a sufficient warrant for this assump- tion; and there seems to be no reason for supposing that the irregularities of distribution of the translatory velocity among the particles of a group should be more serious for the higher than for the lower speeds, or vice versd. For each particle is sometimes a quick, sometimes a slow, moving one :—and exchanges these states many thousand times per second. All that is really required by considerations of this kind is allowed for by our way of looking at the mean free paths for different speeds. I may take this opportunity of answering an objection which has been raised in correspondence by Professor Newcoms, and by Messrs Watson and Bursury, to a passage in § 3 of the First Part of this paper.* The words objected to are put in Italics :— “But the argument above shows, further, that this density must be ex- pressible in the form fa) AYO, whatever rectangular axes be chosen, passing through the origin.” The statement itself is not objected to, but it is alleged that it does not follow from the premises assumed. This part of my paper was introduced when I revised it for press, some months after it was read; the date of revision, not of reading, being put at the head. It was written mainly for the purpose of stringing together what had been a set of detached fragments, and was in consequence not so fully detailed as they were. I made some general statements as to the complete verification of these preliminary propositions which was to be obtained from the more complex investigations to which they led; thus showing that I attached com- paratively little weight to such introductory matters. If necessary, a detailed proof can be given on the lines of § 21 of the paper. The “argument” in question, however, may be given as below. It is really involved in the italicised words of the following passage of § 1:—‘“in place of the hopeless question of the behaviour of innumerable absolutely isolated individuals, the comparatively simple statistical question of the average behaviour of the various groups of a@ community.” Suppose two ideal planes, parallel to =0, to move with common speed, a, through the gas. The portion of gas between them will consist of two qui distinct classes of particles :—the greatly more numerous class being m * In the Phil. Mag., for April 1887, the same objection is raised by Prof. Borrzmann ; who has appended it to the English translation of his paper presently to be referred to. But he goes farthe than the other objectors, and accuses me of reasoning in a circle. ‘ FOUNDATIONS OF THE KINETIC THEORY OF GASES. 253 fleeting occupants, the minority being (relatively) as it were permanent lodgers, These are those whose speed perpendicular to the planes is very nearly that of the planes themselves. The cndividuals of each class are perpetually changing, . those of the majority with extraordinary rapidity compared with those of the minority ; but each class, as such, forms a definite “ group of the community.” The method of averages obviously applies to each of these classes separately ; and it shows that the minority will behave, so far as y and z motions are concerned, as if they alone had been enclosed between two material planes, and as if their lines of centres at impact were always parallel to these. The instant that this ceases to be true of any one of them, that one ceases to belong to the group ;—and another takes its place. Their behaviour under these circumstances (though not their number) must obviously be independent of the speed of the planes. Hence the law of distribution of components in the velocity space-diagram must be of the form S(#) FY); and symmetry at once gives the result above. (Inserted March 5th, 1887.) Another objection, but of a diametrically opposite character, raised by Mr Bursury* and supported by Professor BoLtzMANN,t is to the effect that in my first paper I have unduly multiplied the number of preliminary assumptions necessary for the proof of MAxwELL’s Theorem concerning the distribution of energy in a mzzxture of two gases. In jform, perhaps, I may inadvertently have done so, but certainly not in substance. | The assumptions which (in addition to that made at the commencement of _ the paper (§ 5) for provision against simultaneous impacts of three or more particles, which was introduced expressly for the purpose of making the results applicable to real gases, not merely to imaginary hard spheres,) I found it necessary to make, are (§ 18) as follows; briefly stated. (A) That the particles of the two systems are thoroughly mixed. * The Foundations of the Kinetic Theory of Gases. Phil. Mag. 1886, I, p. 481. + Uber die zum theoretischen Beweise des Avogadro’-schen Gesetzes erforderlichen Voraussetzungen. Sitzb. der kais. Akad, XCIV, 1886, Oct. 7. In this article Prof. Bourzmann states that I have nowhere expressly pointed out that my results are applicable only to the case of hard spheres. I might plead that the article he refers to is a brief Abstract only of my paper; but it contains the following state- ments, which are surely explicit enough as to the object I had in view :— “This is specially the case with his [Maxwstt’s| investigation of the law of ultimate partition of energy in a mixture of smooth spherical particles of two different kinds.” “Tt has since been extended by Botrzmann and others to cases. in which the particles are no longer supposed to be hard smooth spheres.” “Hence it is desirable that Maxwet’s proof of his fundamental Theorem should be critically examined.” Then I proceed to examine it, not Professor Boutzmann’s extension of it. In my paper itself this limitation is most expressly insisted on. 254 PROFESSOR TAIT ON THE (B) That the particles of each kind, separately, acquire and maintain the ‘special state.” (C) That there is free access for collision between each pair of particles, whether of the same or of different systems ; and that the number of particles of one kind is not overwhelmingly greater than that of the other. Of these, (A) and (B), though enunciated separately, are regarded as conse- quences of (C), which is thus my sole assumption for the proof of CLERK- MAXweELL’s Theorem. Professor BoLTzMANN states that the only necessary assumptions are :—that the particles of each kind be uniformly distributed in space, that they behave on the average alike in respect of all directions, and that (for any one particle ?) the duration of an impact is short compared with the interval between two impacts. His words are as follows :—“ Die einzigen Voraussetzungen sind, dass sowohl die Molekiile erster als auch die zweiter Gattung gleichformig im ganzen Raume vertheilt simd, sich durchschnittlich nach allen Richtungen gleich verhalten und dass die Dauer eines Zusam- menstosses kurz ist gegen die Zeit, welche zwischen zwei Zusammenstossen vergeht.” He farther states that neither system need have internal impacts ; and that Mr Bursoury is correct in maintaining that a system of particles, which are so small that they practically do not collide with one another, will ultimately be thrown into the “special” state by the presence of a sinyle particle with which they can collide. Assuming the usual data as to the number of particles in a cubic inch of air, and the number of collisions per particle per second, it is easy to show (by the help of Lapiace’s remarkable expression* for the value of A”0Q™/n™ when m and mare very large numbers) that somewhere about 40,000 years must elapse before it would be so much as even betting that Mr Bursury’s single particle (taken to have twice the diameter of a particle of air) had encountered, once at least, each of the 3.10” very minute particles in a single cubic inch. He has not stated what is the average number of collisions neces- sary for each of the minute particles, before it can be knocked into its destined phase of the special state; but it must be at least considerable. Hence, even were the proposition true, eons would be required to bring about the result. As a result, it would be very interesting; but it would certainly be of no importance to the kinetic theory of gases in its practical applications. I think it will be allowed that Professor BonrzMaNnn’s assumptions, which (it is easy to see) practically beg the whole question, are themselves inadmissible * Théorie Analytique des Probabilités. Livre I, chap. ii, 4. [In using this formula, we must make sure that the ratio m/n is sufficiently large to justify the approximation on which it is founded. It is found to be so in the present case. At my request Professor Cayiey has kindly investigated the correct formula for the case in which m and x are of the same order of large quantities. His paper will be found in Proc. R. S. £., April 4, 1887.] ; FOUNDATIONS OF THE KINETIC THEORY OF GASES. 255 except us consequences of the mutual impacts of the particles in each of the two systems separately. Professor BoLttzMANN himself, indirectly and without any justification (such as I have at least attempted to give), asswmes almost all that he objects to as redundant in my assumptions, with a good deal more besides. But he says nothing as to the relative numbers of the two kinds of particles. Thus I need not, as yet, take up the question of the validity of Professor BoLtzMANn’s method of investigation (though, as hinted in § 31 of my first paper, I intend eventually to do so); and this for the simple reason that, in the present case, I cannot admit his premises. Mr Bursury assumes the non-colliding particles to be in the “special state,” and proceeds to prove that the single additional particle will not disturb it. But, supposing for a moment this to be true, it does not prove that the solitary particle would (even after the lapse of ages) reduce any non-colliding system, with positions at any instant, speeds, and lines of motion, distributed absolutely at random (for here there cannot be so much as plausible grounds for the introduction of Professor BoLtzmann’s assumptions) to the “special state.” If it could do so, the perfect reversibility of the motions, practically limited in this case to the reversal of the motion of the single particle alone, shows that the single particle would (for untold ages) continue to throw a system of non-colliding particles further and further owt of the “ special” state; thus expressly contradicting the previous proposition. In this consequence of reversal we see the reason for postulating a very great number of particles of each kind. If Mr Bursury’s sole particle possessed the extraordinary powers attributed to it, it would (except under circumstances of the most exact adjust- ment) not only be capable of producing, but would produce, absolute confusion | among non-colliding particles already in the special state. Considering what ' is said above, I do not yet see any reason to doubt that the assumption of | collisions among the particles of each kind, separately, is quite as essential to a valid proof of Maxwe tu’s Theorem as is that of collisions between any two particles, one from each system. I have not yet seen any attempt to prove that two sets of particles, which have no internal collisions, will by their | mutual collisions tend to the state assumed by Professor Botrzmann. Nor can | I see any ground for dispensing with my farther assumption that the number | of particles of one kind must not be overwhelmingly greater than that of the ' other. A small minority of one kind must (on any admissible assumption) | have an average energy which will fluctuate, sometimes quickly sometimes very | slowly, within very wide and constantly varying limits. | De Morcan* made an extremely important remark, which is thoroughly | applicable to many investigations connected with the present question. It is to the effect that “no primary considerations connected with the subject of * Encyc. Metropolitana. Art. Theory of Probabilities. 256 PROFESSOR TAIT ON THE Probability can be, or ought to be, received if they depend upon the results of a complicated mathematical analysis.” To this may be added the obvious remark, that the purely mathematical part of an investigation, however elegant and powerful it may be, is of no value whatever in physics unless it be based upon admissible assumptions. In many of the investigations, connected with the present subject, alike by British and by foreign authors, the above remark of DE Morgan has certainly met with scant attention. | In my first paper I spoke of the errors in the treatment of this subject which have been introduced by the taking of means before the expressions were ripe for such a process. In the present paper I have endeavoured throughout to keep this danger in view ; and I hope that the results now to be given wili be found, even where they are most imperfect, at least more approximately accurate than those which have been obtained with the neglect of such precautions. The nature of CLERK-MAXWELL’s earlier investigations on the Kinetic theory, in which this precaution is often neglected, still gives them a peculiar value ; as it is at once obvious, from the forms of some of his results, that he must have thought them out before endeavouring to obtain them, or even to express them, by analysis. One most notable example of this is to be seen in his Lemma (Phil. Mag. 1860, II. p. 23) to the effect that ; ips ale d rails where U and 7 are functions of xz, not vanishing with 2, and varying but slightly between the limits —7 and 7 of x;—and where the signs in the integrand depend upon the character of m as an even or odd integer. This forms the starting point of his investigations in Diffusion and Conductivity. It is clear from the context why this curious proposition was introduced, but its accu- racy, and even its exact meaning, seem doubtful. In all the more important questions now to be treated, the mean free path of a particle plays a prominent part, and integrals involving the quantities ¢, or e+¢é, (as defined in §$9, 10, 28) occur throughout. We commence, therefore, with such a brief discussion of them as will serve to remove this merely numerical complication from the properly physical part of the reasoning. X. On the Definite Integrals 33. In the following investigations I employ, throughout, the definition of the mean free path for each speed as given in§ 11. Thus all my results FOUNDATIONS OF THE KINETIC THEORY OF GASES. 254 necessarily differ, at least slightly, from those obtained by any other inves- tigator. By § 11 we see at once that oO ao -h 2 pt "i wr he gy dy e@ ns? °_av2 © nie ; 0 2 AND ON 1 ae : 0 fi (v2 +,!/30?)dv, + ; g (vv, /3 +0,3/v)dv, é wf: tees NE sae dee, = ae Te suppose. The finding of C, is of course a matter of quadratures, as in the Appendia to the First Part of this paper, where the values calculated are, in this notation, C_, and C,; and Mr Crark has again kindly assisted me by computing the values of C,,C,,C;, which are those required when we are dealing with Viscosity and with Heat-Conduction in a single gas. The value of C, has also been found, with a view to the study of the general expression for C,. These will be given in an Appendix to the present paper. 34. When we come to deal with Diffusion, except in the special case of equality of density in the gases, this numerical part of the work becomes extremely serious, even when the assumption of a “‘ steady” state is permissible. As will be seen in § 28 of my first paper, we should have in general to deal with tables of double entry, for the expressions to be tabulated are of the form— Agt *4¢-" da Ek a BE Dy Sate Poe (o, ae (2a,2 +f Pai 7) =C,= il —1—"_ , suppose. nas" — ree For the second gas the corresponding quantity will be written as .€,. Here @=2 Jh/h, _ mh ()' Sie Nos p ? so that they are numerical quantities, of which the first depends on the relative masses of particles of the two gases, while the second involves, in addition, not only their relative size but also their relative number. It is this last condition which introduces the real difficulty of the question, for we have to express the and 258 PROFESSOR TAIT ON THE value of the integral as a function of < before we can proceed with the further details of the solution, and then the equation for Diffusion ceases to resemble that of Fourrer for Heat-Conduction. The difficulty, however, disappears entirely when we confine ourselves to the study of the “steady state” (and is likewise much diminished in the study of a variable state) in the special case when the mass of a particle is the same in each of the two gaseous systems, whether the diameters be equal or no. For, in that case, we have h,=/ and 2,=2, so that the factor 1/(1+2) can be taken outside the integral sign. Thus, instead of ,€,, we have only to calculate C,. of the previous section. XI. Pressure in a Mixture of Two Sets of Spheres. 35. Suppose there be 2, spheres of diameter s, and mass P,, and , with S, P., per cubic unit. Let s=(s,+ Jie p 0 06 . . . . . . (1 ) js ” Eon pr 0:45 . . . . . . (35 5 | where it is to be remarked that the product pd is independent of the tempera- | ture of the gas. The Conductivity, 4, is defined by the equation ' and thus its value is . ee b=./=, goes where 7, 4) are simultaneous values of 7 and h. | At 0° C. (4¢ 7=274) this is, for air, nearly 3.10~-° in thermal units on | the pound-foot-minute-Centigrade system :—7.e. about 1/28,000 of the con- | ductivity of iron, or 1/3600 of that of lead.t Of course, with our assumption * Pogg. Ann., cxv, 1862; Phil. Mag., 1862, I. } Trans. R. 8. £., 1878, p. 717. 266 PROFESSOR TAIT ON THE of hard spherical particles, we have not reckoned the part of the conducted energy which, in real gases, is due to rotation or to vibration of individual particles. XIV. Diffusion. 45. The complete treatment of this subject presents difficulties of a very formidable kind, several of which will be apparent even in the comparatively simple case which is treated below. We take the case of a uniform vertical tube, of unit area in section, connecting two vessels originally filled with different gases, or (better) mixtures of the same two gases in different proportions, both, however, maintained at the same temperature ; and we confine ourselves to the investigation of the motion when it can be treated as approximately steady. We neglect the effect of gravity (the denser gas or mixture being the lower), and we suppose the speeds of the group-motions to be very small in compari- son with the speed of mean square in either gas. [In some of the investigations which follow, there are (small) parts of the diffusion-tube in which one of the gases is in a hopeless minority as regards the other. Though one of the initial postulates (d of § 1) is violated, I have not thought it necessary to suppress the calculations which are liable to this objection ; for it is obvious that the condi- tions, under which alone it could arise, are unattainable in practice. | CieRK-MAxwELL’s Theorem (§ 15), taken in connection with our preliminary assumption, shows that at every part of the tube the number of spheres per cubic unit, and their average energy, are the same. Hence, if 2, ”,, be the numbers of the two kind of spheres, per cubic unit, at a section x of the tube N,+n,=n=constant,. . : : : j (1.) Also, if P,, P,, be the masses of the spheres in the two systems respectively, h, and h, the measures (§ 3) of their mean square speeds, we have P, /h, = Pp /to= (1, P|, +13 [hin=2pim, - = =) em where p is the constant pressure. Strictly speaking, the fact that there is a translational speed of each layer of particles must affect this expression, but only by terms of the first order of small quantities. 46. The number of particles of the P, kind which pass, on the whole, towards positive 2 through the section of the tube is (as in § 39) am / Ny 4) — Ty, fuels ) where a, is the (common) translational speed of the P,s, and 1/e, the mean FOUNDATIONS OF THE KINETIC THEORY OF GASES. 267 free path of a P, whose speed is v. We obtain this by remarking that, in the present problem, /, is regarded as constant, so that there is no term in vy’. Hence, if G, be the mass of the first gas on the negative side of the section, divided by the area of the section, we have d ' TGA wcivewtiers secula E> If G, be the corresponding mass of the second gas, we have (noting that, by (1), m,/+,/=90) dGy _ “aie —P, (Mtg +1'8,/3) - : . . : (4.) From the definitions of the quantities G,, G,, we have also dG, acs Pin, Ga, aah Py, | dx da? m a ( : ; , 3 (5.) sees HID was ’ dx poet ae os } 47. We have now to form the equations of motion for the layers of the two | gases contained in the section of the tube between w and «+6x. The increase of momentum of the P, layer is due to the difference of pressures, behind and | before, caused by P,s ; minus the resistance due to that portion of the impacts | of some of the P,s against P,s in the section itself, which depends upon the | relative speeds of the two systems, each as a whole. This is a small quantity | of the order the whole pressure on the surfaces of the particles multiplied by | the ratio of the speed of translation to that of mean square. The remaining | portion (relatively very great) of the impacts in the section is employed, as we have seen, in maintaining or restoring the “special state” in each gas, as well | as the MAxweELt condition of partition of energy between the two gases. If R | be the resistance in question, the equations of motion are ”) Sete /L a Pamende) =~ 3 aa( 7, |) Ree, | mn Y) ee. deans | aj Paitan2dx) = = Tal 7, b+ Rew >| | where 2 represents éotal differentiation. 48. To calculate the value of R, note that, in consequence of the assumed } smallness of a, a,, relatively to the speeds of mean square of the particles, the | number of collisions of a P, with a P,, and the circumstances of each, may be treated as practically the same as if a, and a, were each zero :—except in so far VOL. XXXIII. PART II. . 2Q 268 PROFESSOR TAIT ON THE that there will be, in the expression for the relative speed in the direction of the line of centres at impact, an additional term (a, —%)cos yr, where y is the inclination of the line of centres to the axis of x. Thus to the impulse, whose expression is of the form 2PQ peg as in $19 of the First Part of the paper, there must be added the term we seek, v1zZ., . oP,P, ee pa —4a,)COsyp. This must be resolved again parallel to x, for which we must multiply by cos y. Also, as the line of centres may have with equal probability all directions, we must multiply further by sin d/2, and integrate from 0 to z. The result will be the average transmission, per collision, per P,, of translatory momentum of the layer parallel to x. Taking account of the number of impacts of a P; on a Py, as in § 23, we obtain finally 7 1(h, +h) +h pale R= Syn? = whe 2) P, +P, (a — a) . . ° (7.) where s is the semi-sum of the diameters of a P, and a P,. 49. To put this in a more convenient form, note that (2), in the notation of (5), gives us the relation i oes eae Fi Gann eee whence G,/hy + Go/hy=2pu. : : - : : (8.) We have not added an arbitrary constant, for no origin has been specified for z. Nor have we added an arbitrary function of ¢, because (as will be seen at once from (3)) this could only be necessary in cases where the left-hand members of (6) are quantities comparable with the other terms in these equa- tions. They are, however, of the order of aG, dG, dP’ Hoag 2» & and cannot rise into importance except in the case of motions much more — violent than those we are considering. From (8) we obtain Tent lo=0. ae FOUNDATIONS OF THE KINETIC THEORY OF GASES. 269 which signifies that equal volumes of the two gases pass, in the same time, in opposite directions through each section of the tube. This gives a general description of the nature of the cases to which our investigations apply. But, by (3) and (4), we have for the value of ; P, Pom22(%1 — %9) the expression dG, 1 5 dG = Py 5Pim t,) + Pyn,(— 3 1 P ak 5 Pn ,) ) 3 or, by (9), (2), and (5) dG, Bl 0G SU ie dé 3n qe tis ue ar )). Substituting this for the corresponding factors of R in the first of equations | (6), and neglecting the left-hand side, we have finally h PCy +52 mhy+h,) ph, (dG, 1 dG, \ — 3 rer hil oe Aan Fae Maat mG) or dG, _/ 3 P,+P, Ls O(a Jalhy thigh, (mG, +74 «,)\a or, somewhat more elegantly, a7G, +5 (mC, +n, 8 7 is : : - (10.) ws _( 3 Iyths dt ~ \8ns?V ahyhg 50. This equation resembles that of Fourrer for the linear motion of heat ; but, as already stated in § 34, the quantities €, which occur in it render it in general intractable. The first part of what is usually called the diffusion- | coefficient (the multiplier of d?G,/da? above) is constant ; but the second, as is _ obvious from (5) and (8), is, except in the special case to which we proceed, a _ function of dG,/dx ; i.e. of the percentage composition of the gaseous mixture. | 51. In the special case of equality, both of mass and of diameter, between | the particles of the two systems, the diffusion-coefficient becomes eT se pare site ~ 82s? ah * 38n7s? Jh’ - aes C, Ne dene, b= een | where \ is the mean free path in the system. Hence the diffusion-coefficient | among equal particles is directly as the mean free path, and as the square root | of the absolute temperature. Fourier’s solutions of (10) are of course applic- able in this special case. | or 270 PROFESSOR TAIT ON THE If we now suppose that our arrangement is a tube of length / and section S, connecting two infinite vessels filled with the two gases respectively; and, farther, assume that the diffusion has become steady, the equation (10) becomes dG, @G, de ae where the left-hand member is constant. Also, it is clear that, since dG,/dx must thus be a /inear function of «, we have dG, ( T= Pn, =Pn(1—5), so that the mass of either gas which passes, per second, across any section of the tube is SDp/i where p is the common density of the two gases. For comparison with the corresponding formulee in the other cases treated below, we may now write our result as er P yD) p= ars? Ail Ae - Also, to justify our assumption as to the order of the translatory speed, we find by (3) 1:38 3 C=2) Ji Hence, except where /—a is of the order of one thousandth of an inch or less, this is very small compared with h-*. And it may safely be taken as impossible that 7, can (experimentally) be kept at 0 at the section «=/. If the vessels be of finite size, and if we suppose the contents of each to be always thoroughly mixed, we can approximate to the law of mixture as follows. On looking back at the last result, we see that for p we must now substitute the difference of densities of the first gas at the ends of the connecting tube. Let 9,, J be the quantities of the two gases which originally filled the vessels respectively ; and neglect, m comparison with them, the quantity of either gas which would fill the tube. Then, obviously, dG, Pes 1 a ae Uh Yo whence gi alti fou og AA \. htGo Ie This shows the steps by which the initial state (g,, 0) tends asymptotically to. lo? FOUNDATIONS OF THE KINETIC THEORY OF GASES. 271 on Io the final state \e Ea aca When the vessels are equal this takes the simple form g aeDot @=f(ite gl ). n) , in which the gases are completely mixed. 52. In the case just treated there is no transmission of energy, so that the fundamental hypotheses are fully admissible. In general, however, it is not so. The result of § 41, properly modified to apply to the present question, shows that the energy which, on the whole, passes positively across the section z is, per unit area per second, Fae =f =i) - m'(P; 8 ,—P, fs). This, of course, in general differs from section to section, and thus a disturbance of temperature takes place. In such a case we can no longer assume that h, and h, are absolute constants ; and thus terms in ¢, would come in; just as a term in C, appeared in the expression for energy conducted (§ 42). Thus, in order that our investigation may be admissible, the process must be conducted at constant temperature. This, in general, presupposes conditions external to the apparatus. 53. Though it appears hopeless to attempt a general solution of equation (10), we can obtain from it (at least approximately) the conditions for a steady state of motion such as must, we presume, finally set in between two infinite vessels filled with different gases at the same temperature and pressure. For the left-hand member is then an (unknown) constant, a second constant is introduced by integrating once with respect to x; and these, which determine the complete solution, are to be found at once by the terminal conditions 1 dG nm for x=0, Baa %= 10, Seer Kr) And, by a slight but obvious modification of the latter part of § 51 above, we can easily extend the process to the case in which the vessels are of finite size :—always, however, on the assumption that their contents may be regarded as promptly assuming a state of uniform mixture. The consideration of § 52, however, shows that the whole of the contents must be sept at constant temper- ature, in order that this result may be strictly applicable. 54. Recurring to the special case of § 51, let us now suppose that, while the masses of the particles remain equal, their diameters are different in the two gases. Thus, suppose s,>s,. Then it is clear that sy?—s?, and s?—s?, 272 PROFESSOR TAIT ON THE are both positive. In this case, infinite terminal vessels being supposed, (10) gives for the steady state Se 08: if Ta ye Ny dn, , SWAT gt3 ee . ~ Ree whose integral, between limits as in (11) above, is Tg Cn 1 il ate | Al= {4/5 + por at ae 8 s2— a * 2 —syF — 3) log bt op log *) Here A is the rate of passage of the first gas, in mass per second per unit area of the section of the tube. If now we put S,=S+o, 5,=S—oc, then, when o is small compared with s, the multiplier of C,7/3 is (1+?/3s?)/s?, nearly. When og is nearly equal to s, z.¢. one of the sets of particles exceedingly small compared with the other, it is nearly 1:283/s?. Thus it appears that a difference in size, the mean of the diameters being unchanged, favours diffusion. Suppose, for instance, St eee sae eee alee T 2 Fe ie ~ als? ae Th Iw 3+ +95 108 3t5 5 log he P Sug nae ld | ee als? fb $eSuon bee | Riis Pe and we have Compare this with the result for equal particles (§ 51), remembering that now stands for the mean free path of a particle of either gas in a space filled with the other :—and we see that (so long at least as the masses are equal) diffusion depends mainly upon the mean of the diameters, being but little affected by even a considerable disparity in size between the particles of the two gases. Thus it appears that the viscosity and (if the experimental part of the inquiry could be properly carried out) conductivity give us much more definite information as to the relative sizes of particles of different gases than we can obtain from the results of diffusion. FOUNDATIONS OF THE KINETIC THEORY OF GASES. 273 Equation (12) shows how the gradient of density of either gas varies, in the stationary state, with its percentage in the mixture. For the multiplier of dn, - ak is obviously a maximum when 1 al FH ys S +5,7/y’ in which y=%,/n,, is so. This condition gives 1/Ny = ¥Y = 83/5, . Hence the gradient is /east steep at the section in which the proportion of the two gases is inversely as the ratio of the diameters of their particles ; and it increases either way from this section to the ends of the tube, at each of which it has the same (greatest) amount. This consideration will be of use to the full understanding of the more complex case (below) in which the masses, as well as the diameters, of the particles differ in the two gases. 55, Let us now suppose the mass per particle to be different in the two gases. The last terms of the right-hand side of (10), viz., 1 es By MarGr +7, @,) Ages ? may be written in the form P, dm ((n—n)h, [~ Sydy 4 Maly ie Siy)dy eee Nh 9 myhys Ey) +(n—mahys*E(y J) Lhe f (n— nhs PE (y) + mhosF(y JP 5} at where the meanings of fand F are as in § 34. If we confine ourselves to the steady state, we may integrate (10) directly with respect to x, since dG,/dt is constant. In thus operating on the part just written, the integration with regard to « (with the limiting conditions as in (11)) can be carried out under the sign of integration with respect to y :—and then the y integration can be effected by quadratures. The form of the x integral is the same in each of the terms. For 0 7, (n—1,)dn, 2 mdn, 1 ee A 5 bos Li An, + B(n—1,) n A(n—) + Bn, ——_ ree oA This expression is necessarily negative, as A and B are always positive. When A and B are nearly equal, so that B=(1+e)A, its value is ~R(a-gte), so that, even when A and B are equal, there is no infinite term. 274 PROFESSOR TAIT ON THE It is easy to see, from the forms of F(y), and of its first two differential co- efficients, that the equation h hysF(y) = hysPE(y J Ty, ) can hold for, at most, ove finite positive value of y. 56. As a particular, and very instructive case, let us suppose Py: Poti a oe 1 | the case of oxygen and hydrogen. (a) First, assume the diameters to be equal. Then the integral of (10), with limits as in (11), taken on the supposition that the flow is constant, is dG, _ P, Dip rene ooh) “1 fY)- 16/7) FWyfiy)—16E( 2) ¢(4) 16F (4) “dt Ts Jy g Vln eee ls Fre oumer 2) log aye, a As remarked above, the definite integral is essentially negative. For so is every expression of the form —b | Aa—Bb, B A=B (AB ew provided A, B, a, and b be all positive. When A and B are equal its value is A = sae b). I have made a rough attempt at evaluation of the integral, partly by calcu- lation, partly by a graphic method. My result is, at best, an approximation, for the various instalments of the quadrature appear as the relatively small differences of two considerable quantities. Thus the three decimal places, to which, from want of leisure, I was obliged to confine myself, are not sufficient to give a very exact value. The graphical representations of my numbers were, however, so fairly smooth that there seems to be little risk of large error. The full curve in the sketch below shows (on a ten-fold scale) the values of the integrand (with their signs changed), as ordinates, to the values of y as abscissee. The area is about —2:165. Hence we have it, ee =~ eget (4) Suppose next that the diameter of a P, is three times that of a P,, but the semi-sum of the diameters is s as before. The definite integral takes the form f* 0 em FOUNDATIONS OF THE KINETIC THEORY OF GASES. 275 SY) ies 16/(4) fi LEA) oes 640 (T es) ede, S109) w)=02)° Gao Dy (wy —ae(y)y FO The corresponding curve is exhibited by the dashed line in the sketch, and its area is about —3:157. Hence, in this case, dG ss — f ak Ne (c) Still keeping the sum of the semidiameters the same, let the diameter of a P, be three times that of a P,;. The integral is ae é 16 f(+) : FF)f) pee ice : je @)). TFW)— 16F(-4 y ) F(y)— —36F(7 et TFw)—168(4)) Fy) (Fy) —368(f )) 2 | The curve is the dotted line in the cut, and its area is about —1°'713. Hence we have If we compare these values, obtained on such widely different assumptions as to the relative diameters of the particles, we see at once how exceedingly difficult would be the determination of diameters from observed results as to diffusion. (Compare § 54.) VOL. XXXIII. PART II. 2k ya) 276 PROFESSOR TAIT ON THE But we see also how diffusion varies with the relative size of the particles, the sum of the diameters being constant. For the smaller, relatively, are the particles of smaller mass (those which have the greater mean-square speed) the more rapid is the diffusion. And further, by comparison with the results of §§ 51, 54, we see how much more quickly a gas diffuses into another of different specific gravity than into another of the same specific gravity. When the less massive particles are indefinitely small in comparison with the others, the diameter of these is s; and for their rate of diffusion we have dG Ve ae =— ai 4:26. When it is the more massive particles which are evanescent in size, the numerical factor seems to be about 3°48. Hence it would appear that, even in the case of masses so different, there is a minimum value of the diffusion- coefficient, which is reached before the more massive particles are infinitesimal compared with the others. [At one time I thought of expressing the results of this section in a form similar to that adopted in the expression for D in§ 51. It is easy to see that the quantity corresponding to } would now be what may be called the mean free path of a single particle of one gas in a space filled with another. Its value would be easily calculated by the introduction of h, for 2 in the factor v pe 0 while keeping ¢ in terms of 2. This involves multiplication of each number in the fourth column of the Appendix to Part I. by the new factor «~~ h,}/Ai, But, on reflection, I do not see that much would be gained by this. ] FOUNDATIONS OF THE KINETIC THEORY OF GASES. 277 AcPAP EN Dax: The notation is the same as in the Appendix to Part I. aX, /X, oX,/X, Al 000049 ‘000005 ‘000001 000000 2 ‘000758 000152 ‘000030 000001 3 ‘003594 ‘001078 000323 ‘000029 ‘4 ‘010364 ‘004146 ‘001658 000265 5 022505 011252 005626 001407 6 040512 ‘024307 014584 005250 Yi ‘063623 044536 ‘031175 015276 8 089928 ‘071942 057554 036834 9 116712 105041 ‘094537 ‘076575 1:0 141040 ‘141040 ‘141040 141040 Al "160292 ‘176321 193953 ‘234683 1:2 172656 ‘207187 248624 358019 1:3 ‘177229 ‘230398 ‘299517 506184 14 ‘174174 ‘248844 341382 ‘669108 15 ‘164430 ‘246645 369968 832427 16 "149568 ‘239309 382894 ‘980209 eZ 131393 ‘223368 379726 1097407 18 111654 ‘200977 ‘361758 1172098 1g ‘091960 174724 ‘331976 1198432 2:0 ‘073480 146960 ‘293920 1-175680 2-1 057015 ALSO ‘251435 1108829 2:2 043032 ‘094670 ‘208274 1008046 2°3 ‘031579 072632 167054 883714 24 022584 054202 130085 749288 2°5 ‘015750 039375 098438 615234 2°6 ‘010686 ‘027784 ‘072238 ‘488332 27 ‘007074 ‘019099 ‘051567 375926 2°8 ‘004536 ‘012701 035563 ‘278812 29 ‘002871 ‘008326 ‘024145 ‘203063 30 ‘001710 ‘005130 ‘015390 138510 31 ‘001071 003320 ‘010294 098925 32 000629 ‘002014 ‘0064-45 ‘065997 3°3 ‘000361 ‘001192 003935 "042852 34 000211 ‘000689 ‘002344 ‘027098 3°5 000111 ‘000389 ‘001361 016671 3°6 ‘000066 ‘000240 000865 010004 3°7 ‘000037 ‘000136 000505 005839 3°8 ‘000229 ‘003307 39 ‘000118 001798 4:0 000062 | 000985 27095244 2°954862 4:630593 14624154 Thus the values of C,, C,, C,, and C, are respectively 0°838, 1°182, 1°852, and 5°849. -— ( 279 ) XTI.—Tables for Facilitating the Computation of Differential Refraction in Position Angle and Distance. By the Hon. Lorp M‘Laren. (Read 6th December 1886. ) The annexed tables are intended to facilitate the computation of the cor- rections for refraction which have to be applied to differential measures, such as are made with the micrometer or heliometer. Differential measures are of two kinds :—(1) Direct measures of differences of right ascension and declination; and (2) measures of position angle and distance. In either case the observer only seeks to determine the relative positions of the objects under observation; and the correction for refraction consists in the applying to each reading a quantity representing the difference of the separate effects of refraction on the apparent places of the two stars, whose relative positions are to be determined. This might be effected by computing separately the displacement of-each star caused by refraction, and taking the difference between these quantities for the required correction. But, in practice, the correction for refraction is obtained more easily and more accurately by differentiation. When the measures to be corrected for refraction are direct measures of differences of right ascension and differences of declination, the quantities log ae and log = may be tabulated for a given latitude, with the arguments, declination, and hour angle. The numerical values of these co-efficients for unit of arc (or 1’) are to be computed for all possible positions above the horizon ; é F ; : y and then the correction is at once obtained by taking out log = and log = from the table and adding to each the logarithm of the number of seconds of arc in the corresponding measure. It is intended, in a subsequent paper, to submit a specimen of such a table prepared for the latitude of Edinburgh. The correction for refraction in the case of observations of position angle and distance is a more troublesome matter; because the various readings of the position angles and distances for any pair of stars are not all taken at the same elevation above the horizon, and therefore each measure must be separately corrected for refraction before it can be combined with the others into a mean position angle or mean distance. VOL. XXXIII. PART II. 28 280 LORD M‘LAREN ON DIFFERENTIAL REFRACTION. The analytical investigation of these corrections leads to the following expressions :— | If we call 7’ and z the true and apparent position angles; A’ and A, the true and apparent distances of the two stars; ¢, the mean zenith distance of the field of view; », the parallactic angle; and «, the co-efficient of refraction, we have 7 =7—« tan? €sin (7 —7) cos (7—7). A’=A+« A [1+tan? € cos? (7 —x)]. In these expressions all the variable quantities are given directly by the readings, excepting tan’ ¢ and », the parallactic angle. Now, the last men- tioned quantities are functions of the latitude, declination, and hour angle. They can therefore be tabulated for a given latitude, with the arguments, declination, and hour angle. The present tables give for the parallel 55° 56’ (which passes through Edinburgh), and also for 57° 30’ the quantities log tan? ¢ and » for each ten minutes of hour angle, and for each interval of two degrees of declination from 40° north to 90°. The tables include the entire circumpolar region of the heavens visible from the respective latitudes, and one or other of them may be used for observations taken in any part of Scotland, without sensible error. Where great accuracy is desired, a table of differences applicable to the particular observatory may be obtained by interpolating between the two printed tables. The computations for the two tables were made in the following manner :— Calling ¢ the latitude of the place of observation ; II, the polar distance corre- sponding to the interval of declination; and 7 the hour angle—the quantities ¢ and » are to be obtained by solving the spherical triangle, whose vertices are the pole, the zenith, and the star; whose sides are polar distance, zenith distance, and the co-latitude ; and whose angles are hour angle, azimuth and parallactic angle. To adapt the solution to logarithmic computation, the auxiliary angles M and N were computed for each 10 minutes of hour angle by the formule Sin M=cos ¢ sin r. Tan N = cotan ¢ cos 7. The resulting values of N and log cos M were tabulated, and the final compu- tations were made by the formule Cos ¢=cos M cos (II—N). Cos y=cotan ¢ tan (II—N). The quantities log tan’ ¢ and » were directly computed for each alternate column of the tables. The intermediate columns were obtained by interpola- tion, checked by independent computation of a sufficient number of tabular places to ensure substantial accuracy in the last decimal place. PT AB Tok CONTAINING THE LOGARITHM OF TAN’? ZENITH DISTANCE AND THE PARAL AC TIC ANGIE FOR LATITUDE 55° 56’, AND DECLINATION 40° ro 90°. LOGARITHM OF Tan? Z FOR LAT. 55° 56’. 40° H. MM 0 0 |} 89130 0 10 || 8°9220 0 20 | 8:9310 0 30 || 89575 0 40 || 8:9840 0 50 | 9:0220 1 # | 90600 1 10 | 9°1066 1 20 | 9°1532 1 30 | 92013 1 40 | 92494 1 50 || 93092 2 0 || 9°3490 2 10 || 9°3974 2 20 | 9°4458 2 30 | 9°4942 - 2 40 | 95426 2 50 || 9°5888 3 0 | 9°6350 3 10 | 9°6796 3 20 || 9°7242 3 30 | 9°7684 3 40 | 9°8126 3 50 || 9°8557 4 0} 98988 4 10 | 9°9411 4 20 || 9°9834 4 30 || 0°0256 4 40 | 0:0678 4 650 | 01091 5 0 | 0°1504 5 10 | 071921 5 20 || 0°2338 5 30 || 0°2760 5 40 || 0°3182 5 50 || 0°3604 6 0 | 0°4026 8°7840 | 86550 | 84725 | 82900 | 7°9968 | 7°7036 87942 | 86684 | 84961 | 83238 | 80544 | 7:7849 88064 | 86818 | 85197 | 83576 | 81119 | 7°8662 88405 | 8°7235 | 85751 | 84266 | 82260 | 8:0254 88746 | 8°7652 | 86304 | 84956 | 83401 | 81846 89201 | 88182 | 86989 | 85796 | 84493 | 83190 89656 | 88712 | 8-7674 | 86636 | 85585 | 84534 90194 | 8:°9321 | 88393 | 87465 | 86571 | 85676 9°0731 | 8:9930 | 89112 | 88294 | 87556 | 86818 91275 | 9:0536 | 8:9791 | 89045 | 88385 | 87725 9°1818 | 971142 | 9:0469 | 8°9796 | 8°9214 | 88632 9°2361 | 9°1729 | 9-1113 | 9°0497 | 8:°9959 | 8:9421 9°2903 | 9°2316 | 9°1757 | 9°1198 | 9:0704 | 9:0210 9°3419 | 9°2864 | 9:°2337 | 9:1810 | 91348 | 9:0886 9°3935 | 9°3412 | 9:2917 | 9°2422 | 9:1992 | 9:1562 9°4439 | 9°3935 | 9:3464 | 9°2993 | 9°2581 | 9°2169 9°4942 | 9'°4458 | 9°4011 | 9°3564 | 9°3170 | 92776 9°5421 | 94954 | 9°4522 | 9°4090 | 9°3714 | 9°3328 9°5900 | 9°5450 | 95033 | 94616 | 9°4248 | 93880 9°6359 | 9°5922 | 95513 | 9°5104 | 9°4744 | 9:-4384 9°6818 | 9°6394 | 95993 | 9°5592 | 9°5240 | 9°4888 9°7267 | 9°6849 | 9°6455 | 9°6061 | 9°5709 | 9°5357 9°7715 | 9°7304 | 9°6917 | 9°6530 | 9°6178 | 9°5826 9°8109 | 9°7741 | 9°7356 | 9°6970 | 9°6618 | 9°6265 9°8583 | 98178 | 9°7794 | 9°7410 | 9°7057 | 9°6704 9°9005 | 9°8598 | 9°8215 | 9°7832 | 9°7478 | 9°7123 9°9426 | 99018 | 9°8636 | 9°8254 | 9°7898 | 9°7542 9°9845 | 9°9433 | 9°9049 | 9°8664 | 9°8306 | 9°7947 0°0263 | 9°9848 | 99461 | 9:9074 | 9°8713 | 9°8352 0°0673 | 0°0255 | 9°9862 | 9°9469 | 9°9103 | 9°8736 0'1083 | 0°0662 | 0°0263 | 9°9864 | 9°9492 | 9°9120 0°1492 | 0°1062 | 0:0656 | 0:0249 | 9°9871 | 9°9492 0°1900 | 071462 | 0°1048 | 0°0634 | 0°0249 | 9°9864 0'2310 | 0°1860 | 0°1439 | 0°1010 | 0°0621 | 0°0225 0°2720 | 0°2258 | 01829 | 0°1400 | 0°0993 | 0°0586 0°3129 | 0°2655 | 0°2214 | 0°1772 | 0°1357 | 0:0941 0°3539 | 0°3052 | 0°2598 | 0°2144 | 01720 | 0°1296 54° 70560 7°3145 75730 78225 80720 8°2258 8°3796 85021 86246 8°7203 88160 89016 8°9872 90560 9°1248 9°1859 9°2470 9°3025 9°3580 94081 9°4581 9°5051 9°5520 9°5957 9°6394 9°6808 9°7222 9°7621 9°8019 9°8397 9°8775 9°9139 9°9502 9°9854 0°0206 0°0549 0:0893 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. PARALLACTIC ANGLE, FOR LAT. 55° 56’. 283 40° 42° 44° 46° 48° 50° 52° 54° H. M. 135 M. CeeeOn| OO) O2LO') LOMO] Porto | O20") \O2)0') Oe%0"| O20") 24 oO Geno) 5.0) | 5148 | “GeSb |. 6825) Woris |) 13.45 | Tr 1a | 27°66 || 23 - 50 2) | 1Oes0 |) 11035 | 13910 | Wer 5O | 20830 | 27°30 | B4 30) 5412 | 23 40 mse) 14040) | Teka) TONLE | Sars | S719 | 85529 |, 43029) | CIs || 23' 30 eeesGr || 9G20) | SBAT DY | 5th | Boras | S45) 4301 | ba T4 |) 68.0 || 23. 20 Ge50 |) 22059) | B60ea | BOlLy | B8e58' | 88) 49' || 47 251 | 56” Or 69°36] 23 10 LT 0] 26°38 | 29 55'| 33 12) 38 21 | 43 30] 51 38 | 59 46| 71 12 23 OO men) 20etg | Sauer |) Belg7 | 41-3h | eran | 54008 | Gilat | 71° 46 || 22° 50 1 20] 32 56] 36 19] 39 42 | 44 41 | 49 40] 56 38] 63 36 | 72 19 | 22 40 mead) soeih || 88-34 | 456) | 46°40) | 51 24) oF 5a | 64 20'| 72° 12'|| 22 30 40) || 87°26 | 40°48 | 44:10 | 48-39 | 53° 8 | 59 6 | 65: 4) 72 °5 | 22 20 50 || 39°11 | 42°27 | 45.43'| 50° 0 | 5416) 59 47 | 65-17 | 71-51 | 22 10 Be eOr|) 40°56 | 44°°6 | 47° 16 | 51 20'| 55°94 |. 60°97 | 65 30'| 71 36 | 22 oO fend, 42059.) A519} Atta | 5974 | 55°54) |. 6038 | 65° 292'| 71° 0 | 21 50 oee20 || 43°22) 46 17 || 49° 19'| 52.48 | 56 24 | 60 48 | 65 14 | 70 23 || 21 40 Bead) | 44915) | 47°02) 49°48) 53°13) 56 37 | 60 44°! 6452 | 69°40 | 21 30 2 40] 45 8 | 47 46/50 24] 53 37 | 56 50 | 60 40 | 64 30 | 68 57 | 21 20 2 50 || 45 40] 48 11] 50 41 | 53 44] 56 47 | 60 23 | 63 59 | 68 11] 21 10 Be0 |) 46°12)| 48°35)! 50.58] 53 51| 56°44 | 60 6 | 63 28) 67 25 | 21 oO 3 10] 46 29 | 48 46| 51 3] 53 46! 56 29 | 59 40 | 62 51 | 66 31 | 20 50 S20 || 46.46 | 48 57 | 51° 8 | 53 41 | 56-14 | 59 14] -62 14] 65 37 || 20 40 a 30 || 46°53 | 48°58!) 51° 2)| 53°97 | 55°52 | 58 41 | 61 30| 64 45 | 20 30 3 40] 47 0O| 48 58] 50 56! 53 13 | 55 30| 58 8 | 60 46] 63 52 | 20 20 3 50] 4656 | 48 49] 50 41 | 52 51 | 55 0O| 57 30] 59 59 | 62 52 | 20 1o Bea) |), A652) | 48039) | -5O" 96? || 52°28 | 54°-30'| 56 SI | 59 12) 61-52 || 20 oO S10 || 46°40) | 48°91 | 50° °2)| 51 58.) 53°54 | 56° 8 | 58 21 | 60-52 || 19 50 4 20] 46 28| 48 3] 49 38] 51 28 | 53 18| 55 24] 57 30] 59 53] 19 40 4 30] 46 10 | 47 40] 49 9| 50 49 | 52 38 | 54 37 | 56 36 | 58 52] 19 30 4 40 || 45 52 | 47 16 | 48 40| 50 9] 51 58] 53 50| 55 42 | 57 50] 19 20 4 50 || 45 24] 46 45 | 48 5 | 49 34] 51 12| 52 59/| 54 45 | 56 47 119 10 5 0] 44 56| 46 13] 47 30| 48 58| 50 26| 52 7 | 53 48 | 55 43/19 0 5 10] 44 26 | 45 39 | 46 52] 48 16 | 49 39 | 51 15 | 52 50| 54 39/18 50 5 20] 43 56 | 45 5! 46 14] 47 33 | 48 52 | 50 23 | 51 52 | 53 34/118 40 5 30] 43 21 | 44 26 | 45 30| 46 45] 48 O| 49 25| 50 50] 52 27 || 18 30 5 40 || 42 46 | 43 46 | 44 46 | 45 57 | 47 8] 48 28| 49 48 | 5119] 18 20 i t50) |) 49eeGr | 4350s) 4e00) | 4507 | 4613 | £7 29 | 48 45) 50-11 || 18. 10 6 -0 || 4ie26" 42:90") 48°14 iG |) A5mrs | 46°30 | 47 42) | 49743) Ie) 0 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. 44 84 On0c MMO WAIT Way © Oo 0°4026 074454 0°4882 05323 05764 0°6208 0°6752 07115 0°7578 0°8048 0°8518 0°9005 0°9492 0°9993 10494 11009 11524 1°2045 12566 1°3099 13632 14166 14700 15228 15756 16260 16764 17232 17700 18101 1°8502 18817 19132 19314 19496 19581 19666 LOGARITHM 073539 0°3956 0°4363 0°4789 0°5204 0°5626 0°6048 0°6483 0°6917 0°7356 O-7794 0°8243 0°8691 09148 0°9605 10069 10532 10995 11457 11927 12396 12857 13318 13765 14213 14636 15058 15441 15824 16149 16474 16726 16979 17123 1°7266 17333 17399 OF Tan?Z FOR 0°2598 0°2981 03363 0°3747 0°4131 0°4511 0°4891 0°5276 0°5661 06043 0°6425 0°6807 0°7189 0°7570 07951 0°8329 0°8706 09078 0°9441 09804 1:0167 10513 10858 11181 11504 171800 1:2096 1:2355 12613 12826 1°3038 13199 13360 1°3450 13540 1°3580 1°3620 48° 0°2144 0°2513 0:2882 0°3250 0°3618 0°3978 0°4338 0°4702 05066 05423 05780 0°6134 0°6488 0°6837 0°7186 0°7529 07972 0°8203 0°8534 0°8854 09174 0°9477 0°9780 10059 10338 1:0589 1:0840 11059 11278 11454 11630 11762 11894 11969 12044 12076 12108 LAT. 55° 56’. 0°1720 0°2076 0'2432 0°2737 0°3141 0°3484 0°3827 0°4173 0°4519 0°4857 05194 0°5527 05860 0°6186 0°6511 0°6828 07145 0°7449 07754 0°8046 0°8337 0°8611 0°8884 09136 0°9387 0°9611 0°9835 1:0030 1:0224 1:0379 10535 10652 1:0768 10832 10896 10925 1:0953 0°1296 0:1639 071982 0°2323 0°2664 0°2990 0°3316 0°3644 0°3972 0°4290 0°4608 0°4920 0°5232 05534 0°5836 0°6127 0°6418 0°6696 06974 0°7237 0°7500 07744 0°7988 0°8212 0°8436 0°8633 0°8830 0°9000 09170 0°9305 0°9440 0°9541 0°9642 0°9695 09748 09773 09798 0:0893 071226 0°1559 0°1887 0°2215 0°2528 0:2840 0°3102 0°3364 0°3716 0°4068 0°4362 0°4655 0°4939 0°5222 05494 0°57 66 06023 0°6279 0°6523 0°6766 0°6989 07211 07417 0°7622 07799 O97 0°8131 0°8285 0°8406 0°8526 0°8618 0°8710 0°8761 0°8812 0°8831 0°8849 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. . ° CO CO © Oo ow J =I NNT TNT ODD (OO PARALLACTIC ANGLE, FOR LAT. 55° 56’. 43°14’ 46 | 13 58 | 14 10 26 | 12 28 | 12 30 6 | Gos | 200 40 9 40 9 40 14 8 22 8 30 52 tO (os) 30 5 38 5 46 10 4 14 4 18 50 2 50 2 50 25 1 25 1 25 0 0 0 0 0 46° bo bb bo oe i) oo 48° 47° 49’ 46 26 45 10 44 13 43 16 42 5 40 54 39 43 38 32 54° Horizontal Argument, Declination —Vertical Argument, Hour Angle. 286 LOGARITHM OF Tan?Z FOR LAT. 55° 56’. 56° 58° 60° 62° 64° 66° 68° 70° 0 ao — 77420 | 8°0315 | 83210 | 84923 | 86636 | 87936 || 24 .0 0 10 || —— | 7:3788 | 7°7905 | 80598 | 83289 | 84978 | 86667 | 87952 | 23 50 0 20 || 73360 | 7°5880 | 7°8390 | 80880 | 83368 | 85033 | 86698 | 87967 || 23 40 O 30 || 76580 | 7°8240 | 7°9895 | 81904 | 83912 | 85439 | 86965 | 88164 | 23 30 0 40 || 79800 | 8:0600 | 81398 | 8:2927 | 84456 | 85844 | 87232 | 88360 || 23 20 0 50 || 81584 | 82095 | 82605 | 83836 | 85065 | 86307 | 87546 | 88606 | 23 10 1 0 || 83368 | 83590 | 83812 | 84745 | 85678 | 86769 | 87860 | 88851 || 23 0 1 10 || 84637 | 84772 | 84906 | 85640 | 86374 | 87310 | 88245 | 89161 || 22 50 1 20 || 85906 | 85953 | 86000 | 86535 | 87070 | 87850 | 88630 | 89471 || 22 40 1 30 || 86906 | 86891 | 86875 | 87299 | 87724 | 88394 | 89063 | 89811 || 22 30 1 40-|| 87906 | 87828 | 87750 | 88064 | 88378 | 88937 | 8:9496 | 9:0150 || 22 20 1 50 || 88720 | 8°8625 | 88530 | 88761 | 88992 | 89465 | 89938 | 9:0517 || 22 10 2 0 || 89534 | 89422 | 89310 | 89458 | 89606 | 89993 | 9:0380 | 9:0883 | 22 0 2 10 || 90234 | 9:0103 | 8-9971 | 9°0065 | 9°0169 | 9:°0483 | 9:0797 | 9:1235 || 21 50 2 20 || 9:0934 | 9:0783 | 9:0632 | 9°0672 | 9:0732 | 9:0973 | 9:1214 | 9°1587 || 21 40 30 || 9°1549 | 9°1382 | 9°1251 | 9°1224 | 9:°1242 | 9°1429 | 9°1616 | 9°1938 || 21 30 9°2164 | 9°1981 | 91798 | 9:1775 | 9:1752 | 9°1885 | 9°2018 | 9°2288 | 21 20 50 || 9°2722 | 9°2545 | 9°2368 | 9°2296 | 9°2224 | 9:2317 | 9:°2409 | 9:2634 || 21 10 bo bo bk i (2) 0 || 9°3280 | 93109 | 9°2938 | 92817 | 9:2696 | 9:2748 | 92800 | 92980 || 21> ap 9°3777 | 9°3581 | 9°3384 | 9°3260 | 9°3136 | 9°3154 | 9°3172 | 9°3313 || 20 50 20 || 9°4274 | 9°4052 | 9°3830 | 9°3703 | 9°3576 | 9°3560 | 9°3544 | 9°3645 || 20 40 ew woo _ oO 30 || 9°4744 | 9°4513 | 9°4281 | 9-4135 | 9°3990 | 9°3972 | 9:3904 | 9:3971 || 20 30 95214 | 9:4973 | 9°4732 | 94568 | 94404 | 94384 | 9-4264 | 9:4297 || 20 20 50 || 95649 | 95399 | 95148 | 94969 | 94789 | 9°4721 | 9°4602 | 9:-4604 || 20 10 eo ow oo i j=) 0 || 96084 | 95824 | 95564 | 9°5369 | 9°5174 | 95057 | 9°4940 | 94911 | 20 0 9°6493 | 976223 | 9:5954 | 95746 | 9°5538 | 9'°5401 | 9°5263 | 95209 || 19. 50 20 || 9°6902 | 9°6623 | 9°6344 | 9°6123 | 95902 | 95744 | 95586 | 9°5506 | 19 40 He — oO 4 30 || 9°7294 | 9°7008 | 9°6721 | 9°6485 | 9°6249 | 9°6025 | 9°5900 | 9:°5796 | 19 30 4 40 || 9°7686 | 9°7392 | 9°7098 | 9°6847 | 9°6596 | 9°6405 | 9°6214 | 96086 | 19 20 4 50 || 98058 | 9°7755 | 9°7451 | 9°7189 | 96927 | 9°6716 | 96505 | 96357 | 19 10 5 0 || 98430 | 9°8117 | 9°7804 | 9°7531 | 9°7258 | 9°7027 | 96796 | 96628 | 19 0 5 10 || 98785 | 98462. | 9:8139 | 9°7854 | 9°7568 |. 9°7325 | 97081 | 96891 | 18 50 5 20 || 99140 | 9°8807 | 9°8474 | 9°8176 | 9°7878 | 9°7622 | 9°7366 | 9°7154 || 18 40 5 30 || 99483 | 9°9141 | 9°8798 | 9°8489 | 9°8170 | 9°7907 | 9°7635 | 9°7407 | 18 30 5 40 || 99826 | 9°9474 | 9°9122 | 98801 | 9°8480 | 9°8192 | 9°7904 | 9°7660 | 18 20 5 50 || 00158 | 99797 | 9°9435 | 9°9101 | 9°8766 | 9°8464 | 9°8162 | 9°7901 || 18 10 6 0 || 0°0490 | 0°0119 | 9:9748 | 9°9400 | 9°9052 | 9°8736 | 9°8420 | 9°8141 | 18 0 ee ee Horizontal Argument, Declination.—Vertical Argument, Hour Angle. eo ow oO bo bb Oo Oo Co 56° M. 0 180° 0 10 | 90 0 20 || 88 56 30 || 87 46 40 | 86 36 50 | 85 25 0 || 84 14 10 | 83 8 20 || 82 2 30 || 80 56 40 | 79 50 50 || 78 46 O || 77 42 10 || 76 38 20 || 75 34 30 || 74 29 40 | 73 24 50 || 72 23 @ i wl 12 10 || 70 11 20 || 69 O 30 || 67 54 40 || 66 48 50 || 65 40 O || 64 32 10 || 63 24 20 || 62-16 30 | 61 7 40 || 59 58 50 || 58 48 0 || 57 38 10 || 56 27 20 || 55 16 30 | 54 3 40 || 52 50 50 || 51 37 0 || 50 24 PARALLACTIC ANGLE FOR LAT. 55° 56’. 287 58° 60° 62° 64° 66° 68° 70° H. ~M. 180° 0’|180° 0’|180° 0’/180° 0’|180° 0’|}180° 0’|180° 0’| 24 0 137 2 |158 40 |167 30 |168 22 |172 50/174 32 |174 58 || 23 50 118 24 |147 42 |155 O |162 16 |165 40 |169 4 |169 55 || 23 40 111 38 |1385 29 |144 42 |152 59 |157 14 |161 29 |163 29 || 23 30 104 51 |123 6 |134 24 |143 42 |148 48 |153 54/157 3 || 23 20 101 11 {116 56 |127 35 |137 13 |142 49 |148 25 |151 56 || 23° 10 97 30 |110 46 {120 45 |130 44 {136 50 |142 56 |146 49 | 23 O 94 50 |106 32 |115 56 {125 21 |131 39 {137 57 |142 5 |) 22 50 92 10/102 18/111 7 |119 58 |126 28 |132 58 |137 20 || 22 40 90 5} 99 10 |107 29 |115 45 |122 10 |128 34 |133 10 || 22 30 88 0} 96 10/103 51 }111 32 |117 51 {124 10 |128 59 || 22) 20 86 14} 93 41 |100 52 }108 31114 10 |120 17 |125 9 | 22 10 84 27 | 91 12 | 97 53 |104 34 |110 29 |116 24 |121 19 | 22 O 82552) | 89116) |) 9b 23) 1101 39 (107 2P 11138) 3 1117 55 || 21 50 81 17 | 87 O| 92 52 | 98 44 |104 13 |109 42 |114 31 || 21 40 19°49) 85> 8} 90°39 | 96 10/101 26 |106 41. )111 24 | 21 30 78 20 | 83 16 | 88 26 | 93 36 | 98 38 |103 40 |108 17 || 21 20 76 59) | 81 35) 86 26 | 91 14) 96 61100 55/105 25 | 21 10 75 38 | 79 54 | 84 26 | 88 58 | 93 34] 98 10 |102 33 | 21 O 74 26 | 78 15 | 82 34] 87 52] 91 16 | 95 40 | 99 56 | 20 50 72 48 | 76 36} 80 41 | 84 46] 88 58 | 93 10 | 97 18 } 20 40 71 30 | 75 3 | 78 55 | 82 47 | 86 48/ 90 49 | 94 49 || 20 30 70 11 | 73 30] 77 91] 80 48 | 84 37 | 88 28 | 92 19 | 20 20 6GG5D | F2000)} 75°28 || 78056) || 82°36) | $6. Ve |, 89°59) |, 20° 10 67 31 | 7030 | 73:47 | 77 41) 80 34} 84 4) 87 39 | 20 O HOOLL | GOPES | 7251) f worry | FSess8i | SLLS) | 85 26 || 19 50 64 57 | 67 38) 70 34 | 73 30 | 76 41 | 79 52) 83 12 19 40 63 39 | 66 11 | 68 59 | 71 46) 74 49 | 77 50) 81 41] 19 30 62 21 | 64 44] 67 23 | 70 2) 72 56/ 75 50] 78 55 | 19 20 GIeeS | 6G3e19 | 6Ghc51) 68023) || Ti-°9) |) 73154 |, 76 48' | 19’ LO 59 46 | 61 54] 64 19 | 66 44 | 69 21} 71 58| 74 41 | 19 O 58 29 | 60 30 | 62 48) 65 5 | 67 36] 70 7] 72 47] 18 50 57°10 | 599.6 | 61°16) 63°26) 65 51) 68 16) 70-53 | 18 40 55 53 | 57 42 | 59 46 | 61 50 | 64 8] 66 26] 68 52] 18 30 54 34 | 56 18 | 58 16 | 60 14 | 62 25 | 64 36] 66 51] 18 20 53 16 | 54 55 | 56 47 | 58 39 | 60 14] 62 48} 65 21] 18 10 DUPRE) |) 53132))|) SHVLS) |) S004) |, 59020); 610) | 6S°13- |) 18 =O Horizontal Argument, Declination.—Vertical Argument, Hour Angle. ZR VOL. XXXIII. PART II. 288 lo ole oie 2} mm oo aac oe Mis Mie | lor mormor) (© © © LOGARITHM OF Tan?Z FOR LAT. 55° 56’. 0°0490 0°0813 0°1136 0°1451 0°1766 0°2065 0°2364 0°2660 0°2956 0°3242 0°3528 0°3803 0°4078 0°4343 0°4608 04861 05114 0°5349 0°5584 0°5808 0°6032 0°6233 0°6434 0°6621 0°6808 0°6966 0°7124 0°7262 0°7 400 0°7506 0°7612 0°7695 07778 0°7827 0°7876 0'7888 0°7900 60° 62° 64° 66° 00119 00428 0:0737 0°1039 01341 0°1629 071918 0°2199 0°2480 0°2752 0°3023 0°3284 0°3544 0°3793 0°4042 0°4279 0°4515 0°4736 0°4956 05163 0°5370 05557 05744 0°5910 0°6085 0°6229 0°6374 0°6499 0°6625 0°6722 0°6819 06894 0°6969 O-7011 0°7053 0°7067 0°7081 9°9748 0°0043 0°0338 0°0627 0°0916 071194 0°1572 0°1738 0°2004 0°2261 0°2518 0:2764 0°3010 03243 03476 0°3696 0°3916 0°4122 0°4328 04518 0°4708 0°4881 05054 0°5208 05362 05493 05624 0°5737 075850 05938 0°6026 0°6093 0°6160 0°6195 0°6230 06246 0°6262 9°9400 9°9684 99968 0°0245 00521 00786 0°1050 01304 01557 071799 0°2041 0°2274 0°2507 0°2727 0°2946 0°3153 0°3359 03551 0°3743 0°3920 0°4097 0°4257 0°4416 0°4559 04701 0°4823 0°4944 0°5048 075152 0°5232 05312 05374 0°5436 0°5466 05496 0°5512 0°5528 9°9052 9°9325 9°9598 9°9862 0°0126 0°:0377 0°0628 0:0869 0°1110 0°1337 071562 0°1784 0°2004 0°2210 0°2416 0°2609 0°2802 0°2980 073158 0°3322 0°3486 0 3632 0°3778 0°3900 0°4040 0°4152 0°4264 0°4359 0°4454 0°4526 0°4598 0°4655 0°4712 0°4737 0°4762 0°4778 04794 9°8736 9°8995 9°9254 9°9505 99755 99993 00231 0:0459 0:0686 0°0902 01117 0°1323 071528 01721 071914 0°2094 0°2273 0°2439 0°2604 0°2754 0°2909 0°3046 0°3183 0°3304 0°3425 0°3528 0°3630 0°3719 0°3807 0°3873 0°3940 0°3992 0°4044 0°4068 0°4092 0°4105 0°4118 9°8420 9°8665 9°8910 9°9147 9°9384 9°9609 9°9834 0:0048 0°0262 0°0466 00670 00861 071052 071232 01412 0°1578 071744 0°1897 0°2050 02191 0°2332 0°2460 0°2588 0:2699 0°2810 0°2903 0°2999 0°3078 0°3160 0°3221 0°3282 0°3327 0°3372 0°3397 0°3422 0°3432 0°3442 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. PARALLACTIC ANGLE FOR LAT. 55° 56’. 289 56° 58° 60° 62° 64° 66° 68° 70° 50°24’ | 51° 58") (53° S2"| 255° 08") 157° 4") “59° 49°10 | 50°39 | 52° 7 | 5348 | 95-29 | 57 2 47 56 | 49 19 | 50 42 | 52 18 | 53 54 | 55 4 46 40 | 47 59 | 49 17 | 50 49 | 52 20 | 54 3] 55 45 | 57 42 | 17 30 45 24 | 46 38 | 47 52 | 49 19 | 50 46 | 52 23 | 54 O| 55 52 | 17 20 44 7 | 45 17 | 46 26 | 47 49 | 49 12 |} 50 45 | 52 18/] 54 31] 17 10 II 7 4 oO He e oo bo iw bo oo ree ye oo oo ys HS OL j=) re for) i ui ~~ bS ie} q> @ Or oo Or oO bS oO — lor) OL S) I-11 rw [o) oo ~ eo & oo oO bo Or oo te} bo (S) vw oO — (0.2) iw — j—_t for) a bo ow bo nw oo i @ iw Or — S — lor) bo (=) » 330 6 34 | 27 7 7 40 8 17 | 28 54 9236 )| 2908 88") -S1-RO | 15 - 0 oF eLOW 25-10 |- 25-40 | 926) 00 | 26°45 | 27°19 | 27-58 | 28°37 | 29 27 | 14 50 9 20] 23 46 | 24 13 | 24 40 | 25 12 | 25 44 | 26 20 | 26 56 | 27 43 | 14 40 9 30 | 22 16 | 22 43 310 | 23 39 | 24 7 | 24 42 | 25 17 | 26 1 | 14 30 9 40] 20 46} 21 13 | 21 40 | 22 5 | 22 30 | 23 4] 23 38) 24 18 | 14 20 9 M07) 19°23 |-19°46 |.20 <9 |-20°33 | 20°56 | 21.27 | 21-57 | 22 33 | 14 10 Sl 20 Jao) Js 9 16 9 2 9 34 9 48} 10 2); 10 21 | 1 0 Nel 20 7 38 7 40 7 42 7 52 8 2 7 54 7 46 8 21 || 12 50 wl <20 6 16 6 12 Go RS 6 19 6 30 6 0 5 30 6 20 || 12 40 11 30 4 38 40 4 42 4 48 4 53 4 38 4 23 452 | 12 30 Il 40 3. 0 3.8 3 16 3 16 3 16 3 16 3 16 3 23 |) 12 20 11 50 1 30 1 34 1 38 1 38 1 38 1 38 1 38 1 42 10 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. 290 oo oo oO bo bo bo oo © OO ano PP PPP cor or cr 72° 8°9212 89226 8°9240 89362 89488 89665 89842 9°0077 90312 9°0558 9°0804 9°1095 91386 9°1673 9°1960 9°2259 9°2558 9°2859 9°3160 9°3453 9°3746 9°4038 9°4330 9°4606 9°4882 9°5154 9°5426 9°5692 9°5958 9°6209 9°6460 9°6701 9°6942 | 9°7179 9°7416 | 9°7639 9°7862 LOGARITHM 9°0240 9°0251 9°0262 9°0356 9:0452 9°0585 9°0717 90901 9°1084 9°1284 9°1484 brig 9°1950 9°2193 9°2435 9°2686 9°2936 9°3189 9°3443 9°3695 9°3948 9°4201 9°4454 9°4699 9°4945 9°5184 95423 95662 9°5900 9°6128 9°6351 9°6569 9°6788 9°7002 9°7215 9°7419 9°7623 9°1268 9°1276 9°1284 9°1350 9°1416 91504 9°1592 9°1724 9°1856 9°2010 9°2164 9°2339 9°2514 9°2712 9°2910 9°3112 9°3314 9°3520 9°3726 9°3938 9°4150 9°4364 9°4578 9°4793 95008 9°5214 9°5420 9°5631 9°5842 9°6042 9°6242 5°6438 9°6634 9°6824 9°7014 9°7199 9°7384 OF Tan?Z FOR 9°2140 9°2144 9°2148 9°2198 9°2248 9°2319 9°2390 9°2493 9°2595 9°2712 9°2828 92966 9°3103 9°3262 9°3421 9°3584 9°3747 9°3915 9°4083 9°4259 9°4435 9°4615 9°4795 9°4973 9°5150 9°5327 9°5503 95681 9°5859 9°6032 9°6205 9°6375 9°6544 9°6709 9°6874 9°7036 9°7197 9°3012 9°3012 9°3012 9°3046 9°3080 9°3134 9°3188 9°3261 9°3334 9°3413 9°3492 9°3592 9°3692 9°3812 9°3932 9°4056 9°4180 9°4310 9°4440 9°4580 9°4720 94866 9°5012 9°5152 9°5292 9°5439 9°5586 95731 95876 9°6022 9°6168 9°6311 9°6454 9°6594 9°6734 9°6872 9°7010 LAT. 55° 56’. 82° 9°3781 9°3781 9°3781 9°3807 9°3833 9°3869 9°3905— 9°3957 9°4009 9°4071 9°4133 9°4211 9°4288 9°4327 9°4366 9°4508 9°4649 94751 9°4852 9°4958 95064 95178 9°5291 9°5401 9°5510 9°5625 9°5740 9°5855 9°5970 96086 9°6201 9°6317 9°6432 9°6539 9°6646 9°6764 9°6882 9°4550 9°4550 9°4550 9°4568 9°4586 9°4604 9°4622 9°4653 9°4684 9°4729 94774 9°4829 9°4884 9°4942 9°5000 9°5059 9°5118 9°5191 9°5264 95336 9°5408 9°5489 9°5570 95649 9°5728 95811 9°5894 95979 9°6064 9°6149 9°6234 9°6322 9°6410 9°6493 9°6576 9°6665 9°6754 9°5240 9°5240 9°5240 9°5257 95268 9°5279 9°5291 9°5310 9°5328 9°5358 9°5387 9°5422 95457 95495 9°5532 9°5572 9°5612 9°5658 95704 95752 95799 9°5852 95905 9°5957 9°6009 9°6063 9°6117 96177 9°6236 9°6316 9°6396 9°6427 9°6457 9°6512 9°6567 9°6627 9°6686 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. Cll aol a ee © ow co © oO OO bo bo bo Or Or 1 PARALLACTIC ANGLE FOR LAT. 55° 56’. 291 me 7 76° 738° 80° 82° 84° 86° H. M. S00’ 1S0°N0’ | 1807 VO! SOMO" | 180° 0! | 180° “0! | 80° 10’ |180° 0’ || 24. oO 175 50 1175 55 |176 01176 20/176 39 |176 48 1176 56 1177 13 || 23 50 P7040 (171 90 1172 “0 |172 39 1173 1s \1738 35 |173°52 | 174 96 || 23 40 165 29 1166 42 |167 54 |168 54 1169 54 |170 24 |170 53/171 34 || 23 30 160 12 |162 0 {163 481165 9 |166 30 |167 12 |167 54 |168 42 || 23 20 155 27 |157 42 |159 57 |161 29/163 01/164 3/165 5/165 59 || 23 1o 150 42 |153 24 |156 61157 48/159 30 |160 53 |162 16/163 16 || 23 Oo 146 12 |149 43 |152 14 |154 15 1156 15 |158 19 |159 22 |160 32 || 22 50 141 42/146 2/1148 22 {150 41 |153 0 |154 44 |156 28 |157 47 || 22 40 137 45 |141 43 |144 43 [147 12 1149 43 |151 37 [153 30/154 58 || 22 30 133 48 |137 24 |141 0 1143 43 1146 26 |148 29 1150 32 |152 8 || 22 20 130 1/133 46 |137 30 |140 23 1143 16 |145 56 1147 36 |149 21 || 22 10 126 14/130 7/134 01/137 31140 6 |142 23/144 40 146 33 || 22 oO 122 47 |126 43 |130 38 |133 48 1136 57 |139 23 |141 49 |143 50 || 21 50 119 20 |123 18 |127 16 |130 32 |133 48 11386 23 |138 58/141 6 || 21 40 116 7 1190 (7 |\124 6 1197995 1130 43 |133 27 1136 11 |138 24 || 21 30 112 54 |116 55 |120 56 |124 17 |127 38 |130 31 |133 24 |135 42 || 21 20 109 55/113 53 |117 55 |121 20 |124 45 |127 41 1130 36 |133 0 || 21 10 106 56 |110 55 |114 54 1118 23 |121 52 |124 50 1127 48 |130 18 || 21. oO Odea | 108. 7 | 119" 9-158 82 | 119" 1 )1999°3 1125 4 1127 39 || 20° 50 101 26 1/105 18 |109 10 {112 40 }116 10 |119 15 |122 20/125 0 || 20 40 98 48 |102 36 |106 24 1109 54 1113 23 |116 30/119 37 |122 21 || 20 30 96 10 | 99 541103 38/107 7 1110 36 |113 45 |116 54 |119 41 || 20 20 O5142) ("97592 | TON 1 | 104998 OT b> /11 | 0141 116 58 20 10 91 14 | 94 49 | 98 24 |101 49 |105 14 |108 25 /111 36/114 14] 20. Oo 88 53 | 92 22 | 95 51 | 99 15 |102 39 |105 49 |108 59 {111 47 |] 19 50 86 32 | 89 54] 93 18] 96 41 1100 3/103 13 |106 22 |109 19 || 19 40 84 16 | 87 34] 90 52 | 94 10 | 97 28 |100 37 |103 46 {106 45 || 19 30 82710 | 285 131/988 96 1191 39° 9451 | 98 1 1201 10/104 10 |) 19 20 MOO Soo n7 | aso 4 isons |' 92°99 |995030 1998137 | 101 38 i) 19 10 77.40 | 80 41 | 83 42] 86 47 | 89 52] 92 58/ 96 4/99 51119 O Geen |) 78010 | s8ieos | 84096 | 87097 |-9043T 193 34 196135 || 18 50 HOMO) Mioe29 Nie pours | 2825 | So 2 |\Nssies (Toni 4 |i94al 4 | ts 40 71 28 | 73 46 | 76 53 | 79 46 | 82 39 | 85 37 | 88 35] 91 34] 18 30 COm2G. |e en yas | 77227 | Oe (TST | BEUG | 289-4 11 Bs 20 67 26 | 70 27 | 72 98 | 75 12 | 77 56 | 80°48 | 83 40 | 86 36 | 18 10 boe26 |eesnne | Oris | Teea7 | 7Hls6 | 17895 | "BL 14 | 84 o8 | 1s «0 Horizontal Argument, Declination.— Vertical Argument, Hour Angle. 292 OO Cc © lo ole ile 2) =~j “I =F TIT 7 oo wo 9°7862 9°8074 9°8286 9°8494 98702 9°8898 9°9094 9°9278 9°9462 9°9638 9°9814 9°9980 0:0146 0°0298 0°0450 0°0593 00736 0°0867 0:0998 071118 0°1238 071345 0°1452 0°1543 0°1634 01714 071794 071860 071926 0°1979 0°2032 0°2070 0°2108 0°2126 0°2144 0°2154 0°2164 LOGARITHM OF Tan?Z FOR LAT. 55° 56’. 9°7623 9°7818 9°8013 9°8199 9°8386 9°8567 9°8747 9°8917 9°9087 9°9247 9°9406 9°9559 99711 9°9849 9:9987 00119 0°0250 0:0369 0°0489 0°0598 0:0707 0:0805 00902 0:0986 0°1070 01141 0°1211 071272 071332 0°1379 0°1426 01462 0°1497 0°1513 0°1528 0°1538 01548 © 9°7384 9°7562 9°7740 9°7905 9°8070 9°8235 9°8400 9°8556 9°8712 9°8855 9°8998 9°9137 9°9276 9°9403 9°9530 9°9647 9°9764 9°9872 9°9980 00078 0°0176 0°0264 0°0352 0°0429 0°0506 0°0567 0°0628 0:0683 0°0738 00779 0°0820 0°0853 0:0886 0:0899 0°0912 0:0922 0°0932 9°7197 9°7352 9°7507 9°7654 9°7801 9°7944 9°8086 9°8226 9°8365 9°8492 9°8618 9°8739 9°8861 9°8973 9°9085 9:91:89 9°9294 9°9389 9°9484 9°9571 9°9658 9°9735 9°9812 o:98i72 99947 0:0002 0°0056 0°0105 00154 0°0191 0°0228 0°0257 0:0286 0°0299 0°0311 0°0318 0°0324 97010 9°7142 97274 9°7403 9°7532 9°7652 97772 9°7895 98018 9°8128 9°8238 9°8342 9°8446 9°8543 9°8640 9°8732 9°8824 9°8906 9°8988 9°9064 9°9140 9°9206 9°9272 9°9330 9°9388 9°9436 9°9484 9°9527 9°9570 9°9603 9°9636 9°9661 99686 9°9698 2:97.10 9°9713 9°9716 9°6882 9°6988 9°7093 9°7200 9°7307 9°7404 9°7500 9°7600 9°7700 9°7790 9°7880 9°7967 9°8054 9°8136 9°8217 9°8292 9°8366 9°8433 9°8500 9°8564 9°8628 9°8682 9°8735 9°8785 9°8834 9°8874 9°8913 9°8949 9°8985 9°9012 99038 9°9061 9°9084 99090 9°9096 9°9101 9°9106 9°6754 9°6833 9°6912 96997 9°7082 9°7155 9°7228 9°7305 9°7382 9°7452 9°7522 9°7592 9°7662 9°7728 97794 9°7851 9°7908 9°7960 9°8012 98064 9°8116 9°8157 9°8198 9°8230 9°8280 9°8311 9°8342 9°8371 9°8400 9°8420 9°8440 9°8461 9°8474 9°8479 9°8484 9°8490 9°8496 86° 9°6686 9°6739 9°6792 9°6849 9°6905 9°6956 9°7007 9°7059 97112 9°7157 9°7203 9°7251 9°7299 9°7345 9°7390 87429 9°7468 9°7505 9°7541 9°7576 9°7611 9°7640 9°7669 9°7696 9°7722 97744 9°7765 9°7785 9°7805 O57 819 9°7834 9°7849 9°7858 9°7861 9°7864 9°7867 97871 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. 72° PARALLACTIC ANGLE FOR LAT. 55° 74° 76° 78° 68° 5270 70° 180) 72°57" 66 20 | 68 11 | 70 45 63 48 | 66 41] 68 32 61 48 | 63 57 | 66 22 59 47 | 61 50] 64 11 57 49 | 59 49 | 62 4 55 50 | 57 48 | 59 56 53 55 | 55 46 | 57 20 51 59 | 53 44 | 55 44 50 3] 51 42 | 53 38 48 7 | 49 42] 51 32 46 14 | 47 44 | 49 30 44 20} 45 46 | 47 27 42 27 | 43 49 | 45 29 40 33 | 41 52] 43 30 38 41 | 39 55) 41 26 36 48 | 37 58} 39 21 34 57 | 36 3 | 37 22 33 5 | 34 8 | 35 22 31 15 | 32 13 | 33 23 29 24) 30 18 | 31 24 27 38 | 28 32 | 29 29 25 52 | 26 46 | 27 34 23 58 | 24 46 | 25 33 22 3 | 22 46] 23 31 20.10 | 20 47 | 21 30 18 17 | 18 48 | 19 28 16: Bie |) ¥6 SEC" L733 14 37 | 15 4) 15 37 12 46°) 13 7+) 13°38 £O. Soe Et LO 139 ) § 9 22 9 47 7 22 7 34 7 54 5 31 5 42 5 52 3 40 3 50 3 50 1 50 1 55 1 55 0 0 O70 0 0 78° 25/ 76 4 73 42 71 24 bo Re oO — oo - S) S) 56’. 84° 81° 14’ 78 49 76 24 i= i=) bb oT Horizontal Argument, Declination.—Vertical Argument, Hour Angle. 293 > eS ee ww ow co oo Oo OO bo bS bo bo bo bo ll ceil aoe Cr St Or LOG TAN°Z AND PAR. ANGLE (y) FOR LAT. 55° 56’. Dect. 88° Loe Tan?Z Dect. 88° 1] bo kb bo bo bk w M. @ GC 0 GO oO Oo NI an 7 lor mor mer) Cm © Dect. 88° | Dect. 88° Loe Tan?Z 7 96618 Sie Pee” 9°6645 84 34 9°6672 82 6 9°6700 Ce 27 96728 76 48 96757 74 30 9°6786 ha V2 96813 69 44 96840 67 16 96862 64 49 9°6884 62) 22 96910 59 56 9°6936 57 30 96961 55 5 96986 52 40 9°7007 50 14 9:7028 47 48 9°7049 £5 25 9:7070 A 9°7088 40 3 9°7106 Bie 9°7123 35 9 9:7140 Jon 26 9°7152 31 2 97164 238" 3s 97176 26 13 97188 23 48 9°7199 Zl 26 9°7210 19 4 9°7219 16 43 9°7228 ae 22, 9°7237 12 3 9:72.42 9 44 9°7242 7 5 9°7244 A» Did. 97244 2p 12 9°7246 0 0 a ey El CONTAINING | THE LOGARITHM OF TAN’ ZENITH DISTANCE AND THE PARALLACTIC ANGLE FOR LATITUDE 57° 30’, AND DECLINATION 40° ro 90°. VOL, XXXIIJ. PART II. 2U 296 ea ee oo © OO qo oo co or St Or bo bo bo M. | 40° 89979 9°0023 9°0136 9°0324 9°0570 90879 9°1225 9°1607 9°2016 9°2444 9°2881 9°3329 9°3775 9°4227 9°4676 9°5123 9°5566 9°6008 9°6445 9°6871 9°7298 9°7718 9°8138 9°8551 9°8962 9°9369 9°9778 0°0184 0:0587 0:0988 0°1388 071788 0°2190 0°2593 0°2994 0°3398 0°3805 LOGARITHM OF Tan? Z FOR LAT. 57° 30’. 8°8795 88864 88993 89168 8°9529 89902 9°0319 9°0758 9°1220 9°1701 9°2189 92679 9°3154 9°3649 9°4126 9°4598 95064 9°5522 9°5955 9°6364 9°6850 Sar eit rp 9°7705 9°8123 9°8536 9°8946 9°9353 9°9757 00158 0°0554 0°0951 0°1343 0°1736 0°2131 0°2525 0°2915 03311 44° 87611 8°7705 87849 88012 88488 88924 89393 8°9909 9°0424 9:0958 9°1496 9:2029 9°2552 9°3069 9°3576 9°4071 9°4558 95036 9°5505 9°5953 9°6402 9°6840 9°7271 9°7694 9°8111 9°8522 9°8929 9°9331 99728 0:0122 00513 0:0898 071282 0°1669 0°2055 02431 0°2817 46° 8°6046 86159 86376 86706 8°7229 87784 88362 88973 89582 9°0192 90792 9°1379 9°1949 9°2505 9°3044 9°3566 94075 9°4573 95036 95472 9°5976 9°6422 9°6860 9°7287 97706 9°8119 9°8526 9°8927 9°9322 9°9712 0:0099 0:0480 0°0858 0°1235 0°1612 0°1980 0°2354 48° 84480 84612 84904 85400 85969 86644 8°7330 88042 8°87 40 89425 90087 90728 9°1345 9°1939 9°2511 9°3059 93592 9°4109 9°4608 95087 9°5550 9°6004 9°6448 9°6879 9°7301 o7716 9°8123 9°8522 9°8915 9°9301 9°9684 0°0062 0:0434 0:0801 01169 0°1529 0°1890 50° 82073 82379 82823 8°3613 8°4437 85316 8°6226 87086 87907 88686 8°9425 9°0024 90790 9°1422 9°2025 9°2610 9°3151 9°3623 9°4196 9°4684 9°5156 95617 9°6066 9°6500 9°6923 9°7339 97744 9°8142 9°8533 9°8916 9°9293 99715 0:0031 00391 0:0750 071101 0°1453 79667 80145 80743 81826 82916 8°3987 85121 86129 8°7073 8°7945 8°8763 8°9520 9°0234 9°0903 9°1538 9°2159 9°2709 9°3247 9°3784 9°4279 9°4761 9°5229 95683 9°6120 9°6546 9°6961 9°7367 9°7762 9°8150 9°8529 9°8902 9°9267 9°9628 99981 0:0332 0°:0673 0°1016 54° 7°4039 T5547 T7754 T9754 81437 82885 8°4223 85384 86436 8°7383 8°8258 89061 89799 9°0495 91147 9°1723 9°2345 9°2890 9°3435 9°3935 9°4421 9°4908 9°5344 95781 9°6205 9°6619 9°7020 9°7412 9°7795 9°8171 9°8549 9°8896 9°9249 99596 9°9938 0:0272 0:0604 Horizontal Argument, Declination,—Vertical Argument, Hour Angle. PARALLACTIC ANGLE FOR LAT, 57° 30’, 29 40° 42° 44° 46° 48° 50° 52° 54° 39 44 | 40 88 | 41 31 |.42 34 | 43 35 | 44 47 | 45 58 | 47 21) 18 0 Horizontal Argument, Declination.— Vertical Argument, Hour Angle. 298 LOGARITHM OF Tan? Z FOR LAT. 57° 80’. H. M. H. M. 6 O || 0°3805 | 0°3311 | 0°2817 | 0:°2354 | 0°1890 | 0:1453 | 071016 | 0:0604 | 18 O 6 10 || 04217 | 0°3708 | 0°3200 | 0:2724 | 0°2249 | 0°1801 | 0°1354 | 0:0932 | 17 50 6 20 || 04627 | 04103 | 0°3579 | 0°3091 | 0°2603 | 0°2144 | 0°1685 | 01252 | 17 40 30 || 05039 | 0°4498 | 0°3957 | 0°3455 | 0°2952 | 0:°2482 | 0:2011 | 01567 || 17 30 6 40 | 05458 | 04900 | 0°4340 | 0°3822 | 0°3304 | 0°2821 | 02337 | 01881 || 17 20 6 50 || 05879 | 05299 | 0-4720 | 0°4185 | 0°3651 | 0°3154 | 02655 | 02188 | 17 10 7 0 | 06305 | 0°5703 | 0°5101 | 0-4549 | 0°3998 | 0°3485 | 02973 | 02493 | 17 0 é 7 10 | 06729 | 06104 | 05478 | 0°4908 | 0°4338 | 0°3810 | 0°3283 | 02789 | 16 50 7 20 || 0°7163 | 0°6512 | 05861 | 05270 | 04680 | 0°4136 | 0°3592 | 0°3084 || 16 40 7 30 | 0°7603 | 0°6925 | 0°6246 | 0°5633 | 0°5020 | 0°4459 | 03899 | 0°3376 || 16 30 7 40'|| 0°8045 | 0°7336 | 0°6628 | 0°5993 | 0°5359 | 04779 | 0-4200 | 03662 || 16 20 7 50 || 08489 | 0°7749 | 0°7008 | 0°6350 | 0°5691 | 05093 | 04494 | 03941 | 16 10 7 ® 8 0 || 08942 | 0°8167 | 0°7392 | 0°6708 | 0°6024 | 0°5407 | 0°4789 | 04219 | 16 O a 8 10 || 0:9393 | 0°8581 | 0°7770 | 0°7060 | 0°6349 | 0°5711 | 0°5072 | 0-4487 | 15 50 . 8 20 | 0°9855 | 0:9004 | 08153 | 0°7414 | 0°6676 | 06016 | 05357 | 04754 | 15 40 “a 30 || 10312 | 0°9420 | 0°8529 | 0°7761 | 0°6993 | 0°6312 | 05630 | 05010 | 15 30 10784 | 0°9847 | 0°8910 | 0°8111 | 0°7313 | 0°6608 | 0°5904 | 0°5267 | 15 20 50 || 11250 | 1:0266 | 0°9283 | 0°8453 | 0°7622 | 0°6895 | 0°6167 | 05512 | 15 10 [ooo oe 2) i >) ofl O | 11722 | 1:0688 | 0:°9655 | 0°8792 | 0°7928 | O'7177 | 06426 | 05754 || 15 0 12184 | 1:1099 | 1:0014 | 0°9117 | 0°8221 | 0°7446 | 0°6671 | 05980 | 14 50 20 || 12648 | 1:1509 | 1:0370 | 0°9439 | 0°8507 | 0°7708 | 0°6909 | 06201 | 14 40 © © — oO 9 30 | 1°3112 | 1:1916 | 10721 | 0°9754 | 0°8787 | 0°7964 | 0°7141 | 0°6414 | 14 30 9 40 | 13574 | 1:2320 | 1:1067 | 1:0056 | 0°9059 | 0°8211 | 0°7364 | 06620 | 14 20 9 50 | 14020 | 1:2705 | 1:1391 | 1:0352 | 0°9313 | 0°8442 | 0°7571 | 06810 | 14 10 10 O | 14472 | 1:3094 | 11716 | 1:0642 | 0°9568 | 0°8673 | 07778 | 06998 | 14 O 10 10 | 14898 | 1:3460 | 1:2022 | 10910 | 0°9798 | 0°8882 | 0°7966 | 0:7169 | 13. 50 10 20 | 15314 | 1°3809 | 1:2304 | 1:1160 | 1:0016 | 0°9076 | 0°8136 | 0°7329 | 13 40 10 30 | 15690 | 1:°4131 | 1:°2572 | 1:1392 | 1:0212 | 0°9253 | 0°8294 | 0°7470 | 13 30 10 40 || 16064 | 14439 | 1:°2814 | 11608 | 1:0402 | 0°9424 | 0°8446 | 0:7606 || 13 20 10 50 || 16386 | 1-4712 | 1:3038 | 11798 | 10558 | 0°9563 | 0°8568 | 0°7717 || 13 10 11 O | 16702 | 1-4972 | 1:°3242 | 11977 | 10712 | 0°9698 | 0°8684 | 0°7826 | 13 0 11 10 || 16952 | 15183 | 13414 | 1:2127 | 1:0840 | 0°9813 | 0°8786 | 0°7917 || 12 50 11 20 || 1°7192 | 15377 | 1°3562 | 1:2254 | 1:0946 | 0°9907 | 0°8868 | 0°7993 || 12 40 11 30 | 1°7362 | 15518 | 1:3674 | 1:2353 | 1:1032 | 0°9981 | 0°8930 | 0°8048 || 12 30 11 40 | 17506 | 15633 | 1°3760 | 1:2426 | 1:1092 | 1:0034 | 0°8976 | 0°8091 | 12 20 11 50 | 1°7572 | 15688 | 1°3804 | 1:2462 | 11120 | 1:0060 | 0°9000 | 0°8110 || 12 10 12 0 | 1°7612 | 15721 | 13830 | 1:2485 | 11140 | 1:0079 | 09018 | 0°8126 | 12 0 Horizontal Argument, Declination.—Vertical Argument, Hour Angle, aAaD Cmaonm wmnmnnmn NAW WAY © Oo oO 40° 42° 44° 46° 39° 44’| 40°38’| 41°31’| 42°34’ 39 4] 39 54] 40 44] 41 43 38 21 | 39 9] 39 56) 40 51 323% | 38°22 | 390)6 | 39058 36 50 | 37 33 | 38 14 | 39 3 36 2] 36 41] 37 20| 38 6 35 12 | 35-49 | 36 25 | 37 8 34 20 | 34 55 | 35 29 | 36 9 33 27 | 33:59 | 34 30] 35 8 32 32 | 33 2] 33 31] 34 6 31 35 | 32 3] 32 30 | 33 3 30 36 | 31 2] 31 27 | 31 58 29 36 | 30 0| 30 23 | 30 52 28 35 | 28 57 | 29 18 | 29 45 27 32 | 27 52 | 28 12 | 28 37 26 28 | 26 46 | 27 5 | 27 27 25 22 | 25 39 | 25 55 | 26 17 24 15 | 24 30 | 24 46 | 25 5 23 8 | 23 22 | 23 36 | 23 54 21 57 | 22 10 | 22 22 | 22 39 20 46 | 20 58 | 21 9] 21 24 19 34} 19 45} 19 55} 20 8 18 21 |] 18 31] 18 40] 18 52 PL2S | Lhd) | y7L24 |) 1735 hoe ds | LEALO) | 16asH } Wee k7 14 38 | 14 45 | 14 52] 15 0 13 24 | 13 24 | 13 24] 13 34 W2CC4 | 12ek | Tayi | 12620 10 50 | 10 44] 10 38] 10 50 9 16 9 23 9 30 9 30 8 8 SP os San 8 8 6 22 6 36 6 50 6 57 5 30 5 21 5 12 belle 3 54 3 59 4 4 4 20 2 44 2 26 2048 2 26 1 22 1 13 1 4 Tet 0 0 01.50 0 0 0 0 PARALLACTIC ANGLE FOR LAT. 57° 30’. 299 48° 50° 52° 54° H. M. Agaan || Anew || 45° he’) A7° OV 18 42 41 | 43 49 | 44 56 | 46 14117 50 AAG | AQERO | 48054) 45°18 | iF 40 40 50 41 50 | 42 50 | 44 OO] 17 30 29.5 1 40048 | 41 45 | 49-52 || 2% 20 38 52 | 39 46] 40 39 | 41 411117 10 37 51 | 38 41 | 39 31 | 40 30/117 O 36 49 | 37 37 | 38 24 | 39 20/116 50 35 46 | 36 30] 37-14] 38 71|116 40 34 41 | 35 23 | 36 41] 36 531116 30 33 36 | 34 15 | 34 53 | 35 39 || 16 20 32 29 | 338 5 | 33 41 | 34 25 116 10 81590) 82055 | 32°29 | 838re% 1] té6 ko 26 20 | 26 52 | 27 30 | 28 8 | 28 53 | 29 38] 3033] 15 0 24 54 | 25 24 | 26 O| 26 35 | 27 17 | 27 59] 28 50] 14 50 23 29 | 23 56 | 24 29 | 25 21] 25 42 | 26 21| 27 8] 14 40 22 3 | 22 28 | 22 59 | 23 29 | 24 6 | 24 43 | 25 26 1 14 30 2003 || QUVLOr |) 2VLQS) || Bese) || Q20302|| 23063) | Vee || v4 FO 19, 1k | 19°32 | 19 58 | 20°23| 20 54] 21 95 | 22 2/14 Lo ecaR |) 18e34 || TSy4eid || Wed4s) || USEF |) 19240) GoI9T || 1 0 16 15 | 16 34] 16 58 | 17 12 | 17 40} 18 8 | 18 38] 13 50 14 48 | 14 58 | 15 20] 15 42 | 16 6] 16 30] 16 56/13 40 13:19 | 13°34] 13 49] 14° 4] 14 27 | 14.50 | 15 13 | 13 30 ise |) LWRES |) LoeQ |} LOBRE) |) Wes || WSan || P3634) || Vs 20 top |) 10r82) || Toeam | VHate) || Lue!) Weegee) || TAT4eR || 1B: Lo esse || S9nad) | sone || yoNsOr |! soNsi || s9Ra4g|| Noosa || 13 © e342 | GTR40) |) KGS || ESTA eTs.|; ES thay || eses2) || 12t 5O GND |) GOHe2) || UGens || e600) || OF 126i!) “Er44e!) taca7 | Is 40 AAs |) TAGS. |) T4736 |) 14036) e4aiedine |) yan54e |) CHL OS? || I 30 2008) || @anno) || Pens |! Sshlidd |) aseQar! (%280]) esse || 12 20 2349 |) BSsop |) 12342 |) sneser|) eau pea | eressi ||) 128 10 GeaGy |b ORIG? || BORAG!) TOFT5O} |) (O05)! OI Or], ) ie} ~T for) Or bo 30 || 0°0144 | 9:°9666 | 9°9187 | 9°8722 | 9°8258 | 9°7806 | 9°7354 | 9°6924 | 15 30 5 ‘ : 97416 | 9°6971 || 15 20 50 || 00415 | 9°9914 | 9°9413 | 9°8923 | 9°8433 | 9°7954 | 9°7464 | 9°7006 || 15 10 co coo rs (J (=) =) bo lo 2) vse eo} No} ~J ie} ase Sa No} oo oO re No} [e.2} lee) bo lor) Ne} (o2) Ls\) nse Ne} No) ~ [e2) [o2) bo 0 || 0°0544 | 0:0032 | 9°9520 | 9:9019 | 9°8518 | 9°8025 | 9°7532 | 9°7048 | 15 0 : p 9°7580 | 9°7079 || 14 50 20 || 00771 | 0°0239 | 9°9707 | 9°9184 | 9°8661 | 9°8144 | 9°7626 | 97111 || 14 40 © © —_ i=) oO So lor) Or io} oO i=) — eo te} No} Neo) for) — Ne} No} No} _ oO rs Ne} @ Or No} oO te) ez) So (2) Or 9 30 || 0°0877 | 0°0337 | 9:9796 | 9:°9263 | 9:°8730 | 9°8200 | 9°7670 | 9°7140 | 14 30 9 40 || 0:0979 | 0:0430 | 9:°9880 | 9:°9338 | 9°8795 | 98256 | 9:7716 | 9:7172 || 14 20 9 50 || 0°1072 | 0°0514 | 9°9956 | 9:°9405 | 9°8854 | 9°8304 | 9°7754 | 9:7196 | 14 10 10 0 || 0°1166 | 0:0598 | 0:0030 | 9:9473 | 9°8916 | 9°8355 | 9°7794 | 9°7227 | 14 0 10 10 || 0°1242 | 0°0671 | 0°0100 | 9:°9533 | 9°8966 | 9°8395 | 9°7824 | 9°7248 | 13 50 10 20 |} 0°1318 | 0°0740 | 0:0162 | 9°9587 | 9:9012 | 9°8434 | 9°7856 | 9°7271 | 13 40 10 30 | .0°1380 | 0:0799 | 0:0212 | 9°9633 | 9°9054 | 9°8468 | 9°7882 | 9°7287 || 13 30 10 40 || 0°1446 | 0:0857 | 0:0268 | 9°9682 | 9°9096 | 9°8506 | 9°7916 | 9°7308 || 13 20 10 50 || 0°1494 | 0°0900 | 0:0306 | 9°9715 | 9°9124 | 9°8528 | 9°7932 | 9°7318 | 13 10 11 0 || 071544 | 0°0946 | 0°0348 | 9°9751 | 9°9154 | 9°8553 | 9°7952 | 9°7334 | 13 0 11 10 || 0°1586 | 0:0980 | 0:0374 | 9°9777 | 9:9180 | 9°8576 | 9°7972 | 9°7348 || 12 50 11 20 || 0°1616 | 0°1010 | 0:0404 | 9°9804 | 9:°9204 | 9°8596 | 9°7988 | 9:7359 || 12 40 11 30 || 0°1644 | 0°1034 | 0°0424 | 9°9822 | 9:°9220 | 9°8608 | 9°7994 | 9°7365 || 12 30 11 40 || 071666 | 0°1055 | 0°0444 | 9:9838 | 9°9232 | 9°8620 | 9°8008 | 9°7375 || 12 20 11 50 || 0°1674 | 0°1062 | 0:°0450 | 9:9844 | 9°9238 | 9°8623 | 9°8008 | 9°7375 || 12 10 12 0 || 0°1678 | 01067 | 0°0456 | 9°9849 | 9°9242 | 9°8627 | 9°8012 | 9°7377 | 12 +0 Horizontal Argument, Declination.—Vertical Argument, Hour Angle. co CO CO lo oo oe 2} Tu auvw lor or mor) Now eR ie} PARALLACTIC ANGLE FOR LAT. 57° 30’. Horizontal Argument, Declination.—Vertical Argument, Hour Angle. VOL. XXXIII. PART II. 76° 78° 80° 82° 69° 12’) "72° 597) T4845" 07a” 438° 67 51) GO 47 |. 72028 | 75 22 65) OL) G7 36 | 7OUL2 P73 2 62 57 | 65 28 | 67 59 | 70 44 60 53 | 63 39 | 65 45 | 68 25 58 52 | 61 12 | 63 33 | 66 10 56.51 | 59 6} 61°21 | 63 54 54ANSDE 57 (2 |. 59002) -861 39 52 52) 54 58 | 57 3] 59 25 50 53 | 52 54 | 54 54] 57 11 48 56 | 50 51 | 52 46] 54 58 46 59 | 48 49 | 50 39 | 52 47 45 2 | 46 47 | 48 32 | 50 36 43 7 | 44 48 | 46 28} 48 25 41 11 | 42 47 | 44 22 | 46 14 39 16) 40 47 | 42 18 | 44 5 37 21 | 38 47 | 40 13} 41 56 35 27 | 36 49 | 38 10 | 39 47 33 35 | 34 52 | 36 91 37 40 31 43 | 32 54] 34 5 | 35 33 29 48 | 31 4] 32 21 | 33 34 27 54) 28 58 | 30 11] 31 17 26) 2M ai tl | 2iVad brag 12 2410] 25 4] 25 59 | 27 4 22 14] 23 6] 23 58 | 26 O 20 28 | 21 13 | 21 57 | 22 52 18 36 | 19 16 | 19 56 | 20 45 16 42 | 17 20) 17 58 | 18 41 14 56 | 15 29 | 16 2] 16 42 13 Oj 13 28 | 13 56 | 14 32 1020 | U-al | 0b2 Vila 23 9 10 9 34 9 58} 10 26 7 16 7 37 7 58 8 19 5 30 5 49 6 8 6 15 3 54 3 59 4 4 4 20 pay 2 0 2 2 2 15 0 0 0 0 O09 O FLO 84° 80° 40’ 78 16 75 52 73 28 Pe 6 68 46 86° 83° 46’ 81 78 19 53 7 WE 307 308 LOG Tan? Z AND PAR. ANGLE (7) FOR LAT. 57° 30’. Decu. 88° | Dect. 88° Dect. 88° | Dect, 88° Loe Tan 2Z n Loe Tan 2Z n Bese Mil ME H. M. H. M. 0 0 9°5403 T3830” | 0" 24 O 6 0 9°6100 86° 52’ 18 0 OpaLO 95403 in \o2 23 50 6 10 96134 84 22 17 +50 0 20 || 95404 174 48 23 40 6 20 96162 81 54 17 +40 On230 95404 172 8 23 30 6 30 96182 79 24 17 30 0 40 || 9°5412 169 28 23 20 6 40 96216 76 56 17 20 OnF0 95416 166 52 23 10 6 50 9°6236 74 28 17 +10 1 0 95420 164 16 23 100 7 0 9°6270 72 0 17 0 Ue) 9'5430 161 36 22 50 7 10 96298 69 34 16 50 1/320 9°5442 158 46 22 40 7 20 9°6324 67 8 16 40 1b aX0) 95458 156 10 22 130 7 30 9°6350 64 40 16 30 1 40 95472 153 34 22 20 7 40 9°6378 62 14 16 20 1 50 95486 151 2 22 10 7 50 96396 59 48 16 10 z 0) 9°5500 148 18 22 a0) 8 0 96426 DT 2g 16 0 oro 9°5512 145 42 21 ~50 8 10 96448 54 58 15 50 2120 95536 143 4 21 40 8 20 96468 52 30 15 40 20930 9°5550 140 28 21 30 8 30 9°6494 50 8 1b? “30 2 40 9°5572 Site 150 21 20 8 40 9°6526 47 40 15 20 27150 95596 Soe lle: 21 10 8 50 96544 45 16 15 10 3 0 95616 132 40 21 0) 9 0 9°6564 42 56 15 640 3) 1K®) 95632 130 6 20 50 9 10 96578 40 30 14 50 3. 20 95662 127 28 20 40 9 20 96596 38 6 14 40 3) oA) 9°5678 124 54 20 30 9 30 96610 35 40 14 30 3 40 9:5706 122 20 20 20 9 40 96628 33 #18 14 20 3. 50 9°5730 119 46 20 eto 9 50 96644 30. 56 14. 10 4 0 9°5758 the 12 20 O 10 0 96660 28 32 14 0 4 10 95780 114 38 19 50 10 10 9°6672 26 8 13 50 4 20 9°5808 112 6 19 40 10 20 9°6686 23 46 13 40 4 30 9°5836 109 32 19 30 10 30 9°6692 21 24 13 30 4 40 9°5866 106 58 19 20 10 40 96700 18 54 13° 20 4 50 9°5896 104 26 19 10 10 50 96708 16 30 13 10 5 0 95924 101 56 19 0 11 0) 9°6716 14 10 13 «+O 5 10 95952 99 24 18 50 11 10 96724 11 52 12 50 5 20 95980 96 62 18 40 ll 20 9°6730 9 26 12 40 5. 80 96010 94 22 18 30 11 30 96736 7 16 127 £30 5 40 9°6044 91 52 18) #20 11 40 9°6742 4 54 12 20 5 50 9°6072 89 22 18 10 lt 680 9°6742 2 44 123 200) 6 0 96100 86 52 18 0 12 0 96742 0 O 12 0 ( 309 ) XIV.—On a Class of Alternating Functions. By Taomas Murr, LL.D. (Read 7th March 1887.) A glance at the expression (a—a)(a—B)(a—yia—6) , -a)O— B)(b—y)(b—6) (a—b)(a—c)(a—d) (b—a)(b—c)(b—d) git= Mes Bem yie=e) . @—a)e—P\d—y)ad—8) (c—a)(c—b)(e—d) (d—a)(d—b)(d—c) is sufficient to verify the fact that it is symmetric with respect to a, b, ¢, d, and also with respect to a, B, y, 6. It is likewise, although not quite so evidently, an alternating function with respect to the interchange ie be ate a By 6}’ that is to say, if a and a be interchanged, and at the same time b and 8, ¢ and y, d and 6, the function is not altered in magnitude, but merely changes sign. With a little trouble, indeed, the expression can be transformed into (a+b+c+d) — (at+tB+y+0), or say Xa — La This alternating function is only one of a large class to which it is proposed here to direct attention. It may be looked upon as in a certain sense the generator of the other members of the class, because they are derivable from it by prefixing to each of its component fractions a symmetric function of the three variables which occur only once in the corresponding denominator, ¢.g., the symmetric function lc + bd +cd prefixed as a factor to the first fraction, the like function a¢+ad+cd prefixed to the second fraction, and so on. The various kinds of symmetric functions which may be used in this way as prefixed factors are best expressed in the form bm om or 1b B CCN il” Cece? ee dm dm a lad@ eal m, n, 7, having any three values chosen from 0, 1, 2, 3, 4; for example, the above-mentioned instance be +bd + cd is, in this form, | B23 | [B%la2 | * VOL, KXXfII.; PART It. DT, 310 DR THOMAS MUIR ON A The question then is—How can we by transformation set in evidence the fact that |bmera” | eo BYa—y\a—s) , |arerdt| (—a)(b—B)b—y)b—0) | b°cld? | (a—b)(a—e)(a—d) | acd? | (b—a)(b—c)(b—d) ja™brdr| (e—a)(e—B\e—y)(c—3) |ambre"| (d—a)\(d—6)(d—-y)(d—6) * | ald? |” (e—a)(e—b)(c—a) | a°b'? | (d—a)(d—b)(d—c) is an alternating function with respect to the interchange ey a Since [b%cld2} = Gbed) = (d—c)(d—d)(c—d), the fractions evidently have the same denominator, viz., (d—c)(d —b)(d—a)(e—b)(e—a)(b—a) , or G(a be d); so that, if we expand the original numerators in descending powers of a, of }, &c., the expression becomes aS [ =lorendr {ata + a Sa8—aSaBy +086} +|arcrd’ {b!—BSa+ VEaB—IDaBy +aB-ys} —|anbrdr |{ ct —ASa+ eSaB —cDaBy +aBy6} +\a"brer |{d!—d?2a+d?LaB—ddaBby+aBy6 } | : This, when the coefficients of 2a, a8, &c., are collected and condensed, i ST CCRNL aninerd!| — |anbrerd|Za + labrord?| Daf — jarbrerd|XaBy + jantrerd? | SaBy6 | ; and no farther simplification is possible until the special values of m, n, 7 are given. Taking in order the ten aufero sets of special values 0; POO, deseo tee ease, ee and denoting the whole ve by Fin, We see immediately that Fo1,2= ea, i i haa |abe2dt| — ja%bed| Da |, Post : : APSO. Jae] jad? | DaB |. Fo s= AGEow fi —|a Bed |Da + jared? | ZaB | ; CLASS OF ALTERNATING FUNCTIONS. 311 Bios a i; - Fe | abd | = ja°bietd |ZaBy |, Prac 7 - aa eay| lads |S - jar%eetd |Zai8y | Ph ges a ; : aaa [aXDickd®| SaB — |aXBictd!| ZaBy | , P29 age 7 : ee | atbee8dl 4 [abd |ZaBy6 |, Fi, 94= ee a —|albetd?| Xa + | alb%ctd?| ZaBy6 | ; i F;, 34> Ei(a, b, e, d) [ | ab3¢*q2 | La + | ab%ctd® | ZaBy6 | ? F,, 34— —| ab3ctqt | LaBy + | a*b®c*d® | DaByé | ‘ stad ae as f(a, d, ¢, aL Now, from the theory of alternants it is known that | a°blc2d$ | = Ea, b, c, d), | a°ble2dl* | = F4(a, 6, c, d)x Xa, | a°ble8dt | = F4(a, b, c, d) x Zab, | a°D7c%d* | = €2(a, b, ¢, d) x Zabc , | ab°c8d* |= €3(a, b, ¢, d) x Labed ; and thus it follows that Fo1,2=2a—2Za Fo, 1,3= 2ab— LaB , Fo1,4=2ab . La—LaP . 2a. Fo,2,3= Labe—LaBy , Fo,2,4=Dabe . La—LaBy . Xa,- Fo,3,4=Zabe . LaB—LaBy . Zab, Fy 2,3= Zabed—LaBy6 , Fi,2.4=Labed . La—LaGyd . Xa, Fy,3,4= Labed . La8—ZaByo . Zab, Fo 3,4=Zabed . LaBy—LaByd . Xabe , all the expressions on the right being manifestly alternating functions with respect to the interchange (Cg ae Y ) These expressions are seen to be the ten determinants of the matrix | 1 Ya YaB aBy XaGyd | tL Sa, Sab) Dale Dabed and consequently, if we represent this matrix by Gon On Oy Ty F% Sie Str Sy, wiSey Se 312. DR THOMAS MUIR ON A CLASS OF ALTERNATING FUNCTIONS. the results take the form Fo,1,2 = | %o9|5 Fo1,3 = | Soro Il Fo a4 = |Syo31, where the suffixes on the right are got by subtracting from 4 each of those omitted on the left. With the help of this notation, also, we can combine all the results in one statement, viz. :— Ifm, n, vr arranged in order of magnitude be any three of the values 0, 1, 2, 3, 4, and u, v arranged in order of magnitude be the remaining two, then |bmondr| (a—a\(a—BYa—yNa—6) , |arerd"| (B—a)(b—B)(b—y)(b—6) [Ota | (a—b)(a—)(a—d) | actd? | (b—a)(b—c)(b—d) |abnd”| (c—a\(c—By(e—y)e—6)_, lavbre"| (d—a)(d—B)(d—y)(d—8) |abld?|° ~~ (c—a)(e—b)(e—d) | able? | (d—a)\(d—b)(d—c) = | S4—uT4—v | ? where s and o are explained by the examples s,= Zab, o,= YaBy. The case where v=4 has been given by Sylvester, being the subject of his unsolved problem No. 2810 in Mathematics from Educational Times, vol. xlv. p. 129. Of course the foregoing results are not at all confined to two sets of four variables (a, b, ¢, d), (a, B, y, 8). Two sets of w variables have not been taken merely on account of inconvenience in writing. The typical term for the next case (two sets of jive variables) is |bmerdres| (a—a\(a—B)(a—y)(a—6)(a—e) | B°clde8 | ° (a—b)(a—c)(a—d)(a—e) where (m, ”, 7, s) is a set of four values taken from 0, 1, 2, 3, 4, 5. ( 313 ) XV.—Expunsion of Functions in terms of Linear, Cylindric, Spherical, and Allied Functions. By P. ALEXANDER, M.A. Communicated by Dr T. Muir. (Read 20th December 1886. ) The expansion of ¢(z) in terms of G,(z), G,(z), G(x), &c., connected by a given law, being of great importance in mathematico-physical investigation, every method of effecting this expansion must have some interest for scientists. I therefore proceed to propose what I think to be a new method, in the hope that it may prove to be useful. Many special expansions of this nature have been effected by Fourier, LEGENDRE, and others. After I had developed my method, my attention was called to two papers on this subject showing methods ef development of great generality. The titles of the papers are—Konie, J., “ Ueber die Darstellung von Functionen u. s. w.,” Mathematische Annalen, v. pp. 310-340, 1871; and Sonne, N., “Recherches sur les fonctions cylindriques,” &c., Mathematische Annalen, xvi. pp. 1-80, 1879. Konic, assuming that P(X +X) = Eo(t%q) « Go(%) + Fila). Gav) + F(%) . G(x) + &e. where G), G,, G,, &c., are an infinite series of functions of 2, connected by some given law, and also subject to the condition that when z is nearly equal to c, each of them is capable of expansion in ascending integral powers of («—c), beginning in the case of G,(«) with (w—c)’, proceeds to show that the coefficients F,(%>), Fy(x,), &c., are to be deduced from the following— Gy(c). Fy(@) =b(% +2); Ge). Fo (%) = Fo(@) - Go) + Fi@)Gr©@ 5 Go(c) - Fo (%p) = Fo(%) - Go (©) + Fi(@%)- Gi") + F(a). Ga") 5 &e., &e. SoNINE shows that S(¢+z) =A,(a).S,(v) —2{A,(a)8,(~)—A,(a)S,(x) + &e. } if the series is convergent, where §,(z) and A,(a) may be any functions what- VOL. XXXIII. PART IL 3A 314 MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. ever of x and a consistent with convergency, and A,(a), A,(a), A,(@), &c., and S,(x), S,(x), S,(x), &c., are connected by the following relations :— Axa) = — S[ada)], | dA, r Anu t27— An-1=0 : J and S= = , | q a dS, Sntit 2a —Sn-1= 05 J and hence An=(—1)"cosnA,.A and Sn=(—1)"cosnA .S,, where . ah ee and A, and A are operations defined by : d 7 COS A\= 7, and ; d zcos A bre | Konte’s method seems to be much more general than Soninz’s, as K6nta’s functions Gp, G,, G., &c., may be connected by any law, while Sonine’s functions Ay, Ay, Ay, &¢., are connected by one law only. But on the other hand, K6nie’s functions are limited by the condition that G,(«) must, when « nearly equals c, be capable of expansion in ascending integral powers of («—c) beginning with («—c)?, whereas Sonine’s functions are subject to no such condition. Both methods give the expansion of (x) in terms of J,(z), Jy(«), J.(x), &e., BEssEL’s functions. But neither of them give the expansion d(x) = AgJ Akyx) + Ard, (kx) + AQIS, (kx) + &e., where i), &, /., &c., are the roots of some equation of condition. The most general method of expansion I have seen is that of expansion in normal co-ordinates employed by Ray.eieH throughout his Theory of Sound, which is so satisfactory that had I become acquainted with it somewhat earlier, I would probably not have sought after the following method :— The general problem is to determine A,, Aj, Ay, &c., so that when possible p(x) = AyG (x) + AiGi(w)+A,G,(~)+&, . . . . () where G,, Gy, G., &c., are connected by some given law. MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. 315 The solution of this in all its generality has not yet been obtained, but in most of the particular cases which have been solved the method seems to be to operate on (1) with an operator O, such that 0,.G,,=0, Pe) Ps cvriealt> ay) except m=n. And, therefore, O,.6(2)=A,0,. Gar) _ On. $(@) ee ee, fe oe SE A,= 0, Ga) (3) Following this lead, I have found an operator of this nature in the case where G,, G1, G2, &c., are elementary solutions of the equation, (Cate: + sel! CA when g has the values g,, 91, g2, &c., derived from the condition ‘cil ieee et le ror 2) where 6 and o are operations which may have the forms b=X, +X.) +¥( 7.) + be, A) eee o=P,+P,(s-) +P qe) + de. Seb GB Heme 4a") 7) where X,, X,, X», &c., and Po, Pi, P:, &c., may be either constants or functions of wx. The operator I have discovered for the solution of this problem is— 0,= fo(8+9,)° } 2=a (8) The proof is as follows :— 0(8 +9)" Gm=F [Gn In O+9, °° —&e.|G,+G,}. - : (9) But from (4), 6 G2 = =—¢,,.G., a SG = 9 66 = 9.2.G., i é. G., a Im PCy Fy os Ong | &e. &e. y) 316 MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS, o(8+9,) 7G,,= 9) + a ga ma ge re &e. |G,,+ G,} = (G,--Gr) eae oG,, oG.. Oe Gelli, ee —~Im But from (5), [oG,] iS =0=[oG,,] BS Hence (11) gives— {o(8+,)"Gnboe=——— +0 | =0 if m is not=n + : (12) ah ifm=n ; Hence from (8), 4 On 62) * Opes a(S+9,) ~'6(«) 4 = | a(8+9,) 7G, a . . . . . (13) By the method of vanishing fractions, ations Sra ay Jn — 7s (m=n, =a) fd Tah ee i. 7 ay —Jm) (m=n, z= ll 1a ee Gn) f eae - aja Hence (13) becomes (14) =a which is the required solution. As new results, especially when very general, are liable to suspicion, I proceed to test this by a particular example whose solution can be otherwise found. Let EAs be pP+(A—1)p | (15) da} " «\da a pad ld wel * In the proof of (11) it has been assumed that y,, is less than g,. The same may be proved for Im greater than gy, by expanding (8+ g,)~-1 in the reverse order. MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. 317 and d ae Re kn 2 (16) An elementary solution of (4) is in this case G;= (« Apia (« Jn) ay rmeeners. < 5 elo) CO we find also AB- {p= CB. “GB 8° that the direction of the pressure applied at C must also pass through the same point O. Again, on comparing the pressure applied at A with the strain on AB, we find ABC AOC Deaton er ua tor eer whence ai NOC: y= ABE oetie voppmnn ie he. and it follows that the six expressions — BOA.AOC =~, COB.BOA -. AOC.COB OA. ay a? OB : Bia az cC . — VO ar 5 7p ABC.BOA =; ABC.COB 7, ABC.AOC ior TkBeiny x On e opoar tewer Remo he are all of equal value. On multiplying each of the first three by _A0.BO.CO BOA. AOC.COB INSTABILITY IN OPEN STRUCTURES. 323 we get the equalities 7a. BO.00 _ 5 CO.AO _—, AO.BO o (OR NOG eA BOS that is to say, the three external pressures applied at the points A, B, C balance each other just as if they had been applied directly to the point O. When computing the internal strains caused by given external pressures, the area ABC occurs in every case as a division; if, then, the three points were in one straight line, that is, if the area ABC were zero, the internal strains would become infinitely great, unless the applied pressures were all in the same line with them. Here we have the first and very well known example of instability in construction. If the point O be removed to a very great distance, the directions aA, bB, cC of the external pressures become parallel as in fig. 2. The intermediate pressure, in this figure /B, must be op- posed to the direction of the others, its intensity being the sum of those at A and C. The relation of the strain on AC to the external pressure at B is then given by the formula poe et TAK AO SOE ae BX so that if B were shifted along the line BX nearer to X, Fig. 2. the strain on AC would be augmented in the inverse ratio of the new to the former BX ; but the pressures wA, OB, cC, would still remain proportional to the lines XC, CA, AX. Were B brought actually to X the strains would become infinite. It is much to be regretted that, in lesson books on mechanics, the beginner is taught the properties of this impossible straight lever, without a hint of caution in regard to it. The strains on the arms, even that upon the fulcrum, are left out of view. In this way hazy notions are engendered ; the load at A is said to balance that at C, although both be pressing in one direction. The relative positions of four points, as A, B, C, D, fig. 3, are in general fixed by the lengths of the six lines AB, CD, AC, BD, AD, d BC joining them two and two; these form the boundaries of a solid, called in Greek tetrahedron, which may get the English name /ournih, shorter and quite as descriptive ; the potters call it crow/oot: it is the simplest of flat-faced D solids. vy \ As in the triangle pressures applied at the corners can L=5—N balance each other only when their directions meet in one Fig. 3. point; so, reasoning by what is called analogy, we might infer that, of four 324 EDWARD SANG ON CASES OF pressures at the corners of a tetrahedron balancing each other, the directions must all tend to a single point. But this inference does not hold good; it may be that no two of these directions meet at all. At each of the four points we have the equilibrium of four pressures, ' namely, the external pressure and the strains on the members meeting there. These strains can be computed when the direction and intensity of the applied pressure are known. Thus let us continue the direction dD of a pressure applied at D until it meet the plane of ABC in some point O, and let AO, BO, CO be drawn. We have then the equalities a) | DEC 2S DB Mine DO.ABC DA.BOC DB.COA DC. AOB’ The points A, B, C and O remaining as they are, if D were brought nearer to O, tle first of the above expressions would be augmented in inverse propor- tion to DO, and if D were brought actually to O, this term would become infinite, the strains DA, DB, DC also infinite and the structure impossible. When, in such an arrangement as fig. 3, the resistances at A, B, C are ina direction parallel to dD, their intensities are proportional to the opposite triangles, so that D_A_B_& ABC BOC” COA” AOB’ and thus the distribution of the pressure among the ultimate resistances is independent of the distance DO. In fig. 3 the point O is placed inside of the triangle ABC, and a pressure a applied in the direction dDO causes compression in all the three members, DA, DB, DC. In fig. 4 b O is placed outside of the lme AC, and, with Gc Cc = ee D = = D oO B B B A WN A Fig. 4. Fig. 5. Fig. 6. pressure in the direction dDO, the members DA, DC are compressed, while DB is distended. If here the point D were brought down to O, the structure would take the form of a plane tetragon ABCD, with its two diagonals AC and BD, as shown in fig. 5. Such a structure can offer no resistance to pressures inclined to its plane. If the point D were on the straight line AC, as in fig. 6, it might seem that INSTABILITY IN OPEN STRUCTURES. 325 the five distances DA, AB, BC, CD, DB would suffice to secure the straight- ness of ABC; but on consideration we perceive that the two triangles ABD, CBD are merely hinged upon the common line DB. In the cases of two, three, and four points, we have seen that the lengtia of every line joining them in pairs is needed for fixing the relative positions ; this rule does not hold for higher numbers. Thus, if a fifth point E be con- nected with three of the four corners of the tetrahedron ABCD, its relative position is determined, provided always that E be not in the plane of the three points with which it is joined; so that nine lines suffice for five points. The line joining E with the fourth point of the tetrahedron would be redundant. In all such structures, three of the points, as A, B, C in fig. 7, must each have four concurring lines, and the remaining two, D and E, only three; and if no four of the five points be in one straight line, the system is self-rigid. This rigidity will subsist although the two triple points D, E be in the same plane with any one pair of the quadruple ones, as in fig. 8, which is intended to show D, B, E, C as in one plane. The scheme then takes the E Fig. 7. Fig. 8. Fig. 9. appearance of a pyramid, having A for its apex, and the quadrangle DBEC for its base. Thus the flatness of a tetragon may be secured by connecting each of its corners with a fifth point not in the same plane. Moreover, the system still remains rigid although the points D and E be both in one plane with AB also. In this case DBE, the meeting of two planes, must be a straight line, as shown in fig. 9. Thus we see that, for the establish- ment of three points in a straight line, two auxiliary points must be introduced, with seven additional linear members. We have now got a possible straight lever DBE. In order to examine the Jaw of the balancing of pressures applied at B, D, E, we must trace out the strains on the various members, and their equilibriums at the five junctions, subject to the condition that there be no external pressure at A or at C. The result of this examination is, that the strains on the members are eliminated ; that the directions of the applied pressures must all pass through one point ; 326 EDWARD SANG ON CASES OF and that their intensities must be proportional to the sines of the opposite angles,—this result being independent of the positions of the auxiliary points A and C. Does it thence follow that we may omit those points altogether? Assuredly not ; for our whole investigation proceeded on the ground that each member transmits from its one end to its other end the strain with which it is accredited. Each additional point needs for its establishment three new linear members, so that in any self-rigid open structure, if 2 be the number of the points, there must be 52—6 linear connections; this formula failing only in the extreme case, 7=2, Hitherto we have been considering the self-rigidity of structures, and may now proceed to treat of the laws of stability in relation to the ground, taking first the case of a self-rigid structure to be kept firmly in position, In every case the support must be derived from points in the ground, which points necessarily form by themselves a rigid structure, so that our problem assumes the general form of “how to connect one rigid structure with another,” If / be the number of the points in the foundation, and n that of those in the supported structure, we have in all /+2 points in the compound, which must clearly be self-rigid. Hence the total number of linear members must essentially be 3/+3n—6. But of these 3/—6 are virtually included in the foundation, wherefore the number of the members above ground must in all possible cases be 3n. Of these, however, 32—6 are already included in the supported structure, and thus we arrive at the important general law, “ that the number of linear supports must be neither more nor less than six when the supported structure is self-rigid.” This most elementary of the laws of sup- port seems almost to be unknown, the enunciation of it takes even professional engineers by surprise. All our portable direction-markers, our theodolites, alt-azimuths, levelling telescopes, have to be supported above the ground at a height convenient for the eye. It is essential that the stationary part of each be firmly held; yet in every case, with not one exception in the thousand, our geodetical instruments are set upon three slender legs, diverging almost from a point. In such an arrangement the steadiness in direction is derived exclusively from the stiffness of the legs, which, however, are very flexible. The well-known result is, that any strain in handling the instrument, even the pressure of a slight breeze, deranges the reading. More than fifty years ago an instrument maker in London, Robinson by name, placed his beautifully made little alt-azimuths on a new kind of stand. INSTABILITY IN OPEN STRUCTURES. 327 He connected three points in the stationary part of the instrument with three points in the ground by means of six straight rods inclined to each other. In this arrangement, the stability in every direction is derived from the resistance to longitudinal compression, the flexure of the rods having an infinitesimally small influence. It takes essentially the form of the octahedron or sixnib, as in fig. 10; a self-rigid structure having six connected points. This beautiful Robinson stand keeps the alt- azimuth so firmly in position that, even during a heavy gale, the image of the moon may be seen to move without tremour across the cobwebs of the field-bar. Yet it has not been adopted by engineers and surveyors ; the exigencies of the photographer, however, have determined his recourse to it. It is not essential that the supports meet two and two as in this arrangement ; they may be quite detached, connecting six points in the supported Fig. 10. structure with six points in the ground; but in all cases they must be so dis- posed that any dislocation whatever would imply a change in the length of some of them. It might have sufficed merely to remark that this condition excludes the parallelism of the supports; but it is expedient to insist, seeing that, in the deplorable case of the Tay Bridge, the fabric was set upon two rows of upright columns. The opinion is still held that the effective base is equal to the whole breadth of such a structure, whereas the most casual examination may show that, no matter how broad the structure may be, its effective base is only that of a single column. When the superstructure is not rigid in itself, or indeed whether it be so or not, the entire number of the members above the ground must be thrice that of the supported points. If we attempt to do with fewer the fabric must fall ; if we place more we cause unnecessary internal strains. I hope in a subse- quent paper to treat of redundancy, meantime our attention may be confined to structures having the proper number of parts. If the supported points belong to one system they must be mutually con- nected, and, at the least, there must be as many of these connections as there are points, less one, wherefore the number of supports can never exceed twice the number of the points by more than one. Out of the endless variety of cases we may select one class for examination, that in which the supported points are connected so as to form a polygon, not necessarily all in one plane. The number of the connections being already n, 328 EDWARD SANG ON CASES OF that of the supporting members must be 2”, which may rest on 2” separate points in the ground, but which may be brought together in pairs or otherwise. If they be placed in pairs there are as many supporting as supported points. Robinson’s octahedral stand shows this arrangement when there are three supported points ; we shall now take the case when four points are supported from four points in the ground, as in fig. 11, where the connected points A, B, C, D are shown as supported by the eight members AE, EB, BF, FC, CG, GD, DH, HA. In general, that is when there is no regularity, such a structure contains all the elements of stability. The positions of the foundation points being known, if the lengths of the twelve members be prescribed, we shall have twelve equa- tions of condition whereby to compute the twelve co-ordinates of the four points A, B,C, D. Or, viewing the matter from the mechanical side, external pres- sures applied at A, B, C, D may be resolved into their elements in three assumed directions, 2, y, z, and so may the stresses on the various parts ; there must be equilibrium at each of the points, in each of the three directions, and so again we have twelve equations whereby to compute twelve unknown quantities. The algebraist at once perceives that the resulting divisor (or determinant as it is called) may happen to be zero, in which case the stress becomes infinite ; that the dividend may be zero, showing that the particular member has no strain upon it; or even that both the dividend and the divisor may be zero at once, showing the structure to be inde- terminate. But such investigations dis- tract the attention from the objects under consideration to their mere representative symbols, and do not carry intellectual conviction along with them. ‘Their true and highly important office is to deter- mine accurately the various _ stresses, thereby enabling the constructor to ap- portion the strengths of the various parts. The determinateness, that is the Fig. 11. stability, of a structure typified by fig. 11 ceases when we introduce symmetry or even semi-regularity. Let, for example, the figures EFGH, ABCD be rhomboids, having their middle points O and P INSTABILITY IN OPEN STRUCTURES. 329 in the same vertical line, in which case the opposite members are of equal lengths, AE to CG; EB to GD, and so on. Such a structure is clearly instable. Since the lencths HA, AE are fixed, the point A must be on the circum- ference of a circle, having HE for its axis of rotation; and similarly for the points B, C, D. If now we suppose the point A to be pushed inwards, the members AB, AD will push the points D and B outwards, and, consequently, C will move inwards by exactly as much as A; the structure will adapt itself perfectly to its new position. In truth, we have here not twelve, we have only eleven data; for, if one of the connections, say CD, were removed, and the structure thus made obviously mobile, the distance CD would yet remain always equal to AB; that distance cannot be reckoned among the data. So much for the geometrical mobility ; let us examine the strains. Any horizontal pressure at A is decomposable into two,—one in the direction AB, the other in the direction AD. The stress AB transferred to the point B may again be decomposed into two; the one of these in the plane EBF parallel to EF is completely resisted at E and F by the stresses EB, EF, but the other, perpendicular to EF, meets with no resisting obstacle ; it and the corresponding pressure at D may be counteracted by extraneous pressure there, or by a single pressure applied at C, equal to the pressure at A, and in the same direction with it. Thus the distortion of the fabric by an eastward pressure at A is prevented by a like pressure applied at C, not westward, but eastward also ; in respect, however, to the strains on EB, BF, HD, DG, the effects of these counteracting pressures are cumulative. These considerations would seem to warrant the conclusion that all struc- tures of this class are necessarily unstable ; however, before venturing to accept of this conclusion, it may be prudent for us to inquire whether the arguments on which it is founded be strong enough to bear such a weighty superstructure. Now the chief argument was that the longi- tudinal stress on AB, acting at B, tends to E Plan. turn the triangle EBF on EF as an axis; but this tendency exists only so long as AB is out of the plane EBF, and ceases whenever AB comes to be in that plane ; in other words, when- ever AB is parallel to EF. Hence it follows that structures, represented in plan by fig. 12, having the two rhomboids ABCD and EFGH placed conformably, are rigid. The conclusion was not absolutely general. Though every rhomboid be not a rectangle, every rectangle is a rhomboid, and we might hastily thence conclude that these remarks concerning rhom- boidal structures may be at once extended to rectangular ones. But we have VOL. XXXIII. PART II. | 3 C Fig. 12. 3930 EDWARD SANG ON CASES OF just come from seeing an example in which peculiarity precludes generalisation, and thus it is expedient for us to examine specifically the case of rectangular structures. The examination at once shows that the remarks made in regard to figs. 11 and 12 apply when the rhomboids pass into the form of rectangles ; but the rectangle is symmetric, while the rhomboid is not so; the arrangement of the diagonals, as shown in these figures, is unsymmetric, and it thus remains for us to inquire into the laws of symmetry. If, as shown in fig. 13, the rectangle ABCD be placed vertically over and conformably with EFGH, the arrangement is symmetric; it has already four out of the requisite twelve mem- bers, and the eight supports remain to be placed. Fig. 13. These may be inserted symmetrically as the eight diagonals, HA, AF, FC, CH; DE, EB, BG, GD, but then the struc- ture becomes perfectly mobile; the points A B and C move towards or from the central axis, while B and D move from or towards it. In any other symmetric arrangement the four corner parts, EA, FB, GC, HD, must appear, leaving four members yet to be distri- buted. If one of these be placed as the diagonal AF’, symmetry requires also DG, BE, and CH, as shown in fig. 14. — F The insecurity of this arrangement is ob- vious at a glance ; no more need to have been “ said about it, but for its adoption in the & scheme for the central towers of the proposed Forth Bridge. It presents two instances of that most vicious arrangement, the flattened tetrahedron ; vicious because, while incapable of resisting any pressure not directed in its own plane, such a structure as EABF con- verts any twisting pressure into indefinitely exaggerated stress. It also presents two attempts to determine the shape of a quad- rangle by the lengths of the four sides. Were Fig. 14. the three open figures, EFBA, ABCD, DCGH, replaced by flat rigid plates, we should have, turning on the four parallel hinges, EF, AB, DC, HG, a most familiar example of instability. A INSTABILITY IN OPEN STRUCTURES. Jol Among open structures built on a rectangular base, instability is not con- fined to those with rhomboidal tops; for if, as in fig. 15, the triangle HAE be set up equal to GCF and EBF to HDC, so that Pie AC may be parallel to EF and BD parallel to E F is movable. Thediagonals AC 9 [0 8-7 FG, the structure is movable e diagon and BD may be on one level and so cross each = ley, other, or the one may pass above the other at a a distance on the plumb line OP. Since the a G three lines, AO, OP, PB, are mutually perpen- Fig, 15 dicular— AB?= AO?+OP?-+ PB? and CD?=C0?2+ O0P?+PD?; wherefore AB?+CD?=AO?+ OC?+ BP?+ PD?+ 2.0P?. But the sum of BC? and AD?’ is equal exactly to the same quantity, and con-. sequently AB2+CD?=BC?+DaA?, so that one of these four is deducible from the remaining three ; there are then only eleven data in this structure, instead of the twelve needed for rigidity. But it is to be observed that a dislocation must change the horizontality of AC and BD, so that the mutability may be only instantaneous, as in the case of maximum or minimum. Passing now to the case of five supported points, we may remark that, by the introduction ofa fifth point, a symmetric rigid structure may be built on a rectangular base. Thus, if we place, as in fig. 16, the rhombus ABCD vertically over the rectangle EFGH, and complete the construction as in fig. 11, there results a symmetric structure, which, like all those of the same class, is changeable. On assuming, however, a point Z in the vertical axis of the system, and connecting it with each of the points A, B, C, D, we get a fabric both symmetric and rigid. The rigidity is confirmed thus :—If, suppos- ing Z and its connections to be away, the points A and C be brought nearer, B and D would move apart ; now, in virtue of the connections AZ, ZC, Meek the shortening of AB would cause Z to rise, while, in virtue of the connections, BZ, ZD, the widening of BD would bring Z down ; the opposition of these two tendencies keeps Z in its place. 332 EDWARD SANG ON CASES OF Here, in order to support five points, sixteen members are conjoined, and yet there does not seem to be any redundancy ; moreover, the arrangement is quite symmetric. The equilibrium at each point gives rise to three equations — of condition, and these fifteen equations cannot possibly serve to determine sixteen strains. But if we apply a pressure at any one of the five points, the fabric resists it, the various members are strained somehow, the law of equa- tions notwithstanding. The explanation of this paradox may afford an instruc- tive exercise to the student. When the five points are arranged in the corners of a pentagon, each being carried by two supports, as shown in plan by fig. 17, the structure is rigid, pro- vided the polygons beconvex. Of this we easily convince ourselves by supposing one of the con- nections, say EA, to be removed, and by ex- amining the motion of the link system thus left. The point A can move only in a circle, having KF for its axis; let A be moved inwards, the member AB will then cause the triangle FBG to turn outwards on FG as a hinge; BC will draw C inwards, CD will push D outwards, and lastly, DE will draw E inwards; wherefore the distance AE will be shortened, and the member AE can be replaced only when the structure is brought back to its former position. Following this line of argument one step further, we see that in the case of a hexagon the first and last points would move, the one outwards, the other inwards, and that so the distance might remain unchangea. When the hexa- gons are semi-regular or halvable, the distance remains absolutely unchanged, and the structure is indifferent as to position. This va same remark applies to all polygons of an even number aa of sides. If, however, the upper and lower polygons be placed / conformably, as in fig. 18, the structure is rigid, whether a the number of supported points be even or odd. % 40 Cistii, Brongt. . : ; F =| Rosae2tl| ax 41 Bucklandii, Brongt. . ‘ ‘ spare Xe x 42 pteroides, Brongt, . : ‘ GCP 3 Ware | Fen Se" |e Ba Xe 43 crenulata, Brongt. . : ; Fema) w a }o IN SS) ea Sy | x i} x x x x x x d x a x x x x x x Cc x Cc x x x 2 x lae x a x x x x x x c x c x x d x x x x x x x x x c x x x x x a x x d x x Cc x x x b x x x x x x x x c x x x b x b x c x c x x q x x x x x x PS x x x x d x x x x x x x x x x a x x x x x b x x x Cc x x x x x x x x x 3 0 408 MR ROBERT KIDSTON ON THE FOSSIL FLORA OF THE A summary of the results brought out by these columns shows that, of the. 98 species occurring in the Radstock Series— 55 are common to the Coal Measures of France (excluding the Howiller inférieur).* 30 iF , 4wickau Coal Field. 24. ;. Z Lower Saarbriick Series. 30 b a Middle 2 ., 9 i 2 Upper A : 17 7 . f Lower Ottweiler Series. 10 § - Middle ss . 22 . rs | Upper i » The Radstock Series belong to the uppermost beds of the British Coal Measures, with which perhaps the Zwickau Beds are homotaxial. Although the middle division of the Saarbriick Beds contains as many as thirty species in common with the Radstock Series, the floras of the Saarbriick and Ottweiler Series, taken as a whole, indicate a somewhat-higher horizon than that of the Radstock Beds. A characteristic Permian conifer, Walchia piniformis, Schl., sp., has been observed in the Middle Saarbriick Series, and it also probably occurs in the Ott- weiler Beds. No evidence of this genus has been found in the Radstock Series. It is, however, interesting to observe that in the Upper Ottweiler Series (which overlie the Saarbriick Series) no fewer than twenty species occur that are also found in the Radstock Series. Beds of the same age as the Radstock Series appear to be absent from France. Their position is evidently between the Upper and Middle Coal Measures of the French Coal Fields, For the comparative list of the French Superior and Middle Coal Measure plants (Column IT.) I am indebted to M. Zeiller, who has spared no trouble in pro- viding me with the desired information, To elucidate this point more fully, I give some extracts from his letters on this interesting subject. “Your list (Column I.) seems to me to indicate exactly the passage from the Middle Coal Measures to the Upper Coal Measures, something like the highest zone of the Saarbriick Beds of Weiss, or the base of the Coal Field of Saxony. You will see in my text of the Coal Field of Valenciennes (when it appears, which will not be so soon as I would wish +) what I say on this classification. * Tt may be well to note here the probable equivalents of the French and British Carboniferous Rocks :— France. Britain. Upper Coal Measures (Houiller ay Se 3 and b Supérieur inférieur) . Absent. Absent . . . . . ~~. ~~. Radstock Series (= Upper Coal Measures of Britain). Middle Coal Measures (Houiller Moyen) . { Foe te eee ower Coal Measur Carboniferous Limestone Series. Inferior Coal M Houiller inféri ; nferior Coal Measures (Houiller inférieur). Lower passe gd | Caleifotdus ‘Sandstone ene + The Atlas of Plates only has been issued. RADSTOCK SERIES OF THE SOMERSET AND BRISTOL COAL FIELD. 409 “The species in your list that I have not found in our Bassin du Nord, but which occur in our Bassins du Centre, I have marked ‘Superior’; others, which are found both in the Bassin de Valenciennes, but especially in its highest part, and at the base of our Bassins de la Loire and Alais, as Sphenophyllum emar- ginatum and Sig. monostigma, or even extending into the highest beds, as Astero- phyllites equisetiformis, Annularia stellata, Ann. sphenophylloides, and Alethopteris Grandini, I have marked ‘Inferior Superior.’ Others again, that I have only known in the Bassin de Valenciennes and not in our Bassins du Centre, I have marked ‘Jnferior, among these last a certain number occur in the north of France, towards the top of the Basin, that is to say, in the oil or gas-coal beds at the Pas de Calais, which the Abbé Boutay has already shown, and which I show in my turn, to be the highest part of the Bassin de Valenciennes. These are Sphen. neuropteroides, Neur. Scheuchzert, Neur. rarinervis, Dictyopteris Munsteri, Pecopteris crenulata, Aleth. Serli, Ulodendron majus, and Sigillaria reniformis, What I have not marked as to horizon are those which I have not seen in France, but Sphen. Grandini (= Sphen. alata, Brongt.) is from Geislautern, that is to say, from the summit of the middle zone or from the base of the highest zone of the Saarbriick Beds; Corynepteris erosa is from Saxony. These two indications agree with what I have said already. “On the other hand, I remark the absence of several species which with us are abundant in the Upper Coal Measures, particularly Sphenophyllum oblongi- Jolium, Sphenophyllum angustifolium, Sphenophyllum longifolium (to which my Sphenophyllum Thoni should, I now believe, belong), Newropteris cordata, Odon- topteris Reichiana, Dictyopteris Brongniarti, Callipteridium pteridium ( = Pee. ovata, Brongt.=Call. mirabile, Rost., sp.), Pec. arguta (= Fil. feemineformis, Schl.), Sigillaria Brardit, and Sigillaria spinulosa. “ Fyrom this, Radstock represents, in my opinion, a horizon (niveau) that we have not in France, but corresponds to the interval between the end of the Coal Deposits in the north of France and the beginning of the Coal Fields of the centre.” In a later letter M. ZEILLER further says—‘* The mixture they (the Radstock Series) present of Upper Coal Measure species with Lower Coal species is un- questionable, but I do not at all mean to say by that that they are equal both to all or part of our Lower Coal (or more exactly our Middle Coal) and of our Upper Coal; in my opinion they are intermediate between the two, and are situated immediately above our highest beds of the north of France and below our lowest beds of St Etienne, Gard, &c.; perhaps their base is equivalent to the summit of the first and their summit the equivalent of the base of the latter, but on the whole I think their position is between the two,” 410 MR ROBERT KIDSTON ON THE FOSSIL FLORA OF THE APPENDS The Fossil Flora of the Farrington, New Rock, and Vobster Series has not been nearly so fully worked out as that of the Aadstock Series, but still sufficient has been done to make a record of the species known to occur in these horizons of considerable value. In none of these series are fossil plants so plentiful as in the Radstock Series, hence the difficulty of working out their flora,—in fact, it can only be done satisfactorily by those residing in the neighbourhood, who can take advantage of collecting when shafts are being sunk or new roads being driven underground, or when in any other mining operations good fossiliferous shales are met with. Some of the localities at the time of my visit appeared to be very barren, while at other times I know they yielded a very good harvest. Next to the Radstock Series, the Farrington Series is that from which I have collected most, but the time devoted to it has been small compared to that given to the examination of the Radstock Series. I may remark in passing, that palwontologically the Farrington Series cannot be separated from the Radstock Series, of which, in fact, they seem to form a part. The records from the New Rock and Vobster Series are chiefly obtained from the study of specimens in the museums already referred to; but I am also much indebted for information regarding the flora of these series to Mr E. WETHERED, F.G.S., Cheltenham, and to Mr 8. Jorpan, Clifton, who kindly gave me every facility for examining the fossil plants in their collections. The Pennant Rock seldom yields well-preserved examples, owing to the coarse-grained nature of the rock. Any specimens seen were usually coarse casts of Calamites or Lepidodendra. The Red Shales which separate the Radstock and Farrington Series are also very barren, but the plants observed were similar to those of the two series just mentioned. The flora of these various horizons are treated of in descending series :— J. FARRINGTON SERIES. (Upper Coal Measures.) Eucalamites (Calamites) ramosus, Artis. Loc.— Parkfield. RADSTOCK SERIES OF THE SOMERSET AND BRISTOL COAL FIELD. 411 Stylocalamutes (Calamites) Suckowtt, Brongt. Loc.—Farrington Pit, Farrington-Gurney. Annularia stellata, Schl., sp. Loe.—Old Mills Pit, Farrington-Gurney ; Farrington Pit ; Parkfield, Annularia sphenophylloides, Zenker. Loc.—Old Mills Pit. Sphenophyllum emarginatum, Brongt. Loc.—Old Mills Pit; Parkfield ; Farrington Pit. Sphenopteris macilenta, L. & H. (?) Loc.—Old Mills Pit. Sphenopteris neuropteroides, Boulay, sp. Loc.—Old Mills Pit, Sphenopteris, sp. Pl. XIX. fig. 3 shows a small fragment of Sphenopteris having considerable similarity to the fruiting specimens of Sphenopteris (Dip- lothmema) Zeilleri, Stur,* figured by ZEILLER, but is too fragmentary for a satisfactory identification. On the same slab are the indistinct remains of another specimen, which is apparently specifically distinct from that just men- tioned. Loc.—Old Mills Pit. Neuropteris macrophylla, Brongt. Loc.—* Top Vein,” Parkfield ; Old Mills Pit; Farrington Pit. Neuropteris Scheuchzert, Hoffm. Loc.—Foxcote, near Radstock; Middle Pit, Radstock; Farrington Pit ; Old Mills Pit. - Neuropteris flexuosa, Brongt, Loc.—Old Mills Pit. Neuropteris ovata, Hoffm. Loc.—‘ Hollybush Vein,” Parkfield ; Old Mills Pit, Neuropteris rarinervis, Bunbury. Loc.—Old Mills Pit ; Foxcote. Mariopteris nervosa, Brongt., sp. Loc.—Old Mills Pit; Farrington Pit, Pecopteris arborescens, Schl., sp. Loc.—Old Mills Pit; Pucklechurch, near Mangotsfield ; Parkfield. * — Diplothmema acutilobum, Zeiller, Ann. d. sci. nat. Bot., 6° sér., vol. xvi. pl. xi. fig. 2 ; see also Flore foss. d. Bassin houil. d. Valenciennes, pl. xv. fig. 5. 412 MR ROBERT KIDSTON ON THE FOSSIL FLORA OF THE Pecopteris unita, Brongt. Loc.—Farrington Pit ; Old Mills Pit. Pecopteris oreopteridia, Schi., sp. Loc.—Parkfield. Pecopteris pteroides, Brongt. Loc.—Parkfield. Pecopteris Miltoni, Artis, sp. Loc.—Farrington Pit. Dicksoniites Pluckenetti, Schl., sp. Loc.—Farrington Pit ; Foxcote. Alethopteris lonchitica, Schl., sp. . Loc.—Shale over “Top Vein,” Parkfield ; Middle Pit, Radstock. Alethopteris Serlii, Brongt., sp. Loc.—Old Mills Pit ; Farrington Pit ; Middle Pit, Radstock. Alethopteris Grandini, Brongt., sp. Loc.—Old Mills Pit. Rhacophyllum crispum, Gutbier, sp. Loc.—Parkfield; (#) Old Mills Pit. Rhacophyllum Goldenbergi, Weiss. Pl. XX IL ie) 2: Loc.—Pucklechurch, near Mangotsfield. Rhacophyllum, sp. Loc.—Farrington Pit. Caulopteris macrodiscus, Brongt. BA, RORY fia Loc.—Coal Pit Heath, near Bristol. Caulopteris, sp. Loc.—Old Mills Pit. Lepidodendron Worthenti, Lesqx. Loc.—Old Mills Pit. Lepidostrobus. Loc.—Old Mills Pit. Sigillaria monostigma, Lesqx. Loe.—Old Mills Pit. RADSTOCK SERIES OF THE SOMERSET AND BRISTOL COAL FIELD, 413 Sigillaria M‘Murtriew, Kidston, Loc.—Farrington Pit. Sigillaria tessellata, Brongt. Loc.—Foxcote, near Radstock. Sigillaria reniformis, Brongt. Loc.—Coal Pit Heath Sigillaria principis, Weiss. Pl. XXVIII. figs. 6-8. Sigillaria principis, Weiss, Zeiller, Flore foss. du Bassin houil. de Valenciennes, pl. xxix. figs. 1, 2, 1886. Description.—Stem furrowed, furrows straight, ribs smooth ; leaf-scars more or less distant, oval or suborbicular, lateral angles distinct, vascular cicatricules situated slightly above the centre of the scar, the two lateral lunate, the central punctiform. Leaf-scar surmounted by a small cicatrice. Decorticated stem striate. Remarks.—The specimens from Farrington-Gurney agree with fig. 2 of ZEILLER’S plate, where the scars are orbicular, and not so oval as in his fig. 1. In the Somerset examples there is no trace of the downward running lines that proceed from the lateral angles. In some cases the scars are slightly emarginate, in others they are simply rounded at the top. The central portion of the ribs is flattened in my fossils, but this may be due to pressure. Only two specimens were met with, and, from the position in which they were found, they probably belong to the same individual. Loc.—Old Mills Pit, Farrington-Gurney. Sigillaria Voltziw, Brongt. Loc.—Coal Pit Heath. Sigillaria elongata, var. minor, Brongt. Loc.—Parkfield. Stigmaria ficoides, Sternb., sp. Loc.—Parkfield ; Old Mills Pit ; Foxcote, near Radstock. Cordaites angulosostriatus, Grand’ Eury. Loc.—Old Mills Pit; Foxcote. Cardiocarpus fluitans, Dawson. Loc.—Old Mills Pit. Trigonocarpus Noeggerathi, Sternb., sp. Loc.— Old Mills Pit. 414 MR ROBERT KIDSTON ON THE FOSSIL FLORA OF THE Carpolithes ovoideus, Gdpp. & Berger. Loc.—Old Mills Pit. II. Pennant Rock. Stylocalamites (Calamites) Suckowii, Brongt. Loc.—Crewshole, near Bristol. Stylocalamites (Calamites) canneeformis, Schil., sp. (2). Loc.—Crewshole (perhaps only C. Suckowit). Calamitina (Calamites) approximatus, Brongt. Loc.—Downend, near Mangotsfield; Crewshole; Fish Ponds, near Bristol. “ Ulodéndron.” Loc.—Downend, near Mangotsfield. ** Halonia.” Loc.—Fish Ponds, near Bristol. Stigmaria ficoides, Sternb., sp. Loc.—Near Bristol. III. New Rock SeErigs. Stylocalamites (Calamites) Suckowti, Brongt. Loc.—‘ Thoroughfare Seam,” Kingswood, near Bristol. Sphenopteris trifoliolata, Artis, sp. Loc.—Above ‘“ Toad Vein,” Deep Pit, Kingswood. Mariopteris nervosa, Brongt. Loc.—Deep Pit, Kingswood. Pecopteris arborescens, Schl., sp. Loc.—Golden Valley. Pecopteris Miltoni, Artis, sp. Loc.—* Top Vein,” Warmley, near Bristol ; Golden Valley. Lepidostrobus, sp. Loc.—Kingswood. “« Ulodendron” (Sig. major, L. & H., sp. (?) ). Loc.—* Little Toad Vein,” Speedwell Colliery, Kingswood. Sigillaria monostigma, Lesqx. Loc.—Warmley, near Bristol. RADSTOCK SERIES OF THE SOMERSET AND BRISTOL COAL FIELD. Sigillaria tessellata, Brongt. Loc.—Kingswood. Sigillaria mamillaris, Brongt. Loc.—Warmley, near Bristol. Sigillaria mamillaris, var. abbreviata (Brongt.), Weiss. Foss. Flora d. jingst. Stk. u. d. Rothl., p. 165. Loc.—‘‘ Great Toad Vein,” Kingswood. Sigillaria scutellata, Brongt. Loc.—Kingswood. Sigillaria rugosa, Brongt. Loc.— Great Toad Vein,” Kingswood. Sigillaria Schlotheimi, Brongt. Loc.—2 feet above “ Toad Vein,” Kingswood. Stigmaria ficoides, Sternb., sp. Loc.—Kingswood ; Bedminster, near Bristol. Cordaites, sp. Loc.—Deep Pit, Kingswood. TV. VosBstER SERIES. Calamitina (Calamites) varians, Sternb. Loc.—Edford Colliery, near Radstock. 415 Stylocalamites (Calamites) canneeformis (*%), Schi., sp. (or C. Suckowii, Brongt.) Loc.—Edford Colliery. Sphenophyllum emarginatum, Brongt. Loc.—Ashton Pits, near Bristol. Pecopteris oreopteridia, Schl., sp. Loc.—Ashton Pits, near Bristol. _ Lepidodendron aculeatum, Sternb. Loc.—Edford Colliery. Lepidodendron rimosum, Sternb. Loc.—Bed between Stone Rag and Main Seam, Edford. Sigillaria Sillimani, Brongt. Loc.—Ashton Pits, near Bristol. Sigillaria mamillaris, Brongt. Loc.—Ashton Pits, near Bristol. VOL. XXXIIL. PART IL. 3.P 416 MR ROBERT KIDSTON ON THE FOSSIL FLORA OF THE EXPLANATION OF PLATES. Pruate XXIV. Fig. 1. Alethopteris Davreuxi, Brongt., sp.; from Camerton, nat. size. Specimen in the collection of the Bristol Museum. la, two pinnules x 2, p, 386. Fig. 2. Pecopteris unita, forma emarginata, Gopp., sp.; Camerton, p. 367. Fig. 3. Pecopteris unita, Brongt.; New Mills Pit, Farrington-Gurney (Farrington Series). 3a, two pinnules x 2, showing the nervation. Fig. 4. Pecopteris unita, forma emarginata, Gopp., sp.; Camerton. 4a, portion x 2, showing the nervation. Fig. 5. Pecopteris unita, Brongt., forma elliptica (G. elliptica, Font. and White); Old Mills Pit, Far- rington-Gurney (Farrington Series), nat. size. 5a, pinuule x 2, showing nervation. Fig. 6. Pecopteris unita, forma emarginata, Gopp., sp.; Camerton. 6a, portion x 2, showing nervation. Fig. 7. Pecopteris unita, forma emarginata, Gopp., sp.; Radstock, nat. size, showing fructification ( = Stichopteris longifolia (Brongt.), Weiss). Fig. 8. Pecopteris unita, forma emarginata, Gopp., sp.; Upper Conygre Pit, Timsbury, nat. size, showing fructification (= Stichopteris longifolia (Brongt.), Weiss). Fig. 9. Pecopteris unita, Brongt. ; Camerton, nat. size. 9a, pinnule x 2, showing nervation. PuaTe XXV. Fig. 1. Caulopteris macrodiscus, Brongt., sp. ; Coal Pit, Heath, near Bristol (Farrington Series), half nat. size. Specimen in the collection of the Bristol Museum, p. 393. Fig. 2. Caulopteris macrodiscus, Brongt., sp.; Radstock, half nat. size. Specimen in the collection of Mr J. M‘Muttrie. . Fig. 3. Megaphyton frondosum, Artis; Radstock half nat. size. In the collection of Mr J. M‘Murtrie, p. 390. Prats XXVI. Fig. 1. Megaphyton elongatum, Kidston, half nat. size. Specimen in the collection of the Bristol Museum, p. 390. ‘ Fig. 2. Caulopteris, sp., Radstock, nat. size, showing the epidermal layer removed in part, and exhibit- ing the Caulopteris macrodiscus condition. In the collection of Mr J. M‘Murtrie. Fig. 3. Caulopteris anglica, Kidston ; Radstock, half nat. size. Specimen in the collection of the Bath Museum, p. 392. Fig. 4. Megaphyton frondosum, Artis; portion of the outer surface of the stem of specimen figured on Pl. XXYV. fig. 3, nat. size, showing rootlets, p. 390. Fig. 5. Caulopteris, sp., Radstock, nat. size, isolated scar, showing “ horse-shoe ” cicatrice. Piate XX VII. Fig. 1. Macrosphenopteris Lindswoides, Kidston, n. sp.; Radstock, nat, size, p. 353. Fig. 2. Rhacophyllum Goldenbergii, Weiss ; Pucklechurch, nat. size. Specimen in the collection of the Bristol Museum, p. 388. Vigs. 3, 4. Pecopteris oreopteridia, Schloth., sp.; Camerton. Fig. 3, barren pinnule x 3, showing nerva- tion. Fig. 4, fruiting pinnule x 3, showing the villose upper surface, p. 371. Fig. 5. Lepidodendron lanceolatum, Lesqx.; Radstock, Cone, nat. size, p. 394. RADSTOCK SERIES OF THE SOMERSET AND BRISTOL COAL FIELD. 417 Fig. 6. Sigillaria reniformis, Brongt., var. Radstockensis, Kidston ; Radstock, nat. size. 6a, scar x 1} In the collection of Mr J. M‘Murtrie, p. 399. Fig. 7. Lepidophyllum, sp. a, Radstock ; b, Braysdown Colliery, nat. size, p. 395. Fig. 8. Sporangium, Camerton, nat. size. 8a, x 2, p. 400. Fig. 9. Sporangiwm, Upper Conygre Pit, Timsbury, nat, size. 9a, x 2, p. 400. PLate XXVIII. Figs. 1, 2. Pecopteris oreopteridia, Schloth., sp.; Camerton. Two pinne from different parts of the same specimen, nat. size, p. 371. Fig. 3. Lepidodendron lanceolatum, Lesqx. ; Camerton, nat. size, stem showing leaf-scars, p. 394. Fig. 4. Lepidodendron lanceolatum, Lesqx.; Camerton, nat. size, small branches showing foliage. Fig. 5. Sigillaria levigata, Brongt.; Radstock, nat. size. 5a, leaf-scar x 2, p. 398. Figs. 6, 7. Sigillaria principis, Weiss ; Old Mills Pit, Farrington-Gurney (farrington Series), nat. size. 6a, leaf-scar x 2, p. 413. Fig. 8. Sigillaria principis, Weiss, same locality as last. 8a, leaf-scar x 2, p. 413. Figs. 9, 9a, 10, 10a. Stigmaria anglica, Sternb., sp. ; Camerton, nat. size. 100, scar x 2, p. 401. ry Vol. XXXII. Pl. XVIII. Trans. Roy. Soc. Edin. 6 ‘euoJspuLg pory PIO 8 ‘O07 Sa UuIT urez Ono 4 ; : FA) OMO4ST[IP ‘SOMIOG 19ISqOA “9 ) , «, Yooy queuusg “F ‘SOMOG UoySmIIeg * Te & : ‘So[vyg poy “z : : ‘uoIstAr(y 1add 9 | ‘SOllag Yooyspey Tp : SOINSBOTAT [899 (E181 ysn8ny qart ‘smog ye ‘Aqor00g Ar0ystFT [ernye NT pure [eoLsopowyory OTTYSPASIOMLOG OT} o10joq prar srodeg SIT WIM poysiqnd pure ‘gggq ‘TOISSIUNMOG [vog TeAoy ey} toy “gn “A “bsay “ALLAN, W “¢ 4q poaedard Tor}oes Wolf paonpsy) “weg avon ‘UMO(] aq OD 04 TOJMETO ULOIT ‘plang Teo Sie a oe are en ee eS See ee a-9 9 - == =a se, ~ ~ SSS -_-— = —_— _—_ Cl OC LZ, AMT Py lyl TEN — Combe Down Ch. } Radford Pit Paulton Pit. Farrington Pit Dunkerton, Camerton, ‘OD OSTOMY ssorov uoT}0eg Chewton Ch, Trial Pit, ¢ SHCIOTIAHGONAHdS “NNY 41 Std ‘ds ‘SISTIGONTHdS gw (2 98UOag) ‘zerq’ng ‘VITOSINNEL SIUTIdGONIHAG “2 ‘Su “WOISPly “IGUYVMGOOM SIMALGONSHdG “[ Sty AUIPT {4aKT ‘Ounjsag - eMe[e 45 WE “VeU pe pep ‘uoqepryy 4qou ‘ds ‘yj0tHOS “VLVITELS NNY fl Stq “ds ‘z0HU0Z ma Wits. 5 ns. Roy. Soc. Edin” Vol. XXXIII, Plate XX. 4, del. ad. nat. MsFarlane & Erskine, Lith’§ Edin® Fig. 1. SCHIZOSTACHYS SPHENOPTEROIDES, Kidston,n.s. &@.PTYCHOPTERIS ELONGATUS, Kidston, ns. 3. RHACOPHYLLUM SPINOSUM, Lesqx. Pus. Roy. Soc. Edin? Vol. XXXIII, Plate XXI. ‘xdstom@l. ad” nat. M‘Farlane & Erskine, Lith"? Edin Fig]. SPHENOPTERIS GENICULATA, Germar & Kaulfuss. " 2. NEUROPTERIS MACROPHYLLA, Bronét. Fig. 3-5. NEUROPTERIS FIMBRIATA, Lesquereux,(?) 6. DICTYOPTERIS MUNSTERI, Eichwald, sp. ms. Roy. Soc. Edin. Voll XI, Slate p=O+ py; 2 Loy Yr, &e. 0% Ill. a=pig, B=pjq, y=pkq; p( )g opera- tor of rotation. Differentiation of a, By Pw; 50=We, d= We, ,. whence 3p=(o+¥,)e. Expressions of Yow, Yow, Ww, po, . 468 i IV. Various forms of the condition which defines the limiting curve. Hither pe=0, or V.dpWe=0, . 470 V. Both conditions give V.wWjw/k= 0, but the second determines dp also, . 472 5 VI. Abandonment of the solution founded on ye=O alone, . 473 VIL. Method founded on Wee 0 We =0, *. W(=,¢, where (=7z, rs ert 475 VIII. Remank-Ble expression of ~,¢; several successive transformations relating toit, . : 4 , eis, IX. Fundamental equation V.dpye=0 giving y/(Vidp)=0, ¢ being arbi- trary. Treatment of y/(Vidp)= by S. ¢ gives dp parallel to y,¢ what- ever the direction of e, . 479 Treating y '(G)=0 by S.z() anid by S.¢—1u( ), G being a certain per- eaieulad to dp; the result in both 3 n cases is wot Ws; fy, w Wi naan ie h=1,2, 3, - . 482 SECTION a» 99 ” ” 2) XI. Treats w’G by S.dz( )=0, The re- sult of the preceding section being applied, the two equations obtained : : wu will contain “ and “ each equal v w to arational function of Sai= — a, Saj=-a,, &e., &e., the nine co- efficients of a, B, y, : XII, Developments which show the presence of the factor (= VY e= a= 2) _ abc? os in all the terms of the equations. Final form (I.) (II.) of the equa- tion when the factor A is sup- pressed, XIII. Expressions for y and 2, freed from radicals, exceptwu= A/Sip-%i; other- wise also: y expressed proportion- ally to v, and z to w, aN XIV. Expression of the nine coefficients Sat, Saj, &c., in function of three angles A, B, C, with their defini- tion, c | XV. Substitutions into (I.) and (II.), XVI. Rationalising (I.) and (II.) in respect to & ; “| Two resulting equa- vy? w tions : the one of the second degree, the other of the third degree in tgB. : G- PAGE 484 487 491 493 494 497 The question is the following:—We consider the solid angle formed by three planes at right angles to each other, and into the space of this single octant we introduce a given ellipsoid, and cause its surface to be tangent to VOL. XXXIII. PART II. oz 466 DR G. PLARR ON THE DETERMINATION OF each of the three sides of the solid angle. The position of the points of tangence will of course be variable in each plane according to the orientation given to the axes of the ellipsoid, but it is evident that on each of the planes the positions of the point of contact will be unable to outpass certain limits so long as the ellipsoid fulfils the condition of remaining tangent simultaneously to the three planes: these limiting positions of the point of contact in one, as for example, of the planes, will form a certain curve, and the proposed question will be: the determination of that curve, the limiting curve as we shall call it in the sequel. § I. Let a, B, y designate the unit vectors in the direction respectively of the three principal axes of the ellipsoid; let a, b, c designate the lengths of these axes ; then if w represents the central vector of a point on the surface of the ellipsoid, and if we put aSaw , BSBw es ——e Te eo” or briefly aSaw pu=2ae the equation of the ellipsoid will be ‘ Sodo=1; and if w) designates the central vector of a point outside the surface, the equation Sa po =1 will represent that of the tangent plane passing through the extremity of o. Let O designate the apex of the solid trirectangular angle, and 7,7, / the unit vectors in the direction of the three edges. We may assume that O and i,j, & remain fixed in position and direction. From this it follows that the centre O, of the ellipsoid and the direction of the trirectangular axes a, B, y will be the variables of the question. We designate by @ the vector OO, of the centre of the ellipsoid in one of its variable positions, and by P> Oo, T, the vectors of the points of contact of the ellipsoid with the three planes respectively i), ), GS) briefly designated, these vectors having their origin in O. From their defini- tion they satisfy the conditions Spi=0, Soj=0, Stk=0. THE CURVE ON ONE OF THE COORDINATE PLANES. 467 We call, on the other hand, Pr» %> Ty the central vectors of the points of contact, so that we have p=9+p,, o=60+0,. 7=60+7,. § II. At the extremity of p, the tangent plane is by hypothesis the plane (J, /) itself, so that the normal to the plane is parallel to7, At the same time, the normal has the direction of ¢p,, thus we may put gp =iN, N being a scalar to be determined. This equation gives pi=No-%s, where ¢~™ is the inverse of ¢, and is defined by o-w=aa*Saw+ Bb°SBw+je*M7 , or briefly by ow =Zaa*Saw. Having Spga=l, p: being a central vector of a point on the surface, we get by the two vector equations : 1=N?Sig-14. This determines N ambiguously, but we have the equation of the tangent plane to give it its sign: namely, the tangent plane at the extremity of p, passes through the point O ; we may, therefore, for the central vector w, put @)=(— 8). Thus the equation of the tangent plane ’ : So,Pp,=1, owing to ¢p,=7N, becomes es —Nsé@:=1. This gives —1 1 IN SQ /Sig-% If we put O6=w+ju+kw, 468 DR G. PLARR ON THE DETERMINATION OF it will be necessary to assume that the values of wu, v, w will continually be positive in consequence of our admission that the ellipsoid be contained in the one octant of which the edges are in the directions +7, +7, +4. We have thus i. 1 wf Sig=34- A consideration of what takes place in the two other tangent planes will give corresponding values of v and of w. We thus have the three expressions uU?=Sip-le Pa Rip w*=Skp-lk with the restriction that the determination of the square roots of these expres- sions be the positive one exclusively. By the value of N and of analogous values for the cases of o, and 7,, we have now ae Pi anhots i 1 ae ysis da == ok These values are to be introduced into p=9+p,,c= &e, We may at once introduce also the following notations :— H=Sp tk, w=H¢k, Y,=Skp-1t, yy=Skdr, a=Sip-9, m=Sidj. § IL. Let us now calculate the variations of the above vectors when the directions of a, 8, y vary. It will be sufficient for our purpose to consider the variation of p only. We represent the unit vectors a, 8, y by a=piq B=p)4 y=pkq,; THE CURVE ON ONE OF THE COORDINATE PLANES. 469 where p( )q is to express the operator of a conical rotation, in which pg = 1, and Tp=Tq=1. For any position of the point of tangence at the extremity of p (generally comprised within the inside of the limiting curve), we represent the variations of the quantities by the characteristic 6, reserving the sign d of the differential to the increase of p when the extremity of this vector moves on the limiting curve. In this latter case we assume (having Spi=0) p=jy tke ; consequently dp =jdy +hdz represents the element of the limiting curve. In the general case p changes into p+ 6p; putting e=26p.xq we get da=Vea, S58=VeB, dy= Vey Applying these to 6(¢~'w) we get 5(¢-1@) =ZVea. a*Saw + Laa*Sw Vea namely 5(p-1@) = V(ep-1w) + $-1(Vae) This written for » =2, 7, &, gives first 2udu =S8i6(p-2) =Si[Vep- 4+ 71 Vie] = —28.eVig-4 Hence (and mutatis mutandis) : Le eee éu= ae a ee dv= ——S9¢ J» = 1 -1 Sw= —— Sek“. With these expressions we get SO ei; 60= =F Seip-0, eae 8p,=— (Vep-i + g7Vie+ P's cig 't, and the second members evidently represent each a linear vector function of «. 470 DR G. PLARR ON THE DETERMINATION OF We designate them respectively by ye and by ye, so that 50= Wye dpi=Wye. And putting ah Wotyi=y we have now 5p =e. Generally having, for any vector o, ea — 3 Swig-1 i aie! yo= mV (oo ““Y+-1!Vio)+ as Swid-' , the conjugates y,/ and yf,’ will be Wio= —E_Vig-“WSio , hio= — = (Vag 44Vig-'0)+ AVig-Sagp-¥ with yo = pot yo, pyo=potpyoe. eg The condition which must be fulfilled when the extremity of p is to move on the limiting curve may be stated as follows :— Generally the extremity of p is comprised within the inside of the curve in question. In that case two different axes of infinitesimal rotation, say « and ¢«, will produce variations dp differing generally from each other in direction and length. We may assume generally, also, that if « and e, be opposed to each other so that Ue+Ue,=0, (Ue) +y(Ueg)=0. But when the extremity of p is infinitesimally near the limiting curve, then we cannot any more admit this, because if should bring the point m (extremity of p) to the point m’ inside the curve, then the rota- us _ tion e,=(—e) would have to bring the point m to m,’ outside, a circumstance which can- m’ not take place, as it is against the very definition of the curve. We may try the supposition that for a point p on the limiting curve the we will also have 4 m, THE CURVE ON ONE OF THE COORDINATE PLANES. 471 value of ye will vanish for any direction of «, without even excepting that direction of « which might render we parallel to the element dp of the curve. If, for example, we put the question to s determine the maximum of Tp under the con- oe *,8 ‘7 aa - Up=constant, Of then we get the two equations Spdp=0 and Vpdp=0, which result into dp=We=0. Another supposition to be tried will be to assume that for a point on the limiting curve the direction of We will be parallel to dp for any direction of e, so that in that case the possibly not vanishing ye will be invariable as to e. These two suppositions expressed respectively by the condition yre=0, V(dp we) =0, and by the condition in each case for any direction of ¢, are not excluding each other, and we will be able to show that the consequences of the second of the conditions involve the consequences of the first of them. We may remark that the first of the conditions, namely, vwre=0, for any e, will also present itself when we look upon the limiting curve as being the general envelope of all the possible curves which the extremity of p may describe under various limitations. We may further remark that the condition V.dpwe=0, namely expresses the answer to the problem of finding the curve comprising the area- maxima of all the areas comprised by curves described by the extremity of p. In order to conceive how the extremity of p may be caused to describe certain definite curves (both in the case of an envelope and in the case of an area to be considered), we must reflect that the expression of p depends on the three scalar elements of the versor p, and that we may conceive a certain relation between the three to be established, so that only two of them may be considered as independent. But here there arises the theoretical difficulty of our inability of guessing at the true nature of the relations to be established 472 DR G. PLARR ON THE DETERMINATION OF so long as the true nature of the limiting curve is ignored. This our inability of establishing the proper limitations between the three scalar elements of p will force us into the fear that, instead of an outermost limiting curve, we will deduce a possibly self-intersecting curve with many branches, but certainly all evidently contained in the outermost limiting curve. SW. From the equation Vdpie=0 we draw the three scalar equations— Sidpwe =S.eyVido =0, SjdpwWe =S.eW'Vjdp =0, SkdpWe=S.eW'Vikdp=0. As we have assumed Z dp =jdy +hkhdz ‘ we get Vidp=idz , Vikdp = —idy. The two last of the scalar equations will give Seyi=0. But we easily find that 7 is identically zero, because substituting 2 for w in the expressions of yw, x,’ we get Pe ee he a ies a Sed gers and if Pe Sig Mas L ade Ss ait at hii= -[-2Vig- i] +4 which owing to w’=Sid~"z become pita g-% eee so that Wot +, 7 =i =0 identically. There remains the first scalar equation Sey’ Vidp=0, Se[dyWhk—-—dapyj|=0. As this equation is to be satisfied for any direction of « we must have or y'Vidp=0, or, under another form : dyy/k—dap'j =0. a) THE CURVE ON ONE OF THE COORDINATE PLANES. 47 From this second form of the equation we deduce Vij =0. This parallelism of #7 and W& is also the consequence of the condition We=0, for any e, because treating this last equation successively by 8.7, 8.7,5.4 it gives first S West, which is identically satisfied, because y7 is identically zero. Then the two other equations, Sjwe=S.e7=0. SkWe=S. ey k=0 have to be satisfied for any direction of «. This must be effected by the least number of assumptions, lest we get too many conditions between the three scalar elements of p. The only way to reduce the number of conditions is to assume the parallelism between Wj and Wh again. § VI. We cannot refrain from sketching briefly a method of development of the equations w’Vidp=0, V.nivh=0, which we have abandoned owing to the complication arising from the rational- ising the expressions containing the radicals w, v, w together. If we develop namely w7 and w’é into their components parallel to 7, 7, 4, putting (for reasons easily to be accounted for) 7 = —1P,—jQ —kR,, Wh= +iP,+jR, +hQ, and where with the notations we have Ln | Di ae P= ae P,=— i 2 2 ee, w w U 2 2 Zz z v R. el U——? 2 8 mat b VOL. XXXIII, PART 11. 4A 474 DR G. PLARR ON THE DETERMINATION OF then the equation ’Vidp=0 decomposes itself into the three O=P,dy+P,dz, O=R,dy+Qdz, O=Qdy + Ridz. ee 2 ele) nae : The elimination of ay B1ves the equations Y P,Q—P,R,=0, P,Q—P,R,=0, and their consequence RR 60. It is evident that each of these equations contains the three radicals wu, v, w all three at the same time, and their transformation into rational expressions depending on the nine coefficients Sai, Saj, Sak, SBi, S8B7, &., Syi, &e., will lead to hopeless complications from a practical point of view. Of course the expressions of Q, R,, R, may be slightly simplified by the use of the direct function ¢ instead of 47’, but the fact remains that powers of uneven degree of u, v, w will affect together always each of the equations. We will only add the remark that theoretically the two scalar equations above, and the two scalar equations to be derived from ou ? b V PO. will constitute four equations from which the three elements of the versor p may be supposed to be eliminated, so that the problem is theoretically definite. We may also remark that when w’(Vidp)=0 is satisfied, that is when the point of tangence is situated on the limiting curve, then we have for the primi- tive function (Vidp)= —dp(R, + Ry). As V(idp) is directed normally to the curve, the equation shows that a rotation « parallel to the normal produces a displacement of the extremity of p in the (irection of the tangent. From the above equation we draw by the application of the operator w: y*(Vidp) = —dp(R, +R.) . Now as W’7=0 we have also (x) =0, where K=V(jy'k) generally. THE CURVE ON ONE OF THE COORDINATE PLANES. 475 The cubic relative to % becomes a quadric, namely, —M,o+wW’o+KSio=0,* where M, = — Siw’jWk( = RR, —Q’). M,= —Syh') —SkWk = +Q—Q=0. The quadric can also be put under the form vo=V[iV(xo)] . Putting o || Vidp we get W°(Vidp) = V. [iV («Vidp)| =V. i(dpSki—iSxdp) = Vidp(—M,). Hence by the above: wr(dp) = Vidp x ae and M,=0 by W(Vidp)=0. This shows that when the point of tangence is on the limiting curve we have v(dp)=9, but this remark is unessential as to the sequel. § VIL. Let us examine yo more particularly. First we see that w= —EVig i? =ZVig-14, owing to 0=iu+jv+hkw. As dand ¢‘ are self-conjugate-vector functions, it follows that EVig-i=0; thus we have yiO=0. This equation at once gives us the certainty that the primitive function yo will vanish also when » takes a particular direction. (We have shown this neces- sity i a paper published in the Proc. Roy. Soc. Edin. of the year 1881-82, p. 342.) * Proc. Roy. Soc. Edin., 1882-83, p. 342. 476 DR G. PLARR ON THE DETERMINATION OF Let us call Z the vector, for which we have Poo =0. This equation gives the three scalar equations : S.gig-1i=0, S.¢jp*7=0, S.tke-k=0. In developing V. i6-'t=V.jkb-'2, &e., by the formula V(VAnu.v) =ASuv—pSAv, we get easily, employing the notations 7,=S7¢ “kh, w.=Skd"t, ~=Sip-: O=y,Shi—2,Sék , O=2,Stk—a,Sh; hence a Si=y,SG =2,8h= a scalar = —N; c=n(4+445) . my YA hence We assume N=2,¥, 2, introducing thus a factor for the sake of which some precautions are to be taken when two of the three 2,, y,, z, should vanish at the same time. We have thus CS ty +920, + kay, = Diy,2; - By w¢=0, and having generally hothM=yv, we get oat. This shows that when « takes the direction of ¢, then the extremity of 6, namely, the centre of the ellipsoid, remains unmoved, whereas the extremities of pi, 71, 7 displace themselves by the rotation round the instantaneous axis directed parallel to ¢. We have in this particular case dp =pCdt= peat, where dt has a convenient scalar value. THE CURVE ON ONE OF THE COORDINATE PLANES. § VILL. Let us now calculate y,¢ explicitly. The expression of yo gives 1 C20 Ciel a Wié=—| Veo Ut gtVil+ Oe Soph Now ¢ may be written CS ye Fey +h). Having jt = —Wig-%4—jSjp- 4 —kSkp- 4 = — 1? —jz, —ky, , we get C=U(yy2,—% U7) —X,p- 0. The coefficient of 7, namely, Iq — Xu? = Skp-USip-y —Skp-YSip- =S.VhiVE-Yp-%i, =—-SjVo ig 7 eye where we apply the formulas mo "*Vip=V.drAdu mp 1V.¢ No w=V.Nw, hence Vip Now == VN ' . and introduce N97. Thus we get isjok ; Ss LEE agi , Ta ies = ——_— — 2& VU. Sy ae This gives at once S.¢ip-W=0. Then V.gp- tis — Vig aloe, and oVig=—9-(Vig-N).m,. Applying to the second member mp Vr =V. prop for N=1, p=, we get 177% 3 ‘hil o7Vig= mr eet mat ot 477 478 DR G. PLARR ON THE DETERMINATION OF These values give (as 2, =Sj¢7'h): 1 Sigh , VigiSip-%e wé=—| Vig Pon | Ie i: Mv Hence muy, €=V.ipiSip tk—V.ig-USpok. Developing Vidi and Vid-" into their components parallel to 7, * : ping I p Vigi=V.jhpi=jSkpi Sj gi, Vig-M=Vijhp-N=jSkp-—hSjg-U, we get mur, €=J[SkpiSjp—'k—Skp-USjpk] +h —SipiSjp hk + Sjp-MSjpk]. Now ¢ and ¢“' being self-conjugate, we replace Skdz, Skp-“t, respectively by : Sigk, Sip-'k, in the term affected by 7, and Sipi, Sip, Sigh, Sip by Sigs, Sig-'7, Skp-Yj, Shgj, in the term affected by &. This gives murb,€ =j[SighSjp- th — Sip kSjph] +h —SigsShp-Yj + Sig YSkdy]. The coefficients of 7, # are respectively S.VyV.dp-kpk=S.kp- lk. pk and S.ViiVd-Vgoj=S.jb- VY. py. So that, with a change of sign (Shf-'k. 6k= —Skokd'k, &c.), and introducing the notation W,=Sigig-%, W,=Sjgo 9; W,=S.kpko-k, we have muy, (f)= —)W,—kW,. Let us further consider for any vector ): , 2 2) (Saa?Sar) V.grp-r=V.(E% = aBSarspn(— =) + yaSprSan( = = 5) = BySBASp(T —<) ; THE CURVE ON ONE OF THE COORDINATE PLANES. 479 Hence, as aB=y, ya=6, By=a, we get generally Srprp— r= SarSPrSyr x A, a=(G-g)t Gata) It is evident that, if we add and subtract unity, the second member will repre- sent the development of & 2 (f C\/ Go a a a2/\b2 2 a) =(S le Vy(e—a\(@—2) =A. a2b?¢2 where We have thus W,=SaiSBiSyi. A W, =SajSBjSyji.A W,=SakSBkSyk. A. Introducing the notation a=1a,+7a,+kha, B=1b, +70, +kb, YY HU, +)eg +hes, we finally get W, = —a,),,.A, W,=—a,b.¢,.A, W, = —@,b,¢3.A. murp, € =[Jagb,0, + kayb,c,| x A- We put generally for k=1, 2, 3: W,, = Andj.Cy 5 then W,=—W,A and mur §= —(7Ws +kW,)A- oi: If we treat the fundamental equation Ww (Vidp) =0 by 8. ¢, we get S. ob’ Vidp) =S8.Vidpwt=0; and, as alee vo=Wwio= = ad W,+kW,), we get O=W,S.jVidp + W.S.kVidp , 480 DR G. PLARR ON THE DETERMINATION OF namely, O=W,(—Skdp) + W.Sjdp , and for dp =jdy + kdz the result becomes W.dz—W,dy=0. We have, therefore, for the element of the limiting curve, dp=(jW,+kW,)dt’ =—mup(o)dv . Of course, there are two more conditions to be satisfied in order that the extremity of p be actually on the curve. We may remark that ¢ may also be put under a second and a third form, namely, the first being We ; c= eb ts we have also f=— Poy. 4-1), ¢= e -4pk, MN where 2, =ok, y,=Skopi, 2 = Sidy . Hence if we put by analogy do = (Wy tpye, dr = (Wot Ws)e —muyp,f=(kW,+iW,), —murpso= (iW, +jW,), we get, it is true, but these two last expressions are only exceptionally representing do, dr respectively, namely, they do so only when Vjdo, or Vidr, satisfy an equation analogous to W/Vidp=0. We put &=jW,/ +kW, and G=jW,+kW.=—muypf, then we have O= ie, CA ? and also dp=Cdt". If we put FS =ap a, 91, we have & 7; Vig, : THE CURVE ON ONE OF THE COORDINATE PLANES. 481 Further, owing to dp = 4d’, the fundamental equation WVido=0 becomes wVit,=0. Now Vit, = Vive) = 6 i?-iSit, as wi=identically zero, the equation becomes now simply W(C)=0. As we have a) W, +kW,’ ? it follows that Vit =kW,/—jw’,, and the fundamental equation also takes the form O=W,Vk—W,hj. If we replace ws and (—w7) by their expressions (already stated in § VIL), wwe =iP,+jR, +hQ, —wWj =iP,+jQ +hR,, the equation will decompose itself into three vector components parallel respec- tively to 2, 7, 4, which have to be annulled separately, so that we get O=W,'P,+W, Py, O=W,/R,+ W,Q, O=W,Q +W,R,. These reproduce the equations of the same form stated in § VI., in which dy and dz are replaced proportionally by W, and by W.’. These three equations are equivalent to only two distinct ones, because as, for example, the necessary relation R,R,—Q?=0 will help to transform the third of the equations into the second one, W,'Q+W,R,= e (W3'R, + W,'Q) =0. For this reason the equation wW'(G) =0 can yield but two distinct scalar equations. We will get them by treating W'(G)=0 by 8.2) and by S. di ). VOL. XXXIII. PART II. 4B 482 DR G. PLARR ON THE DETERMINATION OF Curiously enough, the scalars S.iw’g, and S.(p-Mi)w's, give the same result, at factors “pres” which cannot generally be annulled; this happens owing to the fact that 3 | yypib man W(p) =" ; | but we shall establish this relation further on. | § X. Let us treat first the case S.ab(G)=S. Ei =0 Wi=Yoit Wye We have, by § III. : ee Oe anil an ; y= ae ao ay kb ae | = aeks —kz,); Peers hence vi=i(O- oa Coe =) , and as =x p74 —2, pt , we deduce Dy @ x i wife (4-2) +n(2 —*)] ge oP : | a(2 v ) TYo € u )] x = para — 21] wy Te [— 2% +%%] 2 at ayia —2Yo] - If now we consider the expressions of W,, W., W;, we get Wi =SVjLV Gib = mYy— AN » W,=SViV pjp-Y = 2%) —%% 5 W,=SVyVoko-k=y,%— Yoh « This gives Styri=n,( wee 24 Ua), THE CURVE ON ONE OF THE COORDINATE PLANES. 483 As 2, vanishes only for particular directions of the system a, 6, y, we must assume W,; W, Ws; Wy = v + 0 ? or briefly Skt. UW Let us form also S.p-ae'G =S.Gv(¢-%) . We have ; WPM) =W( Pt) +, (p>). First, , } ne: age | ; We %)= —1 sg" Jp Y- SoM pk. Applying Ne p a rp a Ly = = gVM and Sip Yo = — + sigh, &e., we get ; Pla Ut w(gr=( L--). Secondly, Wer ')= Liv gig N+ p-"(Vig-')]+ a ‘Sp-liig- Mi 1 ; ; i Se ma ‘= V (gt. 1) , ; oe 1 : = <> [—jSkqpit bgt] =— [Jot he Hence mgr hymj(2— 22) 44( 24 2). Having | G = Xyp- i — aH, we get mE Wo =o | a(S 20) gos a 0) \ —2, { mal 0 2) + yol — \ = “Oy 29 — Yor) = “(_W,) + (az — 2%) + = —W.,) ny “(2x4 — 2/1) ala =a —W,). Hence ; W. mSEW G1) = — EI - 484 DR G, PLARR ON THE DETERMINATION OF By the juxtaposition of yi and y(¢~12), vind) 4a(2—2) a ma(p-M)=7 (ane Ha) 7 234 20) v U | | we remark that we have | . . Y, op a 4 orcrnngie sift) 400%] tifa 8) 40(—24 9] ‘a J¢ ue +)=(e%— 04) | Fah gj PORE el old, om a ih oe = 6d Wa tkWe) = 2 U This accounts for the fact that SG, =8E [movi t map(p-)] becomes identically zero, because {, and & are perpendicular to one another by the definition of &. Hence Sov(e-)= — 0 Se pix 0. HN It follows that for any line o directed in the plane of 7 and of ‘2, namely in the plane normal to the ellipsoid at the extremity of p, and containing the vector p,, the results of S. wy(G)=0 will solely give the equation aM 9. § XI. Let us now form the result S.piW'G=8.GH(pi)=0. As ¢# differs generally in direction from 7 and from $-% we may expect an equation distinct from the preceding one. We have WA pi) = ——Spi.ig-Mi — 1 8$i.jp7j — _ Sei.kp-Me. THE CURVE ON ONE OF THE COORDINATE PLANES. (gi) = Vipig += p7Vigi + app USpi.ig-M. Treating first ¥(¢7) by 8.G, the coefficient of SGz will be ss = Seiig-U = aad Then, remarking that we have 6 =a)¢ 41-2, 91, S67 =%%1—%,%,= —W,, SOk=21 —MHYy= + W,, we get ; a We. WY Ae 3 SO woh? =SCt- + SG IG 779) W. : yp Spi kp th). We will at once eliminate ™s by rao, taking 2, W; 2 ee ube ; W U Vv Thus S6ab (pi) = 2[SGi+ Shp Negi] +2 [sig-Vigi-t Skohgi] As ¢"' is self-conjugate we have for the coefficient of = S(j7@-4y +ko—h) pt = —Sig-lt. gr =+W,. Also SGt+Skp— ki = Su(ayp- ta, 2) +S.VijVp-tkgi =H Sip- 4 — x, Sigi+Sigt.x, —SypiSip- tk = SkqjSip-1i— Siggy Shp-ht =S.VkiVg-lidj = -Yee 4 We have thus, first, Staho($i) = — Aj gig-14 WwW Wa im 485 486 DR G, PLARR ON THE DETERMINATION OF Secondly, in S. Zi the first term of ,¢¢ disappears, because G=ao-4 —2x,¢1. We get Sén(oi) = 78.976 | vigi- Taking the $" of the expression of G Soi (pt) = Sloop "t= ayi)(V.igiu® — iW)) = = [wlw’Sipip-2i— W,Sig-2i] +2,W 2). In the coefficient of X we have Sigig-*¢=4 Sigitm,g—i—myit gi] oe By W,, weary 5 xk a: , tet Sig7i= Flap — mgt +2] Ze Mtns Ms , Sipi m m m Sov (92) = = =| x of we —W,-—W (a cai ae a m \|- —2,W, } W, ange ie i “i (m,+ Sigi) +2, } : This gives Summing together S&(o+y1)¢¢ we have the factor W, affecting all the terms, and the equation SOv(¢7) =0, in which for abbreviating we put “© (my + Sipi) +2, =X | becomes ah ee —wWSipjip-U— x | =0. If, instead of eliminating oy we had eliminated ™. —. from SGo(p7), we would have got U3 Ww — sat Bip But as we have [Wi +S pig" lt] = —[S.(Vigit Vidi)" =+8.kpkp-4, THE CURVE ON ONE OF THE COORDINATE PLANES. 487 the last but one equation becomes W, = Mis + wSkopkp- Me -x | =). au? As the factor W,= —4,),¢c,A vanishes only for particular directions of a, B, y, the general solution will be either Mays =VY=wvSgjd "W4+X, or Shp = Z=wSkphp-i—X. But the equation = me =0 is a consequence of these two, because (Y+Z)=wS (Vid) + Vibh)p- 12 = —uwSigip—! = — Wu? ; : Wu Wu we may therefore look upon the expressions of —— and —*— as con- stituting the two equations sought for. § XII. If we express the quantities Y and Z in function of the nine coefficients @,, 01, ©, A, bo, &e., we will find that the expressions contain the same con- _ stant factor A which we have found to affect the quantities W,, W., Ws. First as to X we have €,=Si/p 1k = Za*aa, e5 = Sigh =D“ Sigi=z—+ 1 = Mg=— 2a? Map X =m, +—0(m, + Sigi) 24272 2 nan {4 a(- 8) a2 Heh Taig? = Baya, | a+ Hf = + oe + 4 |} 02, = Layy | pe —a,”)+02(1—b,2) +0°(1 —¢,?) \ 2 a b2¢? r = Lately | P+ P+ e+ pt —a,?) +(—0b,? — 0c?) { j 488 DR G. PLARR ON THE DETERMINATION OF As 2a,a,=0, the term in a’+?+c? disappears, and as a," +b +¢,°=1, we get b 202 2 x-en[ to —0) +4 Ca] b2 eo X= La,4,b,” x a 1) =e Disa? x oS ae 1) : The first term may be written ae ob = Dayasbie( l a ; the second term = Saye i -4 e is a2 By circular permutation of a, 8, , a,, &c., we move the letters one step forward in the = of the first term (qa into d, b into c, &.), and one step back-. ward in the > of the second term (a into ¢, 6 into a, &c.), the terms become respectively, ey sib By Db,dyera (= _ iil ; and i DeyeyeD( ~ ay 2. Thus we get : X = Ea( — )[PyPyry2e? — 0504 28°]. Thus far as to X. For the other term in Y=wSjdjp-i + X we have b \ “ + a + “ J (awa, + Bb°b, +yc"c,) « V. gig N= ( Applying By=a, yB=—a, &c., we get toe 2 be hs ela & is oy) (a? (oe ae Ba “oy = <1,¢,) }2 a + y( —sayb — plats) ‘ Treating by S.j, and applying Saj= —a,, SBj=—by, Syi= —cs, De 3 , ce h2 Hp w= — Fatah oly + atlas a“ e _ pl ntathy + moe j2 a? — Guat + shot, p2 a we = De S - “) 1 Sl — Sa — =) r My Doly < THE CURVE ON ONE OF THE COORDINATE PLANES. 489 We have now for Y=wSj¢jh 1+ X Y= Ea L— 5 )[atanbae, + e,%.by — BO PC4¢). We have 4 P= a2 +0O24+Cc2 . Substituting this into the [| | we get ae a*a,%b,c, + 07b,7a1,b,c, — 07b,7e,¢, + £076,701, b,6, + 67C;2b 9b, « The terms in the second line are =07b,?c,(a,b.—c¢,) , and as — ¢; =Syk=SVaBVy =Saj8SBi—SaiSBj = a,b, — 4,6, we get ab, —— Cy — apd, D so that the terms in 2? become 6b, °c,a, - The terms of the third line are €76,7b,( ae, + 8, ), and as b,; = —SBk=Sayyj = SajSyi — SaiSry = Mel — Azle , hence G40 + b3 = Aye , the terms in c? become Corbiae. We have now v— Ba(S — 5) ata*bats + 07d Feo + 07¢;3a2) And if we put Vi = a7072boe and consider that by § VIII. we have namely, there we have we have thus Y=—-Y'A. With the notation already indicated in § VIII. as to W,, the equation 3 Uy 7) VOL. XXXIII. PART II. Ac 490 DR G. PLARR ON THE DETERMINATION OF becomes now (v— Ee \a--0, v 8 The equation pale ceding one. We have, namely, by § XL, Y¥+Z=—ww,. This gives Z=(wWy +A Z=X(a7a,"a,b,¢, +07a,8b,0,)A = Da7u,9(b,¢, + b,¢,)A = —(2a7a,3b,0,)A . We put 1S) oF. 7 ZL = Za?a,3b4¢, 3 then the equation in question becomes [z-“SEJa=0. Generally we treat the case in which a’, 2, ¢ are different from one another. The equations are therefore In order to render these equations rational in respect to Sat, Saj, &c., we require only one squaring of both members in each equation. s f W. the case with the equation Sar = We may put the two equations under the form 3 24, 3 ut ye v Ay 3 2g 8 US _ yay a ? w As and remark as a curiosity the relation drawn, as, for example, from the first aD is (aa)? (Zaa,?) im oC) so that the second member expresses a certain mean value of the terms appearing in the first member. This is not —Z=0 may be easily transformed by the help of the pre- (I.) (II.) THE CURVE ON ONE OF THE COORDINATE PLANES. 491 ae i108 We are now able to eliminate » and w from the expression of p. We have, namely, pt U p=O+ =jy the. We have also p= —18ig- i —JSip-s—ESheg “i; hence with O=w+jotkw, we get p=wtjot+hkw— iu—j— lo =jythz. We deduce ca ge a? z=w—4. U Multiplying respectively by UY’, uZ', and applying (I.), (II.), we get yuY' =wY' —2,Y'=utW,'—2,Y' zuZ' =uwZ'—y, Z' =wW, —y,Z'. Having 2 = Leraya,, ¥,= 2AM, and U = dara,,2, we may write yu’ = Xa?a,[u?W,'a,—Y'a,] 2uZ’ = X'a7a,[u? W,'a, —Z'a,). Of the factors between |__|, the first is eee + 070110, 90, + Ca, 6,2b5¢5 —a?a,3 — 62h.3 =e2 @70,°b,6, — 6°b,3c,44 — 67¢,3ab, = A] 67,0, (a,b, — 49) + 00,7, (4,0, — 90,)], a, and the corresponding factor in zwZ/ is the same with the index 2 changed into 3, But = t[ b°b,7c,(a,b, —b,a5) + 07C,2b4(a4C, —3C,) J. ab, — bt, = SaiSBj —SBiSaj = SaBji= —Syk=c,; &c., &e. The factors are respectively aa[ 07b,7C,6, — €7C,2D 5b, | and a[ — b7b,2¢.¢, +.07¢,7b,b, |. This gives ; Betas Tp; yu’ = Xa?a,a,[b7b,7c,c, — c2¢,2b,b.| zuZ! pa Xa7a,a5| = B*b,2Co¢, + 07¢,7bob | . 492 DR G. PLARR ON THE DETERMINATION OF If in the second terms we permute the letters one step forward in the series a, b, ¢, a, &¢c., the terms become respectively — >da7b2 2 Xa7b°b,b,02C.¢, + 21476? ,b,0,7C,C, « Joining them to the first terms we have yur! = Xa7b?a4b,C.¢5[ a, — a,b, 2UuZ' = Xa*b7a,,6,¢,( — b,a@,+ 4b, ]. And as 0,4, — a,b, =SBaji=Syk= —c¢,; — b,a,+4,b, = SaBki=Syj = —c,, we get yuY' = —Xarba,b,¢,c,2= —Y, , 2uZ’ = —La*b?a,b,c,"¢,= —Z,. We may remark that the second members may be put under the form + Xa7b*[a,b,+ ab, |c,¢.2 (in the case of the first equation) fobs C2 2 = wie] WD +W,'> so that yuY' =eb?e|W, Skok +W, 9h). Likewise in the case of the second equation aL’ = OZ] W Skopj + Ws'Sj9i]. Hence dividing the first by W,’, the second by W,', and remarking that Mi UE a VE Wie WN eng we get — v 1 PD aes ) Y= WE: ab?e| Skpk + w,'? pk] w W. é 2=—07b[Sidj 2 Skqy |° satel Sigi + wy Shay As de _ We ay OW we may also write these equations under the form yW, dt’ =v8.dpgk , zW,, dt’ = w8.dpdj dt" being a convenient infinitesimal scalar. THE CURVE ON ONE OF THE COORDINATE PLANES. 495 Ne To express the nine coefficients a@,, b,, ¢, a, &c., which determine the posi- tion of the system a, B, y, we adopt for a the angle A which a forms with 7 and the angle B which the projection of a on the plane (7,4) forms with 7. Thus we have a=icosA+)cosBsinA+ksnBsinA, For abbreviation’s sake, we put cos A=a,, sinA=a’ ; eos B=, sim B=0'. Then comparing a=10,+ja,+ka, , a=tatjabh+kav' , we deduce a, =, a,=a'b,,a,=al’. Calling 8, and y, the directions forming with a a three-rectangular system in such a position that 8, be coplanar with a, and 7, we have then Bo=Yoas and y) being perpendicular to z and to a we have Sy,7—0,ny,a—0'), hence Ny) = Via with AN 2—=——1, Thus i‘ — n= V%ig=S%ia—i?. a2, hence —n=a2—-1=—a?. We put n=a' (not =—7’). Thence ay) = Via=ka'b,—jav' ", =—jbl'+kb,. This gives ns : Bo=(—J' +hby)[iay +.a'(7b) + kb')] , hence Bo=kb'a, —102a! +. jb )%) —ia'b,?, namely, Bo= — 1a! +)b,a)+hd'ay. | We now turn the system a, 8), y. round a as axis and to the amount of an angle C, and we shall have B=P Boa, y=P'y0od': where p =cos$C+asin $C g =cos30—asin $C. 494 DR G. PLARR ON THE DETERMINATION OF This gives, leaving ya in the place of ,, | 8 =(cos $C-+a sin $C)y,a(cos 4C—asin $C), y = (cos $C-+asin 3C)yo(cos 40 —asin 40). Developing 8 =cos*3C. ya —sin? $0(ay,a?) +sin $C cos 4C(ayya—ya2), 7 = C08". — sin? SC(aya) +sin $C cos $C(a%— you). ; Having Sy.a=0 we deduce aya? = — ayy =Yoa aryoa = 2aSyya-—Yyoa? = « Hence 8 =ya(cos*$C —sin?4C) + 2y, sin $C cos aC. he Y= Yo(cos?£C —sin?3C) + 2ary sin $C cos 4C. Replacing — Py for ayo we have B=, cos C+, sin C y=) cos C— 8, sin C; or putting also cos C=¢,, sinCac’, aud replacing for Ry, yo their expressions, we get BS (ta! + yb) +kU'ag)e, +(—30' +hby)c’, Y= (—J8 +hby)e, + (ta! —jagb)—kapb')e’ . We have thus B= —20'C) +5 (Aybycy — B'c’) +.1(agb' cy + Bye’) y= ta'c' +) (—agbye' — b'c9) +k —agb'c' + bocy) . This gives us the table of values, including those derived from a: =, a =ab,, a0. bh=—ae), b,=apbe—b'e', bs = Agb'ey + boc’ GF=a¢, C2= —abe'—V'eg, =F = —ayb'ce' + by. We will now express the quantities u?, 0, w?, Wy’, W,', W,’, Y’, Z’, Yn ie in function of the three angles A, B, C, and their dependents ay, a’, by, U, Cos Oe If we examine the values of @,, b,,¢,, and compare them to those of hs, Og, Cg, THE CURVE ON ONE OF THE COORDINATE PLANES. 495 we see that to pass from an expression in which a,, 4,, ¢, enter alone (exclusive of as, bs, ¢;) to the similar expression in @;, b;, ¢,; we have only to change by into b’, and b' into (-—b,). First w= Zara,? = aa? +a(b%¢o? +07?) which is independent of }), 6’. Then v= zara,? Fe igh ta Be rd!\ v* = a?ab,? + b(aybye,—0'c')? +0(-—abyc' —b'c,)?, namely, v2 =b,[a2a? + 0?a.7e,? + cay?c + 2b,b'[ — Page e’ + a ene’ +b2[bc? + ?c,7]. Introducing cos 2B=B, sin2B=B’, we have 2 ,2=14B, 2b2=1—B,. By analogy we introduce also cos 20=C,, sin2C=C’, but we will leave ¢,’, c” as they are, putting only 2c¢,c’=C’. Thus by multiply- ing both members of the expression of wv’ by 2, 2v2= (14+ B,)[a2a? + a,?(bc,? + €2c'2) | + (1—B,)[b2e2 + c2c,?] —B(0?—)a,C’. 2? = [a?a’2 + a,2(b?e,2 + c?c'2) + (7c? + 0c,?)] + B,[a2a’2 + a,2(b2e,2 + c2c'2) — b?e’2 — c2c,?] —B(P—c)a,C’. By the above-mentioned change of }, into 0’, b’ into —b) we get By =0,2— 6? changed into b'°°—0,2= —B, , B’= 20)6' changed into —20',= —B’. Putting [a?a’? + a,2(b?¢,2+ cc) + Bc? + ec,2] =a, Bol a2a’? + a)°(b7e,? + c2c?) — (Be? + €¢2) | — B(P—C2)ay0' = 2" , we have at once 27 =(@+~2’) 2u? =(a—w’). 496 DR G. PLARR ON THE DETERMINATION OF We remark that by the value of w? and by a?+a%=1, ¢2+¢c?=1, we have W2 +X = aa? + a'(b?e,2 + cc?) tata? + ay?(be)? + 0c) + Bc? +.02¢,2 , wa =P+b24+c? =i, . Then We = Andy Cy W, = ab (aqboe) —U'c’)( —aybye’ — Beg) = abel — a97by"eye' +B? cqc' + aghb' (C2 — 4?) ] 4W, =a'b.[ —a,2(1 + B,)C’+(1—B,)0’— 2a )B'C] ='bfa°C' + ByC'’(—1—a,?)— 2B’a0,] . If we put W'=B,C(1+4,2)+2B'aC,, and by-analogy, W,=a', we have 4W,’=a'b,(W,—W’) 4W, =a's'(W,+ W’). As to W,’=«, 6, ¢ it is independent of B,, B’, namely, we get 4W,’ = —2a,0'2C'. For the calcul of Y’Z’, we may prepare the following values. First, we have alread y 4d,c, =a2C0"— Ww", 4b.c, = a'2C' + W’ Secondly, Colty = —(hoc' + 'cy)a'b, = —a'(dPaye' +,b'c,) 4e,a,= —2a'[(1+B,)a,e' + B'eo] 4eyfty = (—2a')fage’ +(Bye'ay + B'c)] 7 40503 = (—2a')[age’—(Byc'ay+ B'c,)] - Thirdly, yb. = WD (Agbyey —U'C’) = 3 [a ¢(1+ B,)—Be']. Sash, = (20’)[ ge) + (Bolt — B’c’)] : ae 4a,,b, = (2a’)[ age — (Boltge) —B’’)] « Substituting into YS 0704, %D50 +0°, 2c, + CC 5U9b, we get 4Y"=a?a,3.[a’2C’—W’] +0% — a4 4°) — 2a’)[age' + Byclay + Bey] +0°(ae')(2a’) age) + Byte) — B'c']. THE CURVE ON ONE OF THE COORDINATE PLANES. 497 Grouping the terms independent of By, B’ separately from those which are dependent on B,, B’, and putting ay’ =X +X’ AX = x we have (remarking 2¢,c’=C’), X= a2a,3a0'2C' + b?a'4a,0'c,? + cata ,0'c? X= a'2a,C'[a*a,” + a(b2c,2 + c2e'2)] X= —2W,'u?, and X' = —a2a3W' + b?a'4c,?[Bya,0 + 2B'c,?] + 0?a'4c2[B,a,0' —2B'c?] . Replacing W'=B,C(1+ a?) + 2B'a,C, we get X’ = Bay C'[ — aag’(1 +52) + a'4(07e,? + 7c?) ] + 2B'[—a?a,'Cy + 4'4(b7e,* — €?c*)] . Putting Be*t+cc2%=d bc —c*c4#=e, we have Xo =a,C [a?a,2a2 + a'4d] X’ = BoayC[ — a2a,X1 + a,2) +0d] + 2B —a?ay'0, + a'4e] . © vk The equations (I.) (II.) of § XII. put under the form vY’ —wW, =0 wZ’—wW,' =0 when rendered rational as to w, v, w, and when multiplied by 2’, are of course (LY (202) x (24Y2) —u8, 25W,,?=0 (IL) = (2w?) x (24Z) — u8, 29W,?=0. The first becomes (0) +2')(Xp +X’)? — u820b,2(Wo—W'P=0. We group the terms so as to put those of even order in By, B’ together, and those of uneven order together, replacing 20,” by 1+ By. VOL. XXXII. PART II. 4D 498 DR G. PLARR ON THE DETERMINATION OF Thus we have for the first equation (I’) : (Xo? + X?) + 2u’X)X’— wa? W,?+ W?—2B,W’'W, | 4-2! (Xq2+X) +.22,X,X’ — uoa2By(W,2-+ W2)—2W,W']=0. The second equation (II’) will contain the same terms as the first, with this difference, that the terms in the second line will have changed signs because of the change of sign of 2’, X’ and W’, and of By. Thus the terms of the first line will be =zero separately, and the terms of the second line will be =zero separately also. The terms of the first line will be obtained by the sum (2Y?—uwW,2) + (wZ2-wW,2)=0. . . . . (IIL) the terms in the second line in question will be obtained by subtraction (PY? —uoW,”)—(wZ — uv8W,7)=0 . : : Mewes (1'5\') By the application of B?2+B?2=1 we may transform the terms of order zero in (III.) or the terms of the first order in (IV.) in order to render the equations homogeneous as to By, B’. The equations will then be respectively of the forms G)B-+G,B,B'+6,B2=0 , HB? +H,B,2B’ +H,B,B?+ H,B?=0. More explicitly it can easily be shown that these equations are of the forms G,'(ByayC’)? + G,'(B,a,C’)B’ + G,B2=0 H,'B,°(a,0’)? + H,'B,2B'(aC’) + HB, B? + H,'B(a,0')?= 0. As to their degree the terms are complete rational functions of the tenth degree in (III.), and twelfth in (LV.), in respect to both a’, a’, and @,c’. If we look on the whole question from a theoretical point of view, we may eee Se eaters B say that the question is now solved, because the elimination of }7 from the two last equations will give us a relation between the two angles A and C, so that one of them, A as for example, being looked upon as an independent variable, will determine the other angle C in this hypothesis, and consequently B, and mediately y and z will depend upon A. Of course this theoretical result, when put to the practice, will lead to inextricably complicated multiple solutions, ; ; Hi tS Bs B owing to the high degree of the resultant of the elimination of 5. THE CURVE ON ONE OF THE COORDINATE PLANES. 499 For the present we have contented ourselves with the treatment of several particular cases, namely, (a) when B,?=+1, B’=0, that is, when the angle B has its extremity at the end of any of the four quadrants ; (b) when B,=0, B?= +1, when the extremity of B lies in the bisecting lines of the four quadrants ; (c) when the angles A and C answer to the values aeos “A— 1 C2 -—- 1, C= 0, in which case the relation (IV.) decomposes itself into the three linear factors (7? +0?) By +(0?—c)B'ayC’, a? By + (a? + 2c?) B'aoC’, a? By — (a? + 20°) B'a,C, and the equation (III.) becomes an identity. = ” - Loch IY aa) a -thit aVe kee ray vast aly @mlite a —_ | nt ae A ‘ “ SAL G.! isGt @. 80184) X XIL.—On the Partition of Energy between the Translatory and Rotational Motions of a Set of Non-Homogeneous Elastic Spheres. By Professor W. BURNSIDE. (Read July 18, 1887.) At the suggestion of Prof. Tarr, an attempt has been made in this paper to apply the method used by him in § 21 of his paper on “ The Foundations of the Kinetic Theory of Gases” to a case of the question of the distribution of energy in a system of non-homogeneous impinging spheres. The problem may be stated as follows :—Given a very great number of smooth elastic spheres, equal and like in all respects, whose centres of figure and centres of inertia do not coincide, and the sum of whose volumes is but a small fraction of the space in which they move, it is required to find the ulti- mate distribution of energy among the various degrees of freedom when by collisions the system has attained a “ special state.” The received result for the general problem, of which this is a comparatively simple case, is that the energy is distributed equally among the various degrees of freedom. MAxweELt’s original proof of this (Phil. Mag., 1860, ii. 37) is hardly more than a statement ; while the reasoning given by Watson, following BOLTZMANN, is on account of its vagueness difficult either to criticise or to verify. The result arrived at here is submitted with great diffidence, owing to its being directly opposed to the foregoing, but it is hoped that the nature of the reasoning is such that each step may be followed and accepted or rejected with- out doubt. As far as possible the notation of Prof. Tarr’s paper is adhered to. | To specify the nature of the spheres, A, B, C are taken as the principal ‘moments of inertia at the centre of inertia; ¢ as the distance of the centre of figure from the centre of inertia ; and a, 8, y as the direction-cosines of the line | joining these two points with respect to the principal axes. In the special state of the system it is assumed— (i.) That the distribution of the linear velocities of the spheres follows the 'same law as for a system of homogeneous spheres, viz., that the number of i spheres whose speeds lie between v and v+dv per unit volume is 13 + Ne e~yrdy , T jand that the velocities are equally distributed as regards direction. (ii.) That the number of spheres per unit volume whose angular velocities VOL. XXXIII. PART II. 4E 502 PROFESSOR BURNSIDE ON THE PARTITION OF ENERGY IN THE about the A-, B-, and C-axes at the centre of inertia lie between o, and wo, +de,, w, and a, +de,, and a; and w;+ dw; respectively is Ke kok. / Bin. THr0%s = has = Kaoe]en, erodiens . Zz 12@200, [This assumption is made by analogy from the form for the speeds, and can only be justified by results. ] (iii.) That for any sphere all directions of the velocity of the centre of inertia with regard to the principal axes are equally likely. (iv.) That the distance c between the centre of figure and centre of inertia of a sphere is very small compared with the radius. . The last assumption is made for the following reason :—The “ opacity,” or power of intercepting impinging particles, of a layer of such spheres as are being considered will depend both on the linear and angular velocities of the spheres, and the probability of a collision between different spheres will no longer be proportional to their relative speed, but to some function of their linear and angular velocities, which even if it could be expressed analytically would almost certainly be of a most intractable form. If, however, the distance ¢ is assumed to be very small in comparison with the radius, the probabilities of a collision between different spheres, and the mean free path, will be sensibly independent of the angular velocities, and hence the same as for a system of homogeneous spheres, while there will still be an interchange between the energies of trans- lation and rotation at each collision. Let w, uw’ be the velocities of the centre of inertia of a sphere in the line of centres before and after an impact :— @1, Wy, @3, @';, wy, w’, the angular velocities about the principal axes at the centre of inertia before and after an impact :— l,m, n the direction-cosines of the line of centres with respect to the principal axes at an impact :— and let large and small letters be used to distinguish the values of these quantities for the two impinging spheres. Write also for brevity, N8—-My=P, nB—my=p, Ly —Na=Q, ly —na=q, Ma—L@=Rk, ma—lB=r. The spheres are said to be elastic in the sense that the energy of a pair of colliding spheres is unaltered by the impact; and Maxwe.i shows, in the paper already referred to, that in the impact of two such spheres the relative velocity of the points of contact in the direction of the line of centres is simply reversed after impact. ‘ a MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES. 503 Hence the dynamical equations may be written (the mass of a sphere being taken as unity) U'+w =U+u U’—u' —f PO’, +70"; + QO’, +90’, + RO’, + 70's] =u—U+¢ PQ) + po; + QQ, +40. + RO; + vas] A(O'",—Q,) = eP(U—U’') BOQ, —Q,) = eQ(QU—U’) C(,-9,) = «R(U—U’) A(@';—@,) = cp(w’—uw) Bw’, —e2) = eg(u—u) C(w'3;—w3) = er(u'—u). The elimination of the angular velocities after impact from these equations leads to the following equations for U’ and w’:— U1+eK)—w(1+ ek) =u(1 —?k) —-UA—@K) +2ca U’ +y/ =U +U, where again for brevity Pp? GF R2 areal amas plang m ape =” Po, + ULOP) + Ts + PO: + QO, + RQ, =@. Hence the increase of energy of translation due to the impact (=T say) = 4 (U2+u2—U2—w?) sont (u— U+ co)( 2a —ck+K)u- U)) (24+eh+K)) In the special state of the system the mean value of this quantity, as also of the increases in the energies of rotation about the three principal axes, must vanish ; and the three independent results so obtained should give 4,, 4,, # in terms of h. In determining the mean value of any quantity connected with a collision of the spheres here considered, the fourth assumption made above permits the integrations involving the speeds and directions of motion of the colliding spheres and the direction of the line of centres to be performed just as in § 21 of Prof. Tart’s paper above referred to. The additional integrations to be performed in this case will be obtained as follows :—Suppose lines drawn from the centre of the unit sphere (in the figure of the paragraph referred to) parallel to the A- and B-axes of each sphere meet the unit sphere in a, A, b, B; that ds, dS are elements of the surface of the sphere, surrounding a, A; and 504 PROFESSOR BURNSIDE ON THE PARTITION OF ENERGY IN THE that w, V are the angles made by ab, AB with ares joining a, A to some fixed point, Then, to satisfy the third assumption, Ps footy dsdSdypay must be taken between the limits 0 and 27 for % and VY, and over the whole sphere for each of the surface-integrals. _ Finally, as regards the magnitudes of the angular velocities, the integrations are SA J MEAD HAAN) “hoe Oden dO er A Qn Aeag4Qg , and the limits for w,, 2,, &c., are = 0. It might appear that, having taken all possible positions for the principal axes, the limits for w,, Q,, &c., should be 0 and o ; but a little consideration will make it clear that these latter limits would be correct only for bodies symmetrical with regard to three mutually rectangular planes through the centre of inertia. In the process of finding T let the integrations with respect to o,, 2,, &¢., be first performed. The only part of T which will contribute anything after integration is ooo’ + q?0r*, ag rw, a P22, a Q202,+ + R202.) — -(k 4: K)(w = Dy [2-+c%(%+K)]? Performing on this expression the integrations indicated, and dividing by the corresponding part of the denominator of 'T, the result is = 29 EOS) eel GO) 44) [PregtmO) 202 The value, at the same step in the process, of the average increase in ,-energy ; at a collision is similarly found to be a s—U)? 1 1 ort (GP be ferme) ++I(ge-d)}) PY mane Se ON EPO Dr 5 a aN te uh) L F [2+eb+ 1)? and from this, by an interchange of symbols, the values of the corresponding quantities for the other two rotations may be written down. In the four expressions so far obtained, write MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES, 005 they become respectively 2c ¢ Ds ole [2+¢e(&+K)| p+P? ab Bene ae od =a (isi U 4) . [praee KO (uw ) &e., &e. (A —(u— U)’) = 2c? In completing the process of averaging on these new expressions, the integ- rations f, J: Meas dsdSdypav affect only the first factors of each of them, and a consideration of the meaning of the factors shows that in each case the integral is a function of the constant quantities A, B, C, a, 8, y and c. [Neglecting terms in ct as compared with those in ’, the approximate values of the averages of these factors are ea ee ee eerie VNR e eae 3 A 3 B » Hence the three equations are @ solution, and therefore must be the solution, of the problem of the “special state.” The quantity (~—U)? finally is affected only by the integrations of the § 21 of Prof. Tair’s paper already referred to; and indeed its value may be written down at once from the result of that article. For 2 (u—UYsw—uw +U?—-Uu=7- The required result then is or, in words :— The average energies of rotation of a sphere about each of the three principal axes are equal, and the whole average energy of rotation of a sphere is twice the average energy of translation. The forms for the average changes in the rotation-energies at a collision indicate that, if at any time before the special state is attained the three rota- tion energies are equal, they will generally tend to become unequal again; and 506 PROFESSOR BURNSIDE ON THE PARTITION OF ENERGY IN THE therefore the problem of determining at what rate the system tends to reach the special state would be intractable even if it were legitimate to suppose the second assumption to hold throughout. If specially constituted spheres, however, are taken in which Sty _yte_@7?+6 _ P) AOSD ena) (AeSpese % an attempt may be taken to determine the rate in question, for the forms of the average changes in energy at a collision then show that if the equations ean Fe, ig hold at any one instant, they will always hold. Suppose, then, that in this case 2, y are the whole energies of translation and rotation per unit volume; so that 3n 3nA o——— = . he Ale By § 14 of Prof. Tar’s paper in conjunction with the forms found above “= —4 = Y kh APBECE, wh _ f8rnaw 8e's? y—2Qe 3 A+B+C 3 If 3E be the whole energy per unit volume, and if ya Bes? in ai L : 3 A+B+C0 TT /E grt MS fasts t= pal G(u-2) 5 the complete solution of which is = t at +1=constant eT , L—-sN on the supposition that the energy of translation is originally greater than one- third of the total energy. The ratio of the quantity T found here with that in § 23 of Prof. Tart’s paper, supposing there the numbers and masses of the two sets equal, is 6c? ALBEE! Hence it would seem that if ¢ is of the order sx10-*, and therefore this P) MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES. 507 fraction comparable with 10~°, the rate at which the special state would be attained is still extremely rapid, Note.—With regard to the form [Eline ttt for the number of particles per unit volume with angular velocities between given limits, the fact that ,, w,, ; are periodic functions while the particle is moving freely, suggests at first sight a difficulty. The period for each particle is a function of its energy of rotation and ofits angularmomentum, Suppose the particles whose energy of rotation lies between given close limits divided inta sets, the angular momentum in each set also lying between given close limits, Then (compare Krrounorr’s Vorlesungen, p. 64) @=pen(rAt+mu), w,=gsn(t+mu), w,=rdn(At+ym) , where p, 7,7, \ and the modulus of the elliptic functions are constants for any one set. | The individuals of each set are distinguished by the values of ». If then the circumstances are such that » may be regarded as a uniformly varying quautity between limits separated by a period for each set, the number of particles cor. responding to the product dw, dw, dw; will be independent of the time, ( 509 ) XXIII—A Contribution to our Knowledge of the Physical Properties of Methyl-Alcohol. By W. Dirrmar, F.R.SS. Lond. & Edin., and CHARLEs. A. Fawsitt. (Plate XX XIII) (Read May 2, 1887.) Since its discovery by Dumas and PEttcor in 1834, methyl-alcohol has been the subject of a great many researches, and as a result we have long had a perfectly certain knowledge of its atomic composition, and a very accurate knowledge of a great many of its reactions. Yet the physical properties of the substance CH,O have not yet been determined with a satisfactory degree of precision. At this we need not wonder. For the study of the transmutations of a species a very impure specimen may suffice, and a series of such studies may leave no doubt about the correct atomic formula of the species in question, and consequently also, if it is a volatile substance, about its perfect gas density. But no other physical properties can be determined otherwise than by direct experiments on a pure specimen. And pure methyl-alcohol is very difficult to obtain. In whatever reasonable sense we may take the word ‘“‘pure,” as attached to the name of a chemical preparation, “pure” methyl-alcohol must be admitted to have been little more than a chemical fiction until W6uHLER in 1852 discovered his well-known (oxalate) process for its extraction from wood-spirit. As a consequence of this discovery, the properties of “methyl-alcohol ” suffered a remarkable change; what had before been known as a more or less unpleasantly smelling liquid, which boils at about 60°C., and turns brown on treatment with caustic alkalies, assumed the form of an almost inodorous liquid, boiling at or near 66°C., and behaving to caustic alkali pretty much as pure ethyl-alcohol does. And it has since exhibited a fair degree of constancy in its properties in the hands of numerous observers, although the wood-spirits which these have used for their preparations must have been of very different kinds. This tends to show that W6xLER’s alcohol (if carefully prepared) is at least a fairly close approximation to the ideal substance, and this impression is confirmed by a research of KrAmer’s, who prepared methyl- alcohol from purified formate of methyl, and found it to boil at very nearly 66° C. _ The successive application of the oxalate and of the formate process would _ probably yield a very pure preparation, because the former tends to eliminate _ the more volatile, the latter the less volatile of the impurities; and we very VOL. XXXIII. PART II. 4F 510 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON much regret now that this idea did not suggest itself to our minds in the preparative stage of our work. We thought only of the WO6uHLER process ; how we applied it we shall now proceed to explain. The raw material which we started with was a particular fraction of wood- spirit, which had been collected for us in the course of an industrial distillation. 50c.c. of this spirit, when subjected to distillation in a fractionating flask, gave the following results :—- Boiling-point at a given stage of the distillation=?°. Total volume of distillate obtained at that stage =» c.c. t = 67° 69° 70° 72° 75° v= 0 20 30 40 50 c.c, Two analyses by the iodide-method gave 71°8 per cent. of real CH,O; by KRAMER and Gropzxrr’s iodoform test we found 5°6 per cent. of acetone. Before applying the oxalate process we thought we had better subject our alcohol to some kind of preliminary purification. We accordingly tried to remove the acetonic compounds by means of alkaline bi-sulphite; but no modification of the method led to satisfactory results. We then gave a trial to. the now almost forgotten method of Kans (fixation of the CH,O by means of a large excess of fused chloride of calcium, removal of what remains volatile by distillation, regeneration of the alcohol from the CaCl,-compound by dis- tillation with water). In this case we worked quantitatively, and determined the acetone in the several fractions as iodoform. The result was highly dis- couraging ; a considerable proportion of the methyl-alcohol escaped combina- tion with chloride of calcium, and the part fixed by the reagent included a considerable proportion of acetone. After this second failure we decided upon confining ourselves to the use of caustic soda as a preliminary purifier; this agent was sure to decompose at least the acetate of methyl; and, if used dry and in quantity, could be expected to destroy at least part of the acetone. In order to ascertain how far the power of the reagent goes in the latter sense, we heated 100 c.c. of the crude spirit with 150 grammes of powdered caustic soda over a water-bath at the “wrong end” of a condenser for some hours, and next distilled over what could be volatilised by immersion of the flask in a boiling water-bath. We obtained 8c.c. of a distillate smelling strongly of ammonias, and containing 4°6 per cent. of its weight of acetone. The residue was then decomposed with water, and distilled over a naked flame. It furnished a distillate which when tested with iodine and caustic soda gave no todoform. We did not stop to ascertain what had become of the acetone, but at once applied the process to larger quantities of the spirit, with this modification, however, that we used less proportions of caustic soda in order not to lose too much of the CH,0O, which could not be expected to survive the process in its entirety. Not caring THE PHYSICA] PROPERTIES OF METHYL-ALCOHOL. 511 to lose time by a systematic elaboration of the process, we at once carried it out with quantities of 3-4 litres of crude spirit, and in each case adjusted the proportion of caustic soda according to our judgment. A so-called Papin digestor, as sold for the boiling of meat under pressure, served as a retort, and worked well. The best results, on the whole, we obtained in an operation in which 3:5 litres of crude spirit were worked up with 1400 grammes of powdered caustic soda. The crude product, when distilled out of a boiling water-bath, yielded 1°5 litres of a highly acetonic distillate, which was put aside. The residue, when distilled with 1°75 litres of water over a naked flame, yielded 1°9 litres of a strong methyl-alcohol, free from acetone and free of unpleasant odour. From their specific gravity, and assuming them to be aqueous methyl-alcohol, the 1:9 litres of distillate contained 1396 grammes of real CH,O as against the 2354 of CH,O which the 3°5 litres of spirit contained according to the iodide test. In other cases the yield was considerably less, no doubt through partial oxidation of the methyl-alcohol into formate or oxalate, CH,O + NaOH = CHNaO, + 2H, and 2CHNaO, = C,Na,0, + H,. We, of course, do not recommend the process for manufacturing purposes ; but it did good service to us by placing us in possession of some 6 litres of very strong and almost acetone-free methyl-alcohol. From it we prepared crystallised oxalate of methyl (or rather a mixture of this ester and methyl- oxalic acid) by a method which Mr ALExanpER Wart many years ago worked out in Dr Crum Brown’s laboratory, and which we will describe shortly because we believe it is little known to chemists generally. Purified alcohol, : : 5 E : : 400 c.c. Oil of vitriol, . 3 : ; ’ P ; 200 c.e. Oxalic acid crystals, ; : 5 ¢ ; 500 grammes. The oxalic acid is mixed with the vitriol, the spirit is then added, and the whole cautiously heated over a water-bath until the oxalic acid is dissolved. The liquid, on cooling and standing over night, deposits an abundant crop of methylic crystals, which are collected and squeezed out in a powerful press. For the regeneration of the alcohol the crystals were heated with water in a flask connected more directly with an inverted condenser, kept at 70° C. ; what remains uncondensed there passes down a condenser, and is collected in a receiver. The resulting distillate is sufficiently strong to be fit for immediate treatment with carbonate of potash, which latter was applied repeatedly as long as it acted visibly. The resulting product was dehydrated further, first by distillation over quicklime, and then by distillation over baryta-lime, a guasi apology for baryta, 512 PROFESSOR DITTMAR AND MR C, A. FAWSITT ON of which reagent we had not a sufficient stock at our disposal. Nitrate of baryta is heated in a platinum crucible until it ceases to decrepitate. It is then ground up finely, and mixed with its own weight of perfectly anhydrous quick- lime (as obtained by the dehydration of slaked lime at a red heat), the mixture put back into the crucible, and then kept for two to three hours at a red heat, conveniently within a muffle; to be converted into a mixture of BaO and CaO, of which 100 parts require, by calculation, 24:6 parts of water for conversion of the two oxides into monohydrates—4°3 for the baryta, 20°3 for the lime. The platinum crucible remains unattacked, and the product is easily reduced to a fine powder. Our general method for these dehydrations was as follows :— The alcohol to be operated upon is analysed approximately by determining its specific gravity ; it is then mixed with only a little more than the calculated quantity of the respective dehydrator, and next “tortured” with it under an inverted condenser. The mixture is then distilled from out of a water-bath, in which the flask is entirely immersed, and the distillation continued as long as anything comes over. The distillate is again operated upon in the same way until two successive distillates, obtained at the baryta-lime stage, exhibit the same specific gravity at the same temperature. At the later stages of the dehydration the receiver is connected with a “vitriol tower” to keep out atmospheric water, and all unnecessary transvasations are avoided, the flask intended for distillation serving as the receiver in the preceding one. For the determination of the specific gravities we generally used a Westphal balance; in the later stages we determined the difference of specific gravity between two successive distillates by means of the differential method, which one of us described in the Chemical News, vol. xliv. p. 5, some years ago. The idea which guided us was to effect a complete dehydration with the least possible loss of alcohol as alcoholate of baryta or lime. We felt quite sure that a methyl-alcohol which suffered no diminution of specific gravity on renewed distillation over baryta-lime really is anhydrous ; yet some doubts about this arose in our minds at a later stage, and we distilled a presumed to be anhydrous alcohol over dehydrated sulphate of copper. The result was that the specific gravity suffered a measurable diminution. — Possibly the alcohol then operated upon had attracted a little moisture from the air (notwithstanding the care with which it had been protected) since its final treatment with baryta-lime. We did not inquire into this point, but made it a rule henceforth not to accept an alcohol as anhydrous unless it had been made constant in specified gravity by means of, ultimately, dehydrated sulphate of copper. As the result of a long and tedious series of dehydrations, we came into possession of a few litres of baryta-lime proof methyl-alcohol, and the question was to prove its freedom from organic impurities. As a first step towards this THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 513 end, we determined the vapour density of our alcohol by means of an apparatus of our own invention, the description of which we prefer to reserve for an appendix to this memoir; suffice it to point to Plate XX XIII. fig. 1, and to state that it is constructed on the Gay-Lussac principle, in such a manner as to avoid the uncertainties in the variable density of the suspended mercury column, and that the vapour volume in it is measured under very nearly the © prevailing atmospheric pressure at about 100°C. Three determinations made with from 90 to 100 milligrammes of alcohol gave the values 16°17, 16:27, 16°22; hydrogen =1; 7.¢., a little more than the theoretical number 16:00.* We could not help noticing that our three numbers are a little above the value demanded by CH,O; yet as the excesses le almost within the limits of unavoidable errors, we preferred to accept our numbers as showing that our preparation was, at the worst, a fair approximation to the ideal substance. We now regret that we did not endeavour to attain a higher degree of precision in our vapour-density determinations by a suitable modification of our apparatus, and adopt the exact vapour density at a high enough temperature as the final test of purity. At the time it appeared to us better to try, and, if possible, prove, the purity of our alcohol by a series of chemico-physical tests, which all agreed in this, that the given alcohol was subjected to some chemical process of fractionation, and the two fractions were compared with each other, and the mother substance, in regard to some exactly measurable physical property. To give a better idea of what we actually did, let us quote a few special cases. J. 300 c.c. of a certain alcohol (I.) were “tortured ” with 90 grammes of dry caustic potash, and the resulting mass was distilled by means of a water-bath as long as anything came over. The distillate, amounting to 120 c.c¢., was put aside as “ A.” From the residue of alcoholate and hydrate of potash the alcohol was regenerated by distillation with water as fraction “B,” which, according to its volume and specific gravity, contained about 142 grammes of absolute alcohol. Each of the two fractions was dehydrated completely by repeated distillation over baryta-lime until its specific gravity became constant. The final specific gravities, determined by the Westphal balance, and reduced to 0° C. by the same formula, were—that of A=0°8138, that of B=0°8142, 7.¢., the two frac- tions were practically identical. * Kramer and Gropzxt (Berichte der Deutschen Chem. Gres., 1876, p. 1928) determined the vapour densities of synthetically prepared mixtures of methylalcohol with acetone or dimethyl-acetal, and arrived at the curious result that the densities of the mixtures differed from the calculated numbers. I have recalculated the numbers from their own data, and arrived at values which agree quite closely with those demanded by theory, z.c., the assumption that the several vapours mix without contraction or expansion. Anybody who cares can easily satisfy himself that I am right. I am glad to avail myself of this opportunity for disinterring a piece of meritorious work which got lost by an unfortunate lapsus calamt in the construction of a formula.—W. D. 514 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON Even absolute identity of specific gravity of course does not prove chemical identity. What goes a great deal further, as was shown by REGNAULT, is equality of vapour tension at a serves of temperatures ; and as a proof of chemical purity independence of the vapour tension, at a given temperature, of the volume of vapour produced from a given weight of substance. This method we proposed to ourselves to chiefly rely on, and we accordingly employed it pretty exten- sively. The apparatus we used will be fully described in a later section ; mean- while Plate XX XIII. fig. 2 may be referred to as giving a sufficient idea of its construction and of the way it is used. To test a given alcohol, we either charged one limb of the apparatus with only one or two drops, the other with some 2 ¢.c. of the preparation, established a convenient temperature (by means of a water-bath) and external pressure, and took the difference of level between the mercury menisci in the two com- municating tubes; or else we subjected the given alcohol to some kind of chemical fractionation, and compared nearly equal volumes of the two fractions in regard to their tensions at a selection of temperatures. Being anxious to avoid everything that might disturb the proximate com- position of a specimen, we at first expressly refrained from boiling off the absorbed air from the samples to be shut up in the tensiometer ; but we soon found that we thus introduced an error which is in general far greater than we had anticipated. After having recognised this error of judgment, we made an attempt at correction for the absorbed air by measuring the tension of a given specimen at two (or more) widely different volumes. Our vapour density apparatus (suitably modified) lent itself well for this purpose. From 1°9-2 c.c. of a certain alcohol, which we supposed to be very pure, were introduced into the tube over mercury ; a fixed-upon temperature was established by means of a water-bath, and the vapour tension then determined at three different volumes. We did not succeed in maintaining absolute equality of temperature at the three different volumes, but had sufficient data from previous experiments for determining the necessary coefficient Ap/At for reducing the several observa- tions to a standard temperature. Two such series of experiments on the same specimen gave the following results :— First Series: t=16°:0 C. Volume of Vapour Observed Tension in in ¢.¢. mm. of Mercury, (1) 16:8 97-05 (2) é 66°8 83°10 (3) 5 129:0 81:18 Second Series: t=11°-0 C. (1) | . 85 ’ 95°74 (2) 83:2 ; 62:50 (3) 129:9 " 60:08 THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL, 515 From either of the two series it should be possible theoretically to calculate the quantities of air retained by the liquid; but a little reflection, based on the requisite formule, shows that the experimental data do not afford the necessary degree of precision for this purpose, Assuming that at any of the three volumes all the air is in the vapour, and taking 2 as representing its volume in ¢.c, at ¢’, and 1 mm.’s pressure, we have, First Series (16°). By combining (1) and (3) . ‘ . x= 306 » (2) and (3) . s : : x= 266 Mean, ; . x = 286 From this mean and by equation p= -+ py, Where p stands for the total pres- sure as observed, and p, for the partial pressure of the alcohol vapour, we have From (1) (2) (3) Do= 80-0 78:8 79-0. Second Series (11°), By combining (2) and (3) we have Po=5O'79 and «=557'2 ce. From (1) and (3) we have Po=57'60 and 2=322:2 cc. From (1) and (2) we have Py) = 58'73 and «=3004 ce. From these results we clearly saw that a sufficient exactitude for p, could not be obtained in this manner, and we therefore fell back upon the old method of ‘boiling out” the specimens to be operated upon in their respective tubes before shutting them up. From a large mass of notes we extract the follow- ing :— Two alcohols, A and B, were mixed to produce about 300 cc. of “D.” The specific gravity found was such that, according to our present alcoholo- metric tables, the percentage of absolute alcohol was 98°66.* Four distilla- tions over anhydrous sulphate of copper brought down the specific gravity so that it now corresponded to 98°81. This alcohol was distilled by itself, and the distillate collected in two approximately equal fractions. Their specific a ; i II, ; eravities, reduced to per cents., corresponded to 99-02 98-90: 28 We now see ; at the time we took them both as representing absolute alcohol. * We propose, for the convenience of the reader, to quote the results in this manner; at the time we had, of course, to go by the specific gravity as a mere index of strength. 516 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON The two fractions were mixed, again dehydrated by anhydrous sulphate of copper and distilled, the first few drops of distillate being rejected. Two drops of what followed immediately were shut up in the right limb of the tensiometer; the distillation was then carried on to near the end, and two drops of the very last runnings collected in the left limb of the tensiometer. A comparison of the tensions gave the following results :— t =30° 50° 65° C. Ap=1°3 15 30 mm. in favour of the earlier runnings. The alcohol “ D” was now tested, so to say, against itself; the left limb of the tensiometer was charged with only two drops, the right limb with some 2 c.c. of the alcohol; both were boiled before being shut up. A comparison of the tensions at 3 temperatures gave the following results :— i 308 50° 65° C. Ap=6'1 ed 79 mm. in favour of the larger specimen. These results were rather discouraging, because they seemed to show that the alcohol was impure. To show the relevancy of the Ap’s, we add the corre- sponding A?’s, which are At=0"7 0°:4 se It however still remains to be proved that the tension of even pure methyl- alcohol is absolutely independent of the ratio of the mass of the vapour to the mass of the liquid which it is in contact with. Methyl-alcohol, in reference to its conversion from liquid into vapour, exhibits anomalies. It “bumps” badly when distilled out of a glass vessel, 7.¢., it may be heated considerably above its boiling point before it actually boils. According to our experience, it some- times exhibits a similar anomaly in experiments for determining its vapour tension by the statical method. The air-free alcohol, when shut up over mercury under a pressure which is considerably less than its maximum tension at the temperature of the bath, may form no vapour at all until the apparatus is being shaken, when a sudden formation of vapour sets in with explosive violence. We are here referring to a series of experiments in which the two limbs of the tensiometer were charged with nearly equal quantities of the same alcohol; but only one of the two samples was deprived of its air before being shut up (in — the left limb). The temperature of the bath being kept rigorously constant (reading of a sensitive but wxcorrected thermometer 51°:7), the tensions of the two samples were determined at a variety of volumes. The heights of the THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 517 mercury columns were left wncorrected for temperature, as we aimed only at comparisons. ‘The following table gives the results of one of a number of series of observations ; v stands for the volume of vapour in the right limb (where the air had not been removed); the vapour volume in the left was not much different from v in any case :— Vapour Tension. v in c.c. Unboiled. Boiled, mm. mm. 115 412°6 409-7 ) 9°8 412°8 409°5 | 8:1 4130 409°9 | : 3 63 413-0 AMO-tf ee =? 47 412'8 409°8 | 3:1 413°3 410-2 J ical 4162 4118 Max.—Min. 3°6 2°3 | Other similar series of determinations with the identical samples gave ' substantially the same results. There would be no use in troubling the reader | with any further account of our tensiometric and specific gravity tests; we | prefer to give our general conclusion, which was that our methyl-alcohol, | although of a high order of purity, was not sufficiently pure to do justice to even | our (home-made) apparatus for measuring the tension of a given vapour. But | what could we do towards the further purification of our substance? The | only course we could think of was to determine the tension-curve of the alcohol | as it stood up to about its boiling-point; to then subject it to some kind of _ chemical purification (say conversion into formate, purification of the same, ‘and regeneration of the alcohol from the purified ester), to determine the tension-curve of the purified alcohol, and compare it with that of the original | preparation. Our tensiometer would have enabled us to determine correspond- | ing differences of tension with a very high degree of certainty. | Supposing these differences to exceed the limits of observational errors, ' the second alcohol must be purified again, say by the oxalate method; and so on until two successive tension curves coincide practically. The mean curve | could confidently be adopted as the tension-curve, and the last alcohol as | sufficiently pure for any physical determination. | This, indeed, had been our programme from the first ; but at the end of our | pioneering experiments our available time had very nearly been exhausted. | We accordingly decided upon just accepting our alcohol as the best apology for the ideal substance which we were able, under the circumstances, to produce, ‘and using it for what we were forced to let stand as our final determinations. me. VOL. XXXIII..PART IT, 4G 518 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON Our apology for publishing these in the following section is the conviction that the numbers, though not what we would wish them to be, are probably better approximations to the truth than those given in the present handbooks of chemistry. The Tension-Curve of Methyl-Alcohol. In regard to it our programme from the first was limited in the sense that we did not intend to go beyond about 760 mm. as our maximum value. The apparatus which we used is substantially the same as that which one of us employed many years ago for comparing the vapour tensions of the two fatty esters C,H,O,.* Figure 2 on Plate XX XIII. shows the form which it assumed on the present occasion. The part which receives the liquids to be operated upon consists of a glass U-tube A, with a vertical tube soldered in between the two limbs. The two side tubes have an inner diameter of about 1 centimetre ; the section of the middle tube is about equal to those of the two side tubes taken together. The side tubes are contracted somewhere near the upper end, and a well-ground glass stopper is fitted into the neck of the cup at the top. The exit end of the central tube communicates with a large bottle, and through it with a syphon-barometer B, In the latter the close limb is so long that the vacuum can be expanded into very much more than the customary volume. This long limb terminates in a funnel-shaped cup, the neck of the funnel being provided with a well-ground-in stopper. A mercury reservoir R, connected with a short side-branch from the open limb by means of a long piece of capil- lary india-rubber tubing, enables one to bring the two mercury-menisci into convenient positions. A small air-pump constructed so that it may serve for exhaustion or compression, and communicating more immediately with the bottle, serves to establish the required degree of attenuation in the latter. The three limbs of the W-tube and those of the manometer bear etched-in millimetre scales. The close limb of the manometer, from the top down to the lowest occurring position of the meniscus, is calibrated so that the unavoidable residuum of air in that limb can be determined by ascertaining the height of mercury supported by the atmosphere at two widely different vacuum volumes. Before charging the W-tube, a true zero plane for the three scales must be found by charging the apparatus with mercury up to a little beyond the three nominal zero-marks, making the limbs exactly plumb, and reading the three menisci in reference to their respective scales. Supposing the zero in the middle tube to be taken as the standard, this gives the corrections for the mercury columns in the two side tubes. By means of the well-known artifices, it is easy to fill the bends of the tensiometer with ar-free mercury. More mercury is then run into the middle * Chem. Soc. Jour., [2], vi. 477; Annal. d. Chem. u. Pharm., Suppl., vi. 313; Jahresb. f. 1868, p. 500. THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 519 tube through a long-necked stop-cock funnel until the metal comes almost close to the necks of the two cups. Supposing now two specimens of methyl-alcohol to be operated upon, a small quantity of one is introduced into, say, the left limb, boiled there to expel its air, the stopper inserted, a little mercury poured into the cup, a little of the respective alcohol added, and the cup closed by means of a small glass cap fixed on by means of good india-rubber tubing, so that the two glass rims touch each other inside. The india-rubber is well wired on both sides. In a similar manner the right limb is charged with the other specimen, and the greater part of the mercury of the middle tube syphoned out. The charged tensiometer is suspended in an exactly vertical position within a large square water-bath, the front and back of which consist of plate glass, and connected air-tight with the bottle, and thus indirectly with the manometer. A standard thermometer, from GEISSLER of Bonn, suspended in the water-bath, gave the temperatures. The higher temperatures were established by means of a properly adjusted mixture of hot and cold water, and maintained by means of steam sent into the bath through two block-tin pipes which pervaded the bath in its entire height. By properly regulating the current of steam, and perpetual agitation of the water in the bath, we soon learned to keep even the highest temperatures constant to within +0°1C. Immediately before and immediately after each series of experiments the zero correction of the ther- mometer was ascertained, to be allowed for in the ultimate record. To test the thermometer for the exactitude of its calibration, we constructed an air- thermometer—pretty much on the Jolly principle—and determined the true temperature-values for a large number of the marks on the Geissler standard by several series of experiments. As we had no real cathetometer at our disposal, we were not able to bring down the wncertainty of the temperature values, as determined by the air-thermometer, to less than 0°:1 to 0°-2 C., but within these limits the Geissler instrument (as corrected for the zero displace- ment) proved correct. We subsequently procured a large thermometer from Mr CaseEtiA (London), which was made by him out of a long capillary tube which we had calibrated most thoroughly by means of RupBEre’s method. This standard thermometer, by its calibration table, gives temperatures correctly to within about 0°02 C.; but, unfortunately, before we had a chance of using it for standardising the Geissler instrument, the scale of the latter (which is on a separate glass strip enclosed with the thermometer stem within a glass jacket) became loose, and we could not manage to refix it in absolutely its original position. The millimetre scales on all our instruments we made ourselves by means of an excellent screw-engine from Biancur of Paris, which Professor Tart kindly lent to us. In the numerous preliminary tension determinations, which we referred to 520 PROFESSOR DITTMAR AND MR OC. A. FAWSITT ON in an earlier part of this paper, we always worked with two samples of alcohol at the same time, and these determinations included a complete rehearsal of the determination of the entire tension-curve. But in the final series we preferred to charge only one limb to reduce the number of readings by one. The actual routine of the work hardly requires to be described. After having established the desired temperature, we read first the three tensiometer limbs, then those of the manometer, and, lastly, the thermometer a second time, by means of a horizontal telescope. The temperature was in all cases rigorously constant, as far as one could read. To eliminate part of the error arising from unavoidable variations of temperature during a series of read- ings, we found it an improvement to close the open end of the manometer, and thus fix its mercury-menisci in their positions, immediately after reading the limbs in the tensiometer. As a rule, we commenced with the lowest temperature to proceed step by step to the highest, and then retraced our steps, so that each series consisted of an ascending and a descending section. The alcohol used for the final tension determinations was specially dehy- drated (by means of CuSO,), and a sufficiency kept in a sealed-up tube until the tensiometer was ready for its reception. The height of the several mercurial columns was reduced to 0° C., but not reduced to any standard latitude, for an obvious reason. A preliminary survey of the results showed that they fell in approximately with the equation log p=a+0t. We accordingly for a first approximation adopted this function, determined the constants @ and 6b graphically, and from them calculated the values log p, corresponding to the several observational ?s. For a second approximation we laid down the ?s as abscissze, and the corresponding values, “Ay” = (log p as observed) — (log p as calculated), as ordinates in a system of rectangular coordinates, when Ay appeared to be a function of ¢, according to an equation of the form Ay=a+ t+ yt. We then calculated the constants a, 8, and y from three measured ordinates, and thus established an equation, log p=u'+b't+cl?+ d(log p), where 6(log p) stands for the residual correction needed to establish equality between the two sides of the equation. These residuals d(log p), when repre- sented graphically in function of ¢, suggested an equation of the 4th or 5th degree ; but on looking more critically into the matter, we found that this final THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 521 curve registered observational errors rather than anything else, and that a small constant correction applied to a’ did full justice to our results. The final formula adopted was log p=1:4731+40:02649¢ — 0:00007422?. The following table gives the results of our final series of determinations; i.¢., the logg. of the observed p’s, contrasted with the values calculated by means of the formula. The column “ Exp.—Calec.” gives the correction to be applied to the calculated logarithm of p to obtain log (p as found). The last column, under Ap, gives the corresponding difference between the two values p them- selves : Logarithm of Temp. C. Exp. —Cale. HMotbars . Calculated Observed Tension Tension. +. 4°15 P5sL7 15761 —0:0056 0-6 9°15 1:7092 17097 +0:0005 0:07 9°95 17293 1:7318 +0:0025 0-4 14°15 1:8330 1°8373 +0:0043 0-7 ileal 1:9531 1:9542 +0:0011 0:2 2415 2:0695 2:0738 | +0:00438 1-2 29°15 21821 2°1837 | +0:0016 06 oa 15 2:2911 2:2940 +0:0029 13 B15) 2°3963 2°3947 —0:0016 1:0 44°15 24978 2:4988 +0-0010 0-7 49°15 2:5956 2°5936 —0-0020 2:0 54°15 2°6898 2°6895 —0:0003 03 5OP15 27802 2:7792 —(0:0010 15 64°15 2°8669 2°8682 +0:0013 2°4 Gb" kD 2°8837 28847 +0:0010 18 By means of the interpolation formula, given above, the table on the fol- lowing page was calculated. The Specific Gravity of Anhydrous Methyl-Alcohol. For these determinations we used small cylindrical bottles of the form represented in the figure on page 522. The body ¢ holds about 20 c.c.; the stem 5 bears a millimetre scale; 1 mm. corresponds to very nearly 0°01 c.c. To ascertain the capacity at any mark and any temperature which it might be convenient to use, we first determined the capacities for water of 14°-7 (the “15” of a certain delicate thermometer) up to 0, 5, 10, 15,20 mm. We then determined the capacities for water up to 0 mm. (directly or indirectly) at exactly 0° (in melting ice), and at temperatures near 15°, 20°, 30°, 35°, 40°, 45°, 522 Temp. | Tension. | Diff. Temp. | Tension. Diff. Temp. | Tension. Diff. 0° 29°7 ae 25° 122-7 +6:2 50° 409-4 | +17°7 1° 316 +1°9 26° 129°3 6°6 51° 427-7 18:3 2° 33°6 2°0 27° 136°2 69 §2° 446°6 ites) 3° 35°6 20 28° 143-4 72 53° 466°3 19-7 4° 378 2:2 29° 151:0 76 54° 486°6 20°3 5° 40°2 2°4 30° 158-9 79 55° 507°7 21:1 6° 42°6 24 |) 31° 1671 8:2 56° 529°5 21°8 ne 45:2 2°6 32° 175°7 8°6 57° 552:0 22°5 8° 47-9 27 33° 184°7 9°0 58° 5753 23°3 9° 508 2°9 34° 1941 94 || 59° 599-4 241 10° 53°8 30 BD's 203°9 9°8 60° 624:°3 24:9 ia 57:0 3°2 36° 2141 10°2 61° 650°0 25°7 12° 60°3 3°39 oh 224°7 10°6 62° 676°5 26°5 13° 63°8 33 38° 235°8 ele 63° 703'8 27°3 14° 67°5 3°7 39° 247°4 11:6 64° 7320 28°2 15° 71-4 3°9 40° 259°4 12:0 65° 7611 29'1 16° 755 4:1 41° 2119 12°'5 66° Told 30°0 ae 79'8 4:3 || 42° 285:°0 13'1 67° 822°0 30°9 18° 84:3 4:5 43° 298°5 13°5 -————— —_— oe 89:0 4:7 44° 312°6 141 ||\64°°96| '760:0 20° 94:0 50 45° 327°3 14°7 21° 99°2 52 || 46° 342°5 15°2 22° 104°7 55 || -47° 3583 15°8 23° 110-4 57 48° 374-7 164 24° 1165 6-1 49° 391°7 17:0 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON 50°, 60°, 65°. To check the results we also determined the capacities for mercury at a similar series of temperatures. From the sum total of our experiments we calculated a formula C)=C + 4¢ for the capacity at # in grammes of water of the density corresponding to 4°C. (z.¢., the capacity in c.cs.), adopting such values for C, and £ as fell in best with our best determinations. In all these determinations, as also in those of the capacities for alcohol, the temperature 0° was established by means of a bath of melting ice, higher temperatures by means of a water-bath. The alcohol operated upon was protected against atmospheric moisture by means of a dried well-fitting cork inserted in the funnel-shaped tap of the bottle. The exact tare of the bottle was taken without the cork; but in the weighing the cork (compensated for by a piece of metal placed in the other pan) was left on until the still remaining difference of weight could be determined (after the removal of the cork and of its tare) by three consecutive readings of the oscillating needle. The results are given in the table on next page:— gives the Vapour Tension of Methyl-Alcohol in Millimetres of Mercury of 0° C. THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 523 temperature (corrected) ; 2 the level of the alcohol in the stem of the bottle ; “Cap” the corresponding weight of alcohol in grammes ; ,8, the weight of 1 c.c. of alcohol of ?, in grammes, reduced to the vacuum. In these reductions the density of the air was assumed to be at the constant value of 1:2 grammes per litre, which is quite permissible in a case like the present :— Bottle No. I. Bottle No. II. t. h. Cap. 48t. h. Cap. 4St. 0° 20°0 16°3670 0810 24 20°0 16°3845 Q:3 LO; 11 4°-7 20:0 16-2770 0805 69 20°0 16°2940 0-805 53 Sell 7:0 16:0955 0801 20 8:0 1671155 0-801 19 14°-7 19:9 » 16:0985 0°796 66 20:0 16-1195 0-796 69 iS) arg 20:0 16-0088 0°792 09 20:0 16:0250 0-791 94 Zoey 20:0 158217 0°782 64 20:0 158370 0°782 41 39°°7 20°0 156305 0772 93 19°8 15°6455 0-772 82 49°-7 20:0 15-4420 0°763 41 20-0 154580 0°763 28 59°-7 20°0 152380 0°753 12 18-0 152420 Or75a NS 64°°7 10 15-0150 0°748 25 —2°5 14-9910 0747 85 The two bottles were immersed in the same bath. A preliminary survey of the results showed that, in accordance with é ne AS. : MENDELEJEFF’S proposition, a is very nearly constant, so that an equation So—S,=at+b¢ was sure to do sufficient justice to the observed relations. By a proper combination of observations we arrived ultimately at the interpolation formula, S,—S8,=90°53¢-+ 0:0850572 , which, as the following comparisons show, sums up the results satisfactorily :—— Specific Gravity at t°=4Sz. ts Caleulated. Observed. Oy ; ; ; ‘810 18 ‘810 18 4°-7, : : 805 91 805 61 SET, . é : 801 32 801 20 14°-7, ; : ‘796 69 ‘796 67 LOT Ts ; ; 792) 02 “792 02 a ae : , : “782 54 782 52 DOG Is : : : 772 90 12 87 49°-7, : , ; ‘763 09 ‘763 34 RUST : ; ; ‘753 10 oo be 64°-7, 5 : ‘ ‘748 05 ‘748 05 The alcohol used for these experiments had not been rendered air-free by boiling, but we of course took good care, especially in the determinations at higher temperatures, to make sure that there were no air-bells at the sides of the bottle when the level in the stem was read off. To form an idea of the 524 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON error introduced by allowing the dissolved air to remain, we made a determina- tion at 14°°7 with air-free alcohol. The alcohol was boiled in the specific- eravity bottle under an inverted (dry) condenser, then corked up while still hot, brought to 14°°7, &c. The specific gravity of the air-free alcohol was found equal to 0°79683, z.¢., by 0°00016 higher than that of the original prepara- tion. The difference barely emerged from the limit of unavoidable errors, and we thought that we should probably risk more than we could possibly gain if we adopted the more refined method as the method. Being in possession of a sufficient quantity of the kind of highly purified alcohol which had served for the above determinations, we thought we ought to supply to the chemical community what has hitherto been felt as a desider- atum, namely, a set of tables giving the specified gravity of any aqueous methy]l- alcohol as a function of its percentage and temperature. We accordingly pre- pared, by exact gravimetric synthesis, a series of aqueous methyl-alcohols, containing as nearly as possible 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95 per cent. by weight of absolute alcohol, and then determined their specific gravities in each case at 0°, 9°:7, and 19°'7 C. The modus operandi was exactly the same as that for the anhydrous alcohol. All the weighings were reduced to the vacuum, and water of 4° adopted as the standard substance at all tempera- tures. The numbers given as specific gravities accordingly may be read as giving each the weight of 1 c.c. in grammes. All determinations were made in duplicate—one with bottle No. I., another with bottle No. II. In the follow- ing table of results, ““p” gives the percentage of real CH,O; the three columns D give 100,000 times the weight of 1 c.c., @.¢., the value ,S, for <=0°, 9°°7, and 19°-7 respectively as a mean of two determinations. ‘‘A” is the deviation of the two determinations from the mean :— Specifie Gravities 8, of Aqueous Methyl-Alcohols. At 0". At 9°°7. At 19°*7. p - = = D A D A |i D A 95:062 823 82 45 || 815 30 20°5 806 30 65 89-990 837 46 10°5 829 00 8°0 820 44 8:0 79°959 863 54 45 || 855 48 2'5 847 24 10:0 70°063 886 78 6°0 879 45 4:5 871 58 65 60:020 908 95 50 901 95 35 894 75 6:0 50°022 928 62 0°5 922 3 0°5 915 73 5) 40°028 945 85 5°0 940 45 7:0 934 67 0°5 30°023 960 39 aos) 956 11 75 951 58 0:0 20°032 972 46 2°5 970 00 6°5 966 67 30 10°018 984 22 4:0 || 983 42 0°5 981 54 2:0 5008 991 41 60 | 991 18 Be) 989 61 15 THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 525 With these data before us we began by formulating the relation for the several kinds of aqueous alcohol operated upon, between specific gravity on the one hand, and temperature between 0° and 20° as limit values on the other. With the stronger alcohols the simple formula S,—S,=at afforded quite a satisfactory degree of approximation; the actual function, it is true, proved obviously non-linear, but, when the formula S,—S,=at+¢? was substituted, b assumed, in general, so very small a value that, with its own a, the simpler function was practically as correct as the more complex one. Thus we found for absolute alcohol :— At 0° ee 1927 gst = 81018 80120 79202 S,-Si= 0 898 1816 Boa Sr 92°58 92:17 whence s,-S ; *=92-98-—0-041t; or, if we adopt the linear function S)—Se re 92°37 + 0°20, which latter function gives, for 20°, S,—S,,=1847'4 + 4:0, z.¢., an uncertainty of +0:00004 only. Our general formula for S,=/(¢), however, gives calculated values for 0°, 10°, and 20°, which when used as a basis for the calculation of the a, of the linear function, assigns to it the value 91°75 + 0:43. We found subsequently that this value falls in better with the general relation between @ and per- centage than the value 92°37. For the aqueous alcohols we found the following values for the constant of the linear function :— Percentage of CH,0. Value of a. 100 91°75 + 0:48 95 89:40 + 0°48 90 86°81 + 0°41 80 82°86 + 0:12 70 76:35 + 0°79 60 72:17 + 0°09 50 65°26 + 011 40 56:20 + 0:53 30 44°39 + 0:27 20 27°30 + 2°04 10 8:25 for 9°7 and 13°60 for 19°-7 VOL. XXXIII. PART II. 4H 526 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON From 20 per cent., or rather somewhere between 20 and 30 per cent. downwards, the value a as we see becomes izconstant. Retaining the second term in the formula 8,—S,=at+0?, we found the following values for this « and for b :— Percentage of CH,0. a. is 100 to 80 a Some mall 70 7403 +0158 60 7243 —0-018 By 6494 +0-022 2 5463 +0°107 a sie 43-59 +0-055 = 2130 +0408 a 306 +0535 | S —405 +0°672 | iH —565 +0685 . Seeing that the equation 8,—S,=a affords a sufficient approximation for all alcohols from 30 per cent. to 100 per cent., we tried to calculate an inter- polation formula for the relation between the per-unitage p of CH,O and the value of @ which should cover the whole of this interval, but found that an equation of the second degree did not establish sufficient agreement between experiment and calculation. We ultimately divided the interval into sections as follows :— I. From p=1 to p=0°6 (z¢., from 100 per cent. to 60 per cent.). Adopted formula | a=a,+bp+ep?; the constants were calculated from the six experimental values by the method of the least squares, and found to be 35°018 ; log a,=1544 39 68:379; log b =1:834 92 —11°718; loge =1:068 84 b C Hoi dl Contrasting the calculated with the experimentally determined values of a, we have : p. By eas By Direct Det. Mg 91°68 91°75 0:95 89°40 89-40 0:90 87:07 86°81 0°80 82°22 82°86 0°70 77:14 76°35 0°60 71°83 72:17 * By Rosetti’s Table for the specific gravities of water. Our constants give +4°:1 as the temperature of maximum density. THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 527 II. From p=0°6 to p=0°30. Again adopting the function a,+bt+ ct, and calculating by the method of the least squares (which in this case, it is true, is almost out of court), we found— @=— 6388; log a,=0°805 37 =++207°53, ; log & =2:317 08 e =—12788 ; loge =2:106 82 aS which gives the following values :— Dp By wonaula, By Direct Det, 0°6 72:09 (DAT 0:5 65°41 65:26 0:4 5616 56°20 0:3 44:36" 44:39 IIL For alcohols of less than 30 per cent. we adopted the function S,—8,=at+ 00, and accordingly had to establish the relations between a and p and between 2 and p. No doubt the best mode of procedure would have been to bring the equation into some form like S$) —-8,= (ap +apt+yp’)t+ (B+ 2p + wp)t?, and to calculate the constants directly from all the experimental data by means of the method of the least squares ; but we shrank from the very troublesome calculations which this would have involved, and satisfied ourselves with estab- lishing the relations a=/(p) and b=4¢( p) by separate graphic interpolations. The results were as follows :— p Const. a. Const. 6. Curve. Exp. Curve. Exp. 0 — 60 — 5:65* +0°705 +0°685* 0:05 — 22 — 405 +0°648 +0°672 0°10 + 3:3 + 3:06 +0°581 +0°535 0:20 +20°0 + 21°30 +0°398 +0:408 0:30 +440 +43°59 +0'060 +0:055 The values a and 6 for p=0°01,0°02...... up to 0°30 were read from their curves and tabulated (see the alcohometric table below). The agreement between experiment and calculation is not as perfect as we * Rosetti’s Table. 528 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON should have wished; but we could not see our way towards doing better, and consequently left the subject on one side to proceed to calculate interpolation formulee for The Relation between the Per-Unitage p of Methyl-Alcohol and the Spectjic Gravity So at 0°. As the specific gravity for p=0 must be that of water at 0°, which we will call W,, we at once adopted the difference ,W,—,S, as our dependent variable, but found it convenient to take ,W,=1000, and adopted “y” as a symbol for the value which the difference then assumes; while ‘‘x” was substituted as a handier symbol for the per-unitage of CH,O. A preliminary graphic interpolation showed that there is a change of curva- ture somewhere about 2=0-20 (corresponding to 20 per cent.), showing that if a parabolic formula worked at all it must at least be of the third degree. Warned by MENDELEJEFF’S experience with ethyl-alcohol, we never attempted to obtain one formula for the whole curve, but at once decided upon dividing it into sections. As the part from #=025 upwards exhibited no change of curvature, we tried a variety of functions, including the general equation of the second degree (Ay’+ Bay+ Ca’, &c.), for summing up the relation y=/(2) for z=03 to 1:0 in one formula, but arrived at no satisfactory result. After a deal of pioneering, we ultimately came to divide the curve into the following sections :— I. From z=0 to xz=0°4. MeL Me=0Stor—0 74 TEE, Cea = Ob tow =: First Interval. We began by bringing the formula y = az + ba’ + ca’ into the form a+ bx + cx? — £ =0, and then proceeded to calculate the constants a, b, c by means of the method of the least squares. This, as we now see, was an error of judgment, because it is the error in y and not that in y+ which must be brought to its minimum value ; yet the result was satisfactory all the same. We found a=+185:079; log a=2:267 357. b= — 348°682 ; log b=2'542 429. c= + 559542; log c=2°747 833. THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 529 affording a sufficient approximation, as seen by the following comparison. “Ay” means the uncertainty in the y as determined experimentally (see p. 524). aD y calculated, y by exp. Ay. 0:05008 8:465 8:46 +0:060 010019 15-609 15°65 0:040 0-20033 27°582 27°42 0°025 0°30025 39:281 39°49 0:035 0:40029 54104 54:02 0:050 ” As a glance at the two columns “y” shows, the error of the formula amounts to —0°209 in the case of x=0°3; the other errors, arithmetically, range from 0°005 to 0°162. Second Interval. x%=0°3 to 0°7. By proceeding exactly as in the ease of the first interval, we found a=+129°871; log a=2:113 512. b= — 265315; log b=1:423 762. e=+10271; loge=2:011 629. Having found before that a general formula of the third degree for the whole interval from x=0°'3 to 1:0 gave fair, though not quite sufficient approxima- tions, we made only two comparisons between y calculated and y found. iif y calculated, y found. Ay. 0:30 39°332 39°49 0:035 0-70 113021 113°09 0-060 Third Interval. x=0°6 to 1:0. Again, operating as before, we found a=-+ 89:001; log a=1:949 395. 6=+109:012; log b=2-037 476. C= G6 2501; Sos ¢c—0 918 aie We made only one comparison this time : For z=1; y=a+b+c=189°724. By experiment, y=189°69. After having made sure of perfect arithmetical correctness all round, we next calculated by means of the formule the values ,S, for all the experimental 530 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON values 2, using each formula for the full range which it-had been calculated for. We thus found that the best agreement between experiment and calculation was obtained by using formula II. for the interval z=0-4 to x=0°6, and formula I. and III. for their entire intended ranges. But the actual utilisation of the formule in this manner would have produced unpleasant, though by no means alarming, discontinuities in the final table at z=40 and x=60; we therefore ultimately decided upon exhausting each formula, so as to obtain duplicate values for the y’s corresponding to 2=0°3 to 0°4, and to =0°6 to 0°7, and for each such x took the mean of the two competing values. This is the history of the entries for ,S, in the following tables. From these values ,S, and the values a=(S,—S8,)~+¢ as calculated by means of the interpolation formula, pp. 526 and 527, we calculated the specific gravities ‘Si55s for 60° F. for all the percentages from 31 to 100. The corresponding values ,8,;.. for the alcohols from 30 per cent. down- wards might have been calculated similarly from the values a and 6 in formula S.—-S,=at+0? as obtained graphically from the directly calculated values (see p- 527), but we had no perfect faith in these interpolations, and therefore preferred to calculate the specific gravity at 60° F. of each of the alcohols experimented on from the results obtained at 0°, 14°°7, and 19°-7, and, from the set of values ,S,,.;, thus obtained, to calculate the coefficients of a formula y=aa +ba’ +cx’ by the method of the least squares. In this case y = 4W45.s¢—4Si5-56, Where the first symbol stands for the specific gravity of water at 60° F.; water at 4° being taken as = 1000. Calculating on the basis of a+ bxa+cx?— : —0, we found a=+180°522; log a=2-256 530. b= —191:450; log b=2°:282 055. e=+318-220; log ¢=2'502 727. To test the equation, we calculated the y’s for 20 per cent. and 40 per cent., and found Value of y by x. Formula. Experiment. 0°200383 31:039 30°92 0:40029 61:995 61:98 The values given in the following table for ,8,,., up to 30 per cent., are calculated from this formula ; those calculated in the same way for 31,32, .. . 40 per cent, agreed very well with those deduced from the values ,S, and the equation S1;.5;=S)— 15°56 a; but we preferred to let the latter stand uncorrected. As some of our readers may consider this an error of judgment, we give the two values in the following table :— THE PHYSICAL PROPERTIES OF METHYL-ALCOHOL. 531 Value of ,S,;.., calculated Percentage of From 4S) and From general equation 40. linear equations. HOO risers 30 953 67 953 55 dl 952 11 952 03 32 950 53 950 48 33 948 94 948 91 34 947 32 947 32 35 945 67 945 70 36 943 99 944 05 37 942 28 942 37 38 940 55 940 66 39 938 77 938 91 40 “936 97 937 13 TABLE giving the Specific Gravities of Aqueous Methyl-Alcohol at 0° and 15°56 C.; Water of 4° =100,000. I. From 0 to 30 per cent. of CHsO. 9 4S:= 489 —(at+ #2). Penepiage ot | Sector] inerence pou, step .| an 'boay, | PEST | Diseeee 0 999 87 bes —6'0 +0°705 999 07 2 1 998 06 —18r | 54 694 997 29 178 2 996 31 175 4-8 ‘681 995 54 as 3 994 62 169 3°9 ‘670 993 82 172 4 992 99 163 30 659 =| 992 14 168 5 991 42 157 2:2 ‘648 990 48 166 6 989 90 152 1:2 634 988 93 TSS t 988 43 147 —0:2 ‘621 987 26 167 8 987 01 142 +0°9 609 985 69 157 9 985 63 138 21 596 984 14 155 All Positive. 10 984 29 134 +3°3 0581 982 62 152 tal 982 99 130 4:8 569 981 11 151 12 981 71 128 62 "oe 979 62 149 13 980 48 123 78 536 978 14 148 14 979 26 122 9-5 “pg 976 68 146 15 978 06 120 11:0 ‘500 975 23 145 16 976 89 117 12°5 ‘480 973 79 144 ify 975 73 116 14:5 ‘461 S725 a5 144 18 974 59 114 16-2 ‘440 970 93 142 19 973 46 ie 18°3 ‘420 969 50 143 20 972 33 113 20:0 398 968 08 142 21 971 20 r3 22:2 373 966 66 142 22 970 07 1a 24:3 350 965 24 142 23 968 94 113 26-4 coll 963 81 143 24 967 80 114 29:0 ‘291 962 38 143 25 966 65 115 ol3 ‘261 960 93 145 26 965 49 116 33'8 ‘230 959 49 144 27 964 30 119 36-0 191 958 02 147 28 963 10 120 38°8 Tod. 956 55 147 29 961 87 123 411 106 955 06 149 30 960 57 130 44-0 063 953 55 I51 or oo bo Percentage. PROFESSOR DITTMAR AND MR C. A. FAWSITT ON II. From 30 to 100 per cent. sS,—,8,= ena ee 100° 0. 200°C. 350°C. *43 1 0° — 100° 0-0 =O 432 0°— 100° 0:0 —O01 433 0° — 100° 0-0 0-0 434 0° — 100° 0:0 0-0 435 0° — 100° 0:0 +01 436 0° — 200° 0°0 00 0:0 437 0° — 200° 0-0 +02 +01 438 0° — 200° 0:0 +01 +01 439 0° —200° 0-0 0°0 0:0 440 0° — 200° 00 +0°2 +01 441 0° — 350° 0:0 +015 +01 ¥ 4.4.2 0° — 350° 0-0 +0715 +0°25 443 0° — 350° 0:0 +01 +0°25 “eh ~ 444 0°—350° 00 + 0°25 sie +03 ¥4A5 0° — 350° —0'1 —O1 ass +01 446 0° —350° 0-0 +0:2 Binia +03 44:7 0° — 350° 0-0 +015 ed +01 #449 0° — 350° 0:0 +01 a +01 *4.50 0° — 350° 0:0 +0°05 ane +0°05 * Not used in the experiments. But after several exposures to high temperatures, sometimes nearly 300°, the thermometers began to show an error, and that one of increase, in the indications. The amount of this error in each instrument was found by comparing the reading taken when the long bar had cooled down overnight after the previous day’s heating, with the simultaneous reading of a thermo- meter in the short bar standing in the vicinity of the long bar. This latter thermometer was only used for temperatures which did not exceed that of the air, and had never been subjected to any such treatment as had those it was employed to correct. The difference of the two readings named was taken as the error of the first, and was applied in the reduction of the experiments. In the case of the thermometers used in the hottest hole in the bars, this error amounted to about seven degrees at ordinary temperatures. A series of corrections for the different thermometers was thus obtained, and in order to test whether the readings obtained from one day’s work, and after the application of these corrections, were consistent with those of another day, the thermometers in the first three holes were interchanged among each other, and also with the thermometer which was always used to indicate the temper- ature in the cooling experiments on the short bar. It was found that the different sets of readings when so tested were consistent with each other. This, however, is a source of less error than that which arises from the circumstance that the temperature (especially at any point near the hot end) of the long bar during an experiment is given by a thermometer with an OF IRON, COPPER, AND GERMAN SILVER. 541 unequally heated stem. In connection with this, Prof. Tarr calculated that for a temperature of 250°C. “the utmost error that can be introduced in the indications of the thermometers used is somewhere about 10°C. That is to say, the highest temperatures were read, at the most, 10° Jower than they would have been if the whole thermometer had been exposed to the same temperature. This correction of 10° at 250° diminishes at lower temperatures and increases at higher, nearly as the square of the excess of the temperature above the freezing point ” (7rans. R. S. E., 1878). Wishing, if possible, to obtain some experimental verification of this com- putation, I had two thermometers constructed, with stem, bulb, capacity, bore, and length of degree similar to each other and to those used in determining the temperature towards the hotter end of the long bars. They were tested in the following manner. One was placed vertically, with bu/b only immersed, in a bath of melted paraffin wax; the other, placed horizontally, was wholly immersed in the same bath. The paraffin was maintained at a steady temperature throughout. After their indications had become stationary, the thermometers were read, and the excess of the reading of the wholly immersed, over that of the partially immersed, thermometer noted. From several ex- periments of this kind, it was calculated that the error in question amounts, at 250° C. to 9°°5, an exceedingly close verification of the previously calculated estimate. In the reduction of the experimental readings, and in the deduction of conductivity this error has not been taken into account, chiefly because the thermometer used in the experiment on the cooling of the short bar was almost exactly the same in construction as those used in the long bar, and that therefore the difference in this respect between the two experiments must be small. In Appendix II., however, I have endeavoured to correct for this source of error. Statical Experiment.—The bar to be experimented upon had one end inserted into an iron crucible containing melted solder, which is heated by a powerful Bunsen burner. Heat conducted along the bar raises the temperature of each portion of it, and this is carried on until the flow of heat across each section has become steady, a state indicated by the steadiness of the thermo- metric indications along the whole length of the bar. This condition was attained after (about) 7 hours in the case of iron, 6 in German silver, and after 5 hours in copper. This steadiness of state is then maintained for some time, generally at least an hour, after which the distribution of temperature along the bar, as well as the temperature of the unheated short bar in the vicinity, is carefully determined. Initially, therefore, the success of this part of the experimental work depends upon the maintenance of this particular distribution of temperature, 542 MR A. CRICHTON MITCHELL ON THE THERMAL CONDUCTIVITY after it has been reached. This means that (besides such minor details as draughts of air, and a steady or aslowly changing temperature in the laboratory) great regularity in gas pressure has to be ensured. The gas regulator, devised by Prof. Crum Brown, was employed in this inquiry also. Its working is as effective as its construction is simple; and its automatic action, hour after hour, relieves the experimenter of what would otherwise be the necessity of constantly noting and regulating the temperature of the melted solder in the crucible. The following extract from my note-book shows sufficiently how perfect are the results obtained by employing it :-— German Silver. July 9, 1886. Gas lit 6°50 a.m. Hour p.m. 2,30 2.45 3.0 3.15 3.30 38.45 4.0 4.15 4,30 ‘Doce pear | 27085 2709 270°7 2706 2707 2706 2707 2707 270-75 This, showing as it does only an extreme variation in the temperature excess of -09 per cent., bears out what has been said. The room in which the experi- ments were conducted was lighted from the north alone, the south windows and the doors were kept closed, and the air temperature throughout the day did not change so rapidly as to cause any uncertainty in the results, A slight modification of the experiment on the long bar was tried, enabling the method to give results with more certainty. Where, as in iron, the flux of heat across any section at the lower temperatures (7.2, below 20°) is small, a small error in the determination of the temperatures makes a comparatively large difference in the estimate of conductivity. In order to avoid this, a suggestion of Prof. Tarr’s (first given in Trans. R. S. E., 1878, p. 734) was adopted. While the bar was heated in the ordinary manner, a cold water bath was placed halfway up the bar towards the heated end, and through it a stream of water was continually passed from below. This had the effect of “steepening ” the temperature-gradient, and thus allowing the measurement or calculation of the tangent to be made with greater exactness. This process is almost necessary in the case of metals the change in whose conductivity with increased temperature is so small that the ordinary experiment of ForBEs is insufficient to detect its sign and amount. An idea of the effect of this midway cooling on the distribution of tempera- ture along the bar may be obtained from the following table, in which are compared the temperature excesses at the various holes in the bar, both in the ordinary experiment and in that with the cooling bath. ee i OF IRON, COPPER, AND GERMAN SILVER. 545 TRON. Temperature Excess °C. Distance from origin ONE ae 11/6/86. 13/4/87. Bar not cooled. Bar cooled 46 in. from origin. 0 229°3 229°35 3 162:'5 1633 6 118°45 119°85 9 87°6 89°5 15 50°35 51:0 21 29°8 29°95 PH | 18:25 17°65 33 114 9°6 45 46 —1:05 + * The hole in the bar nearest the heated end is taken as origin. + The — sign indicates that the actual temperature of the bar at this point was below that of the short bar standing in the vicinity. Cooling Experiment.—The chief difficulty hitherto in this part of the experimental method has been the oxidation of the short bar when heated to a high temperature. This introduces an uncertainty into the final results owing to the necessity of applying a value of the “rate of cooling” of the short bar at a particular temperature, to a portion of the long bar whose surface at that temperature is generally in a different state from that of the short bar at the same temperature. And although a correction may be applied to the ordinates in the curve of cooling in order to remove this, its application must in many cases be a matter of doubt. In this work, however, this uncertainty has been almost entirely removed by the nickel-plating of the bars ; the difficulty arising from an oxidised surface never even presented itself, for at the end of each experiment the surface of the short bar on cooling had retained its original brightness, and had, except in a slight manner to be yet noted, in no way been affected. So that the condition which is necessarily involved in the deduction of the conductivity, viz., that the surface of the long bar at any particular temperature should be the same as, or, at least, strictly comparable with, that of the short bar when at the same temperature, was very fully realised. Each of the short bars experimented upon was so placed, on bearings, in a rack, that it was possible to rotate it, on its long axis, while it was being heated. The temperature of the bar was raised by placing beneath it a row of 50 very small Bunsen burners. The heating was proceeded with as cautiously as possible, and generally (where it finally reached about 270°) occupied about 2 hours. This, so far, guaranteed an equable distribution of temperature through- VOL. XXXIII. PART II. 4. 544 MR A. CRICHTON MITCHELL ON THE THERMAL CONDUCTIVITY out the bar. But, for a reason to be shortly stated, the results thus obtained were not used. To this method of heating there is an objection. There is always a slight smoking of the Bunsen burners, and this, together with moisture deposited at the commencement of the heating, dulls the bright plated surface of the bar to some extent. At the instance of Prof. Tarr, who was surprised to find the rates of cooling greater with the nickelised than with the plain bars, these experiments were repeated in a form calculated to avoid any deposition of moisture, and to avoid the smoking as far as possible. The bars were heated to over 100° C. before a clear fire, and then as quickly as possible raised to a high temperature over the row of Bunsen burners. The results of this set of experiments have been used in the calculations. In Appendix I. a comparison between the results of the two methods is given. The temperature of the bar during cooling was observed by means of a particular thermometer, which was used almost exclusively (with an exception already stated) for this purpose. As might be expected, the zero of this instru- ment rose after repeated exposures to such high temperatures as it was employed to indicate. But this error was carefully estimated. As already detailed, the short bar during the progress of the statical experiment was placed near the long bar, while its temperature (practically that of the surrounding air) was recorded by a thoroughly trustworthy thermometer. The thermometer used in the cooling experiment was simultaneously placed in one of the holes in the short bar, and its readings compared with those of the other thermometer in the bar. A continuous process of correction was thus established; and by always placing the thermometer in this position when not used for the cooling experiment, the gradual change which the amount of the error underwent as the experimental work proceeded was carefully noted. Prof. Tart, in the paper already referred to, pointed out the importance of raising the short bar to a temperature considerably higher than that actwally required for the observation of the rate of cooling at any particular temperature. The exact reason for this is, that although the mercury in the bulb of the thermometer is heated almost at once, the column of mercury in the stem is not so heated for some little time after, and that the cooling of the two regularly together also does not ensue for some time. For example, in order to obtain a determination of the rate of cooling at 200° C. (as accurately as the method will allow), it is necessary to raise the bar to a temperature of 250° or 260°, insert the thermometer as quickly as possible, and allow the bar to cool down through the temperature required. For a full experimental proof, see Trans. R. S. £., 1878, pp. 730, 731. This affords a complete explanation of the curious result Forbes arrived at, viz., that the curve showing rate of cooling in terms of temperature excess exhibited a point of flexure about 150° C. OF TRON, COPPER, AND GERMAN SILVER. 545 By adopting this precaution I have verified Prof. Tarr’s results, and although in some cases the temperature excess was as much as 280°, no such result as that obtained by ForBeEs was even indicated. In addition to the usual cooling experiments, a series was conducted at temperatures as high as it was possible to safely observe with a mercurial thermometer, generally about 330° C. These experiments were of great use in accurately estimating the rates of cooling which are necessary for the higher portions of the statical temperature curve. They were not made until all the normal experimental work was over, as they involved some risk of the safety of the thermometers employed. II. DEDucTION oF CoNDUCTIVITY. Statical Curve.—Although the manner in which this curve was constructed was similar to that in previous work, it may yet be well to describe it in detail. From among all the statical experiments on the metal bar for which the curve was required, that one was chosen which, in its approach to, and maintenance of, the final thermal distribution, appeared to be the most steady, ‘and when considered in all respects, the most successful. This was termed the Standard Experiment. The readings taken during the steady state of the bar were then examined, and one special set of readings, apparently the most trustworthy, was selected. The appropriate corrections having been applied, the true temperature excess at each hole in the bar was obtained by subtracting from the corrected reading the reading taken simultaneously of the temperature of the short bar. The series of numbers thus obtained were then laid down as points on a curve, in which the abscisse represented distance along the ex- perimental bar, and the ordinates denoted temperature excesses. The other experiments on the same bar were then taken in the order of their apparent merits, regarded in the same way as in selecting the standard experiment. The temperature excesses were plotted separately for each experiment, on a sheet of tracing paper superposed upon a blank part of the sheet of divided paper on which had been marked the points obtained from the standard experiment. The axes on both sheets were coincident. After each experiment had been so represented on the tracing paper, it was carefully drawn along, keeping the horizontal axes still coincident, over the sheet beneath, until the points marked upon both lay in one smooth curve. Each point on the super- posed tracing paper was then transferred to that, immediately covered by it, on the sheet below. In this manner a series of points was obtained, through which a smooth curve was drawn, exhibiting the relation existing between temperature excess and position along the bar—what Fores termed the statical curve of temperature. 546 MR A. CRICHTON MITCHELL ON THE THERMAL CONDUCTIVITY It may be remarked that once the standard experiment, along with, say, two- other decidedly trustworthy experiments, is chosen, and the three represented by a curve in the manner already described, we have therein furnished a test of the correctness of the others ; indicated by the extent to which they agree with the first. By this means, the calculator is enabled to set aside as less trust- worthy, certain of the experiments. The only discrepance, which falls to be reported in this connection, between the several experiments on the same bar, lies in the uncertainty connected with the representation by the curve of the temperature excess at the cool end of the bar. Here it was somewhat difficult to trace the curve accurately, and to determine the point on the bar where, for all practical purposes, at least, the temperature excess disappears. An error of a tenth of a degree is quite sufficient to cause this doubt. This circumstance argues strongly in favour of the adoption of that modified form, already detailed, of the long bar experiment, viz., where the bar is cooled midway by the application of a cold water bath; for in this case there is no dubiety what- ever as to the point where the horizontal axis is crossed by the curve of temperature-excess, Regarding the equation representing the curve, two -formule were employed :— log v=log A- eo Le eee Jog o=log A+ en POLIS OE che. mae where v=temperature excess, g=distance along bar, reckoned from any arbitrary origin ; and where A, b, c, and e are constants. The formula (A), involving three constants, originally employed by REGNAULT (Mém, Ac. Sct., vol. xxi.) to represent the relation between the temperature and pressure of saturated water-vapour, was that used by Forses and also by Prof. Tarr, to represent the curve of temperature-excess in iron. But the differences between the calculated and observed values of v, both as shown by the tables in ForseEs’ last paper, and also when used by myself, were such as to lead to the construction, for use in this work, of the empirical formula (B). It was constructed by Prof. Tarr to suit as closely as possible, by four disposable constants, the curve plotted from the logarithms of the temperature excesses obtained in one of my earlier experiments on Forses’ iron bar. For the iron curve, it has been found to work very well, and certainly much better than its predecessor, But in the case of both copper bars and German silver, as well as the iron bar cooled midway, the equation (A) has been found to be more applicable, and has accordingly been used. In some cases it was found better OF IRON, COPPER, AND GERMAN SILVER. 547 to take the curve in two or more sections, employing for each the proper corresponding values of the constants. A conception of the relative value, for this purpose, of the two formule may be obtained from the two tables subjoined, showing the degree of approxima- tion of the value of v, the temperature excess, as calculated by the formula, to the value as given by the observational curve. The first, headed (F), is extracted from ForseEs’ paper already quoted ; the second, headed (M), is the record of one of my own experiments, along with the values of v calculated from formula (B) ; while the third table, also headed (M), shows in how far the same experi- mental results are approximated to by formula (A). These tables refer, of course, to iron. I. (F). . v by formula a, in feet. v by empenmental eae Fer veaa l+cx 0 °C. a : Be 272: cle 0°5 134°7 135°9 +12 0°75 97:3 98:23 40:93 — 53°6 —* i ae 242 23°58 —0°67 2°5 148 ire ~ nae oe 1-96 +016 60 09 “Re eee 8:0 0:28 0:36 0-08 IT. (M). a9, fin (ae » by experimental by fprmale me 7 , curve, log v=log A+——— - ex ifference, C+ez 0-0 247-2 2466 _06 0:25 1728 1738 +17 05 125-25 125°5 40:25 1:25 52:0 52-0 0-0 1-75 30°35 30°35 0-0 Dee 18:2 18:2 0:0 2°75 11:15 111 —0-05 3°75 4:3 43 0:0 475 1°85 17 —0-15 548 MR A. CRICHTON MITCHELL ON THE THERMAL CONDUCTIVITY IIL. (M). 4 v by formula x, in feet. » by aoa log abe Wee: 7 Difference. + cx 0:0 247-2 24.5'4 —18 0:25 1721 1721 00 05 125-25 12545 +0°2 0°75 92:5 92:3 00 1:25 52:0 52°0 00 1°75 30°35 30°6 +0°25 2°25 18:2 18°72 +052 2°75 11:15 11°65 +05 3°75 43 4°35 +0°05 475 1°85 2°53 +0°68 5°75 07 1:34 +064 The next table shows how far the formula (A), on the other hand, suits the case of the iron bar cooled midway. x, in feet JSS Teo RG. l ‘ H ee Difference y ; curve. Be oe cae : 0:0 206°35 206°35 0-0 0:25 150°85 150°75 —01 05 111°4 6m a —02 0°75 83°45 82°85 —06 1:25 ATT 47°25 — 0°45 175 28°15 27-9 —0°25 2°25 165 16°95 +0°45 The agreement shown here, though not so close as that shown in IT. above, was much closer than that which the use of formula (B) gave. LAMBERT (Pyrometrie, Berlin, 1779) showed, from experiment, that the curve of stationary temperature along a bar could be represented by an equation of the form v=Ae* whence log v=log A—pz. So that the curve representing, at each point of the bar, the logarithms of the temperature excesses should be a straight line. This result was shown to hold for the case of copper (C) by Prof. Tarr (Trans. R. S. E., 1878). OF IRON, COPPER, AND GERMAN SILVER. 549 Diagram 2 shows the degree of approximation to it in this work. With the exception of that of German silver, the curves for the other bars did not exhibit the result in question to the same extent. In connection with the consideration of the statical curve, it will be well to compare in this respect the results of this work with those of Principal Forses and Prof. Tarr. The curves for iron, according to these three different sets of experiments, are shown in Diagram 1. From the disposition of these curves one to another, it will be seen that (1), that given by ForBes (marked F), shows a greater rate of change of temperature along the bar than either of the other two; (2) that my own experiments, 7.¢., on the nickel- plated bar (marked M), show a slower rate of change than those of Prof. Tarr (marked T); (3) that all three agree with comparative closeness at the lower temperatures. The first of these remarks is confirmed when the values of dv/dz at different temperature excesses are plotted. The values which Fores used are then seen to be all greater than those used in the deduction of conductivity in this work ; on the average they are about 10 per cent. larger. No details of this kind (z.., values of tangents used, &c.) are printed in Prof. Tarr’s paper, otherwise they would have formed an interesting comparison. In the calculation of the tangents the vaiues obtained by differentiation of the formula representing the curve have mainly been relied upon, except at the lower temperatures, where graphical measurements have been made and used. But some allowance must be made for the discrepance between the calculated and observed values of v. It was effected in this way. By the sign and amount of the difference between these two values, it can be determined whether at any given point the curve (by calculation), in bending away (below or above it) from the observational curve, makes the calculated values of dv/dz too small or too great. By taking other values of the constants, a value of dv/dx may be obtained with a probable error, at any given point, of opposite sign to that obtained by the first value of the constants. Thus it is known between what two limits lies the accurate value of the tangent at the given point. Excepting the remark made that the value of the tangents graphically measured agreed satisfactorily with the value calculated from the formula, Forses seems to have used alone, and without any such modification as is men- tioned above, the values of dv/dz as they are given directly by one particular set of values of the constants,—viz., that set which appeared to give the best agreement with the observed temperature excess along the bar. Treating the curve in sections, as he did, partly avoids the difficulty ; still, it cannot be doubted that without any allowance of the kind mentioned, the calculated values of the tangent to the curve, constrained as it is to pass through three 550 MR A. CRICHTON MITCHELL ON THE THERMAL CONDUCTIVITY points (or four, as the case may be), cannot be received and adopted without involving the possibility of occasional large error. Some assistance may be obtained from the graphic representation of the value of dv/dz for different values of v. It sometimes gives a means of casting out untrustworthy estimates, and at all events prevents the entrance into the calculations of any great error. Diagram 3 illustrates its use in the case of the iron bar. Curves of Cooling.—In the reduction of the observations, the same methods were employed as in previous work. The deduction of the rates of cooling was performed mainly by constructing a curve whose ordinates represented succes- sive differences of temperature of the cooling bar (or 4, 4, &c. difference, where the interval of observation was 2, 3, &c. times the time-unit), and whose abscissze represented the mean temperature excesses during the interval. It is not accurate to say a curve was thus constructed; in reality, there was thus obtained (as is shown in Diagram 4) a number of irregularly disposed points, through which was drawn a smooth curve, the ordinates of which were the rates of cooling at the temperature excess given by the respective abscisse. In order to check the results derived in this manner, the following method of “grouped” readings was adopted. Six readings of the cooling bar were selected, taken at equal intervals. Correcting for thermometric errors, and subtracting the air temperature, we obtain six successive temperature excesses. Divide their sum by six, and there results the mean temperature excess during the five time-intervals which elapse between the reading of the first and last. Next take the five successive differences of temperature observed by the six readings ; divide their sum by five, and we get the mean rate of cooling at the mean temperature excess during the interval. We thus have obtained, by taking many such groups of readings, a series of estimates of the rate of cooling at different excesses of temperature. When so plotted, the points nearly all lie on a smooth curve which may easily be drawn through them. It was usual to go over each cooling experiment twice in this manner. All the readings were grouped into sets of six, and the results deduced. Going over them a second time, the readings were again placed in groups, so that one group in this second reduction overlapped equally two groups in the first; thus giving a second series of intermediate points on the curve. In Diagram 5 is shown the curve thus obtained, in the case of one of the cooling experiments on iron at the higher temperatures. In order to show how satisfactory is this method, the following comparison, made between the ordinates (i.¢., rates of cooling) of this curve, obtained from “grouped” readings, and those of the curve obtained from “single” readings, may be considered :— OF IRON, COPPER, AND GERMAN SILVER. 5d1 Rate of Cooling. Temperature Excess. From ‘‘ Grouped” Readings. | From ‘‘ Single” Readings. = OT ah ar t Tyee sae “ale - rowers ry ry a9 2 * tA se oe “ae 7 ‘ eww Ramat 8 - r y : ° ret ™ ¢£ y Atma? a "eg ag ? awe 7 fase ’ a | . 4 ety awe ~ ‘ re he ate — , ~ : rr o- itr Wi ro creer raja F : < + 4 dy PS er ee en 3 ded ey BR ee ed = ; Ais } re \ ‘ Pee BITTE F. “e » ’ - 4 oF < r tf ’ <<»? + “7 + . ~~ at < . . . - = * e Cd e “ “yr — > we Py = > a “se : s see 4 . Ly - “ sees — ~ Aled te A ssf a , . one a ~ Berd £al. oe - 2 a * Shae i > pat es tr ( " . ” . 4 i ‘pers ‘} ‘ * P f ‘ ees ‘ i 4 4 . < 3 : \ rie 304 - 4 “ eee ‘ - : 7 i 3 s ‘ ¢ Fm . : i } , w byl 4 of hae PRR . ws tw ey é ) ‘ . vi 7 , iJ a bya. .0in7 IRON Rate of Cooling at different Excesses of Temperature Diagram 6. | 8 emperature Baxcess : Re o 80 0 ] :. 200 240. » if | : | | | boo § | ee | a zl \ IRON Stationary State of Temperature Excess along Long Bar Diagram I |80 ° +- " 60 4] ex a iL | | 40 U wt a L 20 : Peck Engravers Edin® _+ Vol XXXL Pl XXXV =F as | COPPER. C. Shewing approximation to Lambert's Result oe, ean 2. ng Experimental Bar 2 Distance in feet alo 7 Soc. Edin? | IRON me Diagram Z. & 4 i nS 23 205 b- £3 8 q a oy Sy 5 i IRON Cooling Experiment at Higher Temperature Reduction trom "Single Readings Diagram #. ( 561 ) XXV.—Critical Experiments on the Chloroplatinate Method for the Determination of Potassium, Rubidium, and Ammonium; and a Redetermination of the Atomic Weight of Platinum. By W. Dittmar and Joun M‘Arruour. (Read 18th July 1887.) The analytical methods referred to in our heading are infected with numerous sources of error, amongst which until lately the uncertainty of our knowledge of the combining constant “ Pt” played a prominent part. This uncertainty, it is true, has been removed to a great extent by SEUBERT’S investigation “Ueber das Atomgewicht des Platins,” Liebig’s Annalen (for 1881), vol. cevii. p. 1 et segg. The value 194°8* for Pt, which he ultimately adopted, falls in well with his analyses of the chloroplatinates of potassium and ammonium, and there can be no doubt that his chloroplatinates were close approximations to the ideal substances. Hence his atomic weight 1948 must be admitted to be nearer the truth than the value 198 of ANDREws, which, until lately, was so generally employed by chemists in the calculation of their analyses. But it does not follow that in, for instance, the determination of chloride of potassium “as metallic platinum,” the factor 2K Cl : Pt=0°7657, calculated from SEuBERT’s Pt, gives a more correct result than even the factor 0°75252, which follows from K = 39 : Cl = 35°5, and Pt = 198. The expe- rience of practical analysts might almost be said to point the other way. A forcible illustration is afforded by FINKENER’s test-analyses in support of his own form of the chloroplatinate method for the determination of potassium. FINKENER separates out the potassium as chloroplatinate (in admixture with the sulphates of the foreign metals present), and weighs it as platinum. In reducing his results, he used ANDREWS’ value, Pt=198 (or one not far removed from it), and obtained close approximations to his syntheses. Had he calculated with SruBERT’s number, 194°'8, his results, if reduced to K,0O, would have been too high by nearly 2 per cent. All this, of course, goes no hair’s-breadth towards invalidating SEUBERT’s result; it only shows that those analytical factors which are, by theory, equal to K, : Pt; K, : PtCl,K,, &c., must be determined directly by standard experiments ; and separately for the several methods. This is what we have attempted (in a limited sense) to do. But a purely empirical determination of these factors by even the most exact experiments would have been of little use. If, for * Let us state at once that in this memoir all atomic weights are referred to O=16. For the atomic weights of potassium, ammonium, and chlorine we adopt Stas’ numbers, as calculated by LorHar Meyer and Ssuvsert; K=397136; NH,=18-056; Cl= 35-455. VOL, XXXIII. PART If, 40 562 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON instance, the analyst’s factor for K, : Pt differs from the ratio 2 x 39°136 : 194:°8, the causes of this difference must be ascertained, and this necessarily involves an inquiry into the true value for “Pt”; Sras’ numbers for KCl, &c., can, of course, be taken for granted. We accordingly did take up this inquiry, and in this sense our investigation joins on to SEUBERT’s, which, of course, was welcome to us, as affording a most valuable basis for this branch of our work. Before proceeding to detail our experiments, let us state that, unlike our predecessors, we did not merely analyse our chloroplatinates, but in their productions used exactly known quantities of standard solutions of chloro- platinic acid and (for instance) chloride of potassium ; so that the weights of platinum and alkaline chloride contained in the precipitated chloroplatinate could be ascertained by determining the small quantities of these which passed into the mother-liquor, and deducting them from the weights of platinum and chloride of potassium employed in the precipitation. In the preparation and application of these standard solutions (and of standard solutions generally) we used a system which was introduced some years ago by one of us in connection with a series of analyses of ocean water,* and which consists in this, that the solutions are standardised both by volume and by weight. The volume-titre serves only for calculating the number of cubic centimetres of solution containing a predetermined quantity of reagent: the exact weight of the latter is calculated from the weight of the measured off quantum and the weight-titre. The necessary special reagents were prepared in the following manner :— Chloroplatinic Acid. For the preparation of this reagent, we utilised two supplies of what had been sold to us as “pure platinum.” One of these came from the St Petersburg Works, through the kind agency of a St Petersburg gentleman, who happened to be in Glasgow at the time; the other, from Messrs Johnson, Matthey, & Co. of London. The St Petersburg metal, when examined, turned out to be contaminated with iridium so largely that we presume a wistake must have been made in the transmission or execution of our order. To purify the metal we dissolved it in aqua regia,as far as possible,t and subjected the crude chloroplatinic acid thus produced to the process of ScHNEIDER, which is so fully described by SeuBERT in his memoir. The pure chloroplatinate of ammonium obtained was reduced to metal by ignition, the platinum washed with hydrochloric acid, then with water, dissolved in aqua * “Challenger Reports,” Physics and Chemistry, vol. i, published by order of Her Majesty’s Stationery Office, &c, Adam and Charles Black (and others). + A good deal of black iridium remained. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 563 regia, and eliminated from the solution by means of pure hydrogen in the wet way. The metallic platinum was then filtered off, was washed successively with hot water, hot hydrochloric acid, and again with water, dried, and ignited ina porcelain crucible. When 23°338 grms. of this metal were dissolved in dilute aqua regia, with the view of preparing a standard solution, a black, sooty-looking residue remained, which was removed by filtration, ignited, and weighed. It amounted to 25°7 mgs. The solution was freed from nitric acid by repeated evaporation on a water-bath with hydrochloric acid, the ultimate residue dissolved in water, diluted to 471 c¢.c. in a tared bottle, and the solution weighed. It weighed 509°969 grammes, and, taking the part soluble in aqua regia as pure platinum, contained 23°3164 grms. of this metal. For roughly quantitative work, it was assumed to contain 49°5 mgs. of platinum = + x 198 mgs. per ¢.c¢. ; Strictly speaking, we had no right to assume that all the metal contained in this solution was pure platinum; but as the solution exhibited @ normal colour, and gave light yellow precipitates with chlorides of potassium and ammonium, we thought it might safely be used as pure chloroplatinic acid of the above titre, the more so as the atomic weight of iridium is only a little less than that of platinum, and a small admixture of the former metal in a chloroplatinate could not appreciably affect the percentage of metal in it. We accordingly did use this solution in some of our experiments. Matthey’s Platinwm.—To examine this metal for impurities, 10 grms. of it were dissolved in hydrochloric acid, with the aid of the least sufficiency of nitric. The solution was very light in colour, and no visible residue remained. The solution was transferred to a large platinum basin, mixed with a large excess of caustic soda,* boiled for 20 minutes, the hypochlorite destroyed by alcohol, and the resulting liquid acidified strongly with hydrochloric acid. No permanent precipitate was produced. The solution was precipitated with sal-ammoniac, the chloroplatinate (which was light yellow) filtered off, and the filtrate concentrated to a small volume, and allowed to stand to recover a second instalment of chloroplatinate. The filtrate from this was evaporated to dryness, the sal-ammoniac burned off, the residue washed with water, ignited, and weighed. It weighed 123 mgs. On treatment with aqua regia, a white residue (silica ?) remained, which weighed 32°6 mgs. after ignition. Hence metal in solution = 90°4 mgs. This solution was mixed with a weighed quantity of a standard solution of chloride of potassium, containing 64°79 mgs. of KCl (corresponding to about 0°95 time 90-4 mgs. of platinum), the mixture evaporated to a very small volume, and mixed, after cooling, with absolute alcohol, to precipitate the “chloroplatinate” produced. The precipitate, after * Specially made from pure (Trommsdorff’s) carbonate-crystals, in a nickel basin. 564 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON drying at 110° C., weighed 217°9 mgs.; 193°5 mgs. of it, when reduced in hydrogen (dry-way), yielded 77:8 mgs. of metal, or 87°61 mgs. per 217°9 mgs. of precipitate. The filtrate from the “chloroplatinate” contained 2°4 mgs. of “platinum,” and 0-49 mg. of chloride of potassium (determined and weighed as PtCl,K,). Hence chloride of potassium in the precipitate = 64°79 (weight of KCl used) —0:-49=64'3 mgs. Platinum (1) by synthesis, = 88°0; (2) by analysis, = 87°61; hence platinum per 2K Cl by (1) = 2042; by (2) =203°3. These numbers are certainly higher than the true atomic weight of platinum, but, considering the small scale on which the analysis was made, and that all the impurities of the 10 grms. of original metal were concentrated in that small remnant of 904 mgs. of metal, the original metal, we thought, could be accepted as sufficiently pure. To prepare a standard solution, 49°5327 germs. of Matthey’s metal (sponge) were ignited in a porcelain crucible, which brought down the weight to 495302 grms. These were dissolved in a Berlin basin, under a funnel, in aqua regia, made from specially prepared acids, the nitric acid expelled by repeated evaporation with hydrochloric acid, the ultimate residue dissolved in water, diluted to 1 litre, and weighed. A small quantity of an insoluble residue separated out on dissolving the chloroplatinic acid crystals in water; it was filtered off, ignited, and weighed, and its weight, 3-4 mgs., deducted from the original weight of platinum taken, as a correction. The solution was preserved as containing very nearly 495 mgs. of metal per c.c., and exactly 0°45930 grm. per.10 grammes of solution. The vast majority of our quantative syntheses and analyses were made with this or some other standard solution derived from Matthey’s metal. Wherever St Petersburg metal was used, this is specially stated. Chloroplatinic acid, as prepared by means of the aqua regia process, is liable to be contaminated with the nitroso-compound PtCl,(NO),. After having made a considerable number of syntheses and analyses of alkaline chloroplatinates, we found that one of our solutions at least was thus contaminated. It is difficult to see how a small admixture of the nitroso-body could affect a synthesis of potassium or rubidium chloroplatinate, yet its presence is at best no improvement; hence, after that discovery, we tried to make platinum solution by means of hydrochloric acid and chlorine gas, @.¢., the method which SeuBert used once (for his Darstellung IIL, p. 16 of his memoir), to return to the old process. We soon saw how it came that Sruperr tried the process only once; the solution of the metal takes place with an exas- perating degree of sluggishness. In the following form, however, we found the method to work satis- factorily. The platinum, preferably in the spongy form, is introduced into a light-coloured glass-stoppered bottle of, say, 2 litres’ capacity; a sufficiency of fuming hydrochloric acid is poured on it, and the bottle filled with chlorme en be a Pe Be Ant eee << CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 565 gas (purified by washing with water, and filtration through asbestos), the stopper put on, and the whole allowed to stand. After about twelve hours the chlorine is mostly absorbed. The bottle is then refilled with chlorine, again allowed to stand, &c., until the metal is all, or mostly, dissolved. The solution is decanted off, again chlorinated to make sure of the absence of platinous salt, evaporated on a water-bath to expel the surplus chlorine and hydrochloric acid, and diluted to the proper volume. Of course, if it is meant to be used as an exact standard solution, the remnant of undissolved metal is collected, washed, ignited, weighed, and allowed for. With a “chlorine Kipp” (see Dirrmar’s Ezer- cises in Quantitative Analysis, p. 137) at hand, the method is not so tedious as it appears at first sight. We have long come to adopt it even for ordinary laboratory purposes. When used in connection with work here reported on, it is referred to as the “ chlorine process.” Chloride of Potassium. A few of our earlier experiments were made with a chloride obtained from recrystallised chlorate, but in the majority of cases we prepared our chloride of potassium from recrystallised perchlorate. The perchlorate is heated in a platinum basin until the bulk of the oxygen is removed. The rest is then expelled by fusing the residue in a platinum crucible, and keeping it at a dull-red heat until every trace of gas-bells has disappeared. The fused salt is poured into a platinum basin, allowed to cool, broken up, and used in this condition. The neutrality of the fused salt was made sure of by dissolving 2 grms. in water, and titrating the “alkali” with very dilute standard solutions of hydrochloric acid and caustic potash, in the presence of aurine as an indicator. The “alkali” was found equivalent to 33, x+}K,O mgs. = 0:2 mg., or rather nd. Pure Sal-Ammoniae. See the section on chloroplatinate of ammonium, page 627. Rubidium Chloride. See the section on its chloroplatinate, page 618. Standard Solution of Nitrate of Silver. As a rule, we prepared this reagent from pure nitrate, in which the water had been determined immediately before use. A quantity, containing 7 x 17 grms. of dry nitrate, was dissolved to m litres. For the standardisation of the solution a standard solution of chloride of potassium was made synthetically by dissolving ;4 KCl grms. in water, diluting to 1 litre, and weighing the 566 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON solution produced before the last mixing. 50 (or 100) c.c. of this solution were weighed into a tared phial, 51 (or 101) ¢.c. of silver solution added, the mixture reweighed to obtain the weight of the silver solution added, then shaken, and the chloride of silver allowed to settle in the dark. The supernatant liquor was decanted off, iron-alum added, and the exact quantity of dissolved silver determined by to-and-fro titration with decimal (é.e. centi-normal) solutions of sulphocyanate of ammonium and nitrate of silver. These solutions were standardised only by volume, and measured off in burettes. From the result and the specific gravity of the stronger silver solution, it was easy to calculate the exact number of grammes of the latter which precipitate ~, KC1=74°59 germs. of chloride of potassium. This number was recorded as the “ weight- equivalent” of the reagent, and utilised in the calculation. In a perfectly similar manner we proceeded in our determination of chlorine by titration. The given solution of chloride was diluted to a known weight. A small fraction, weighed out, served for a preliminary titration of the chlorine in which the reagents* were merely measured by volume. In a larger aliquot part then the exact weight of the chlorine was determined by a method of gravimetric titration closely analogous to the one explained for the standardisation of ‘the silver solution. This particular modification of VoLHARD’s method having been worked out by one of us (long ago) for the determination of the chlorine in sea-water (see ‘‘Challenger” Memoirs, Physics and Chemistry, vol. i. in Dirrmar’s Report on the Composition of Ocean Water), it will be referred to as the Challenger method. In most of our analyses we determined the chlorine by means of this titri- metric method ; sometimes we checked the results by gravimetric determina- tions in other aliquot parts of the respective solutions. In a few cases, when we had only little material at our disposal, we combined VoLHARD’s method with the ordinary gravimetric method. The chlorine was precipitated. by means of a known sufficient weight of the standard silver solution, and the chloride of silver collected and weighed. In the filtrate and wash-waters the still unprecipitated silver was then determined by VoLHArD’s method, with volumetrically adjusted decimal (centi-normal) solutions. As each such dupli- cate analysis was preceded by a preliminary titration of the chlorine, we had no difficulty in so adjusting the weight of silver solution to be added, that the excess of silver left unprecipitated could be conveniently determined in the way explained. After this introduction, we will proceed next to report on a series of experiments for determining the composition of chloroplatinate of potassium as produced under conditions similar to those prevailing in certain forms of the corresponding analytical method. * The silver solution and a volumetric deci-normal solution of sulphocyanate. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 567 J. EXPERIMEN''S ON THE COMPOSITION OF CHLOROPLATINATE OF POTASSIUM. (First Series.) The general modus operandi was as follows :—A known weight of chloride of potassium (weighed out in the form of a gravimetrically standarised solution) was mixed with, in general, a slight or moderate excess of standard chloro- platinic acid solution, whose weight was determined and recorded likewise. After production of the chloroplatinate (in some way or other), the mother- liquor was decanted and filtered off, the chloroplatinate washed (as a rule first with small instalments of water, then with alcohol), dried at a certain temperature, and weighed. The mother-liquor, after removal of the alcohol from the alcoholic portion, was diluted with water, placed in an Erlenmeyer flask, and the platinum reduced out by purified hydrogen in the wet way at 80 to 90° C. After complete reduction, the hydrogen was displaced by carbonic acid (to avoid explosions), the platinum filtered off, ignited, and weighed. The filtrate and wash-waters were diluted to a certain weight, and aliquot weighed parts used for the repeated determination of the fixed chlorine, i.¢., the chlorine present as KCl. For this purpose the respective liquid was evaporated to dryness, the residue dried further at 130°, then moistened with water, the solution re-evaporated, and the residue heated again to 130°, so that the free hydrochloric acid could be assumed to be away, which point, however, was always made sure of by the application of a thread of delicate litmus-paper to the last solution. In the chloride of potassium thus isolated, the chlorine was determined ; generally by means of the “ Challenger” method. In the later experiments a very small quantity of fixed chlorine present in the chloroplatinic acid used was determined in a large quantity of the respective reagent, and allowed for. By deducting the weight of platinum and chloride of potassium found in the mother-liquor from the weights originally taken, we obtained the weights that had passed into the precipitate. In addition thereto, the precipitated chloroplatinate, as a rule at least, was analysed, more or less completely. Our method of analysis, in the earlier experiments, was to reduce the chloroplatinate in a boat standing within a combustion-tube, in hydrogen gas. To avoid loss the out-going gas was made to bubble through a little water contained in a bulbed U-tube. The contents of the U-tube were evaporated to dryness, the residue united with the aqueous extract of the ignited mass and the washings of the combustion-tube, the whole evaporated to dryness, the residue re-dissolved in water, and the small quantity of platinum, which invariably separated out (even from perfectly clear liquors) collected and weighed. In one case the chloride of potassium thus obtained was weighed directly; as a rule, we relied for it on the 568 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON synthetical data. The platinum was washed, ignited in a small porcelain crucible, and weighed. We soon, however, came to discard this method, and to reduce our chloroplatinates in the wet way. The chloroplatinate, after having been dried at a certain temperature and weighed, was placed in an Erlenmeyer flask, along with a sufficient quantity of water to dissolve the chloroplatinate in the heat, the platinum reduced out in the wet way, filtered off, and weighed,* and aliquot parts of the filtrate used for the determination of (in general) the fixed and the total chlorine. In the following reports on the several experiments, the symbol A stands for the weight of chloride of potassium used in the synthesis; P for the approximate weight of platinum used, as chloroplatinic acid, per 2KC1=149:18 parts of chloride of potassium; M for the exact weight of chloroplatinate which, according to analysis or synthesis, contains 2K Cl parts of chloride of potassium. Experiment I. (St Petersburg Metal). A=0°74 grm. (about); P=201°6. Platinum solution poured into that of the chloride of potassium. Total volume of mixture, 40 c.c. The whole was evaporated to 2-3 c.c., and, as no excess of platinum was visible, another 1 c.c. of platinum solution was added (and its water evaporated away); hence, finally, P=209°3. The residue was washed with absolute alcohol (which became distinctly yellow); the chloroplatinate dried at 110°, weighed, and analysed in the dry way. The results, referred to 2KCl parts, were as follows (in the analysis of the precipitate only the platinum was determined) :— Platinum. Chloroplatinate, M. Synthesis, . ; 5 t : i ‘ . 196°26 491:07 Analysis of precipitate, : : - ; - lOGBL Hence Pt=196'29 ; and thence, by calculation, PtCl, K, =487:29=M — 3°78; hence water, &c., in the chloroplatinate =0°700 per cent. Experiment II. (Matthey’s Metal). C=1°85495; P=218-4. Platinum poured into potassium salt. Evaporation to about 5 c¢.c., then addition of absolute alcohol, as in Experiment I. Precipitate washed with absolute alcohol, and dried, first at 110°, then at 130°. Reduction by hydrogen effected in the dry way. Combined chloride of * Sometimes a film of platinum adheres firmly to the sides of the flask. This is easily recovered by dissolving it in agua regia, evaporating to dryness in a porcelain crucible, and igniting the residue. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 569 potassium determined only synthetically; the platinum both ways. Results, referred to 2K Cl parts, were— Platinum: or er aaa, dried ae Synthesis,. . . 195-71 488-43 (= M’) 488-28 (=M”). Analysis, . ‘ : 19563 Hence, by calculation, P=195°67, and PtCl,K,=486'67. Hence water, &c. in M’, =1°76 parts, or 0°360 per cent. in M”, =1°61 . 0°330 5 Experiment III. (Matthey’s Metal). A=1'85556 ; P=206-4 parts. The two reagent solutions were evaporated separately to about 30 c.c. each, and then mixed ; the platinum being poured into the potassium salt. Mixture evaporated to 5 c.c., and allowed to stand over night. The precipitate washed by decanting filtration with three instalments of water, each=3 c.c.; 15 ¢.c. of absolute alcohol were then added, and the precipitate washed with absolute alcohol. The precipitate was dried at 130°, first for 5 hours, and then for other 24 hours; and in part of the final residue the water was determined directly (see page 578). The combined chloride of potassium was determined only by synthesis; the platinum both ways. Found per 2KCIl parts, Platinum, by synthesis, = 195-75 Platinum, by analysis, = 195°56 \ 19566. and for the weight of the chloroplatinate :— Dried at 130° for Anhydrous, by water 5 hours. 29 hours. determination, M’=487°78 M’ =487:29 M’’ = 486:°63 From Pt=195'66 ; by calculation, PtC]l,K,=486°66 ; hence water, &c., 1:12 0°63 — 0:03 or 0:230 0-129 nil per cent. Experiment IV. (Matthey’s Metal). This experiment was wrought side by side of Experiment III., on the same scale, and in the same way, except that the chloride of potassium was poured into the platinum, and that the excess used of the latter was a little greater. The two chloroplatinates were dried, side by side, in the same chamber, only while, after 29 hours at 130°, No. III. was taken out to be analysed, No. IV. was dried for an additional 12 hours at 150°. The water in the finally dried substance was determined directly (see page 578). During the heating process involved in the determination of the water, the salt decrepitated so much, that a reduction of the residue in hydrogen could not have been effected with VOL. XXXIII. PART II. 4P 570 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON quantitative precision. Hence we relied entirely on the synthesis for both the fixed chlorine and the platinum. Platinum taken, per 2 KCl parts of chloride of potassium, = 214°3 parts. Found per 2K Cl parts. Platinum = 195-79, and chloroplatinate Dried at 180° for ft 56 houks. 29 i rae 150° Anhydrous Salt. M = 489°67 488°67 488:07 487-05 Now, for Pt =195:79 : PtCl,K, = 486°79. Hence for water, &c., InM parts 2°88 1:88 1:28 0:26 or 0°588 0°385 0:262 0:0534 per cent. Experiment V. (Matthey’s Metal). A=3'7822 ; P=195-0. The chloride of potassium was dissolved in 20 c.c. of water, and the platinum solution (107°6 cc.) added, cold. Mixture allowed to stand over night, liquor decanted off through a small filter, precipitate washed with small successive instalments of water, until the small excess of either reagent, which could be presumed to be present, was sure to be washed away (wash- water used, =4x5=20c.c.). The small portion of the precipitate which had gone on the filter was then washed with a little absolute alcohol, but these washings were collected by themselves. The aqueous filtrate and washings (some 130 cc.) were evaporated to dryness over a water-bath in a platinum basin ; the residue, along with the small quantity of chloroplatinate from the filter dissolved in boiling water (about 150 c.c.), and the solution cooled down finally in ice, to cause as much as possible of the chloroplatinate to crystallise out. The crystals “I.” obtained were allowed to settle, the liquor decanted off through the original filter, and the filtrate evaporated down to 20 ¢c.c. The solution was cooled down, and allowed to stand over night, when another smaller crop of crystals “II.” came out. The mother-liquor was decanted off through the filter, and the three portions of recovered chloroplatinate washed systematically with small instalments of cold water, each equal to about 2 c.c. Each 2 c.c. of water went first on crystals “I.”, then on crystals “II.”, and lastly on and through the filter. Four such washings were carried out. The last mother-liquor and these washings were analysed as usual to find the weights of chloride of potassium and platinum to be deducted from the quantities originally taken, in order to find how much of each was contained in the three precipitates of chloroplatinate, namely, the bulk of the original precipitate (it, after drying at 130°, weighed 9°6244 grms.), and the two crops of chloroplatinate recovered from filter and mother-liquor. The weights of CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 571 these latter were not determined, but from the total combined platinum in the three portions we can calculate the total weight of chloroplatinate synthetically analysed. The total platinum obtained amounted to 4°88266 grms., whence, if we go by Experiment “III.”, and the “M” for 5 hours’ drying at 130° given there as corresponding to 195°66 of platinum, we have 12°1724 grms. for the total chloroplatinate, whence, by difference, 2°5480 grms. for the two minor portions conjointly. Found, synthetically, per 2K Cl parts, platinum=196°64. Hzact data for “MM” absent. Note.—The aqueous filtrate and washings contained 60°4 mgs. of platinum, which need 46:2 mgs. of chloride of potassium to be made into chloroplatinate ; instead of these 46:2, we found 77°9 mgs. Experiment Va. (Matthey’s Metal). A=4'5385 grms. weighed directly, and dissolved in 100 c.c. of water. P=203'4 parts, calculated from the volume of platinum solution used, which was 125 c.c. The chloride of potassium was poured into the platinum solution, and the mixture allowed to stand over night. Next morning the mother-liquor was decanted off through a small filter, and the precipitate washed three times, each time with 10 c.c. of water ; 20 c.c. of absolute alcohol were then poured on the precipitate in the basin, the small portion from the filter washed in with absolute alcohol, and the whole allowed to stand for 4+ hour, to effect at least a partial dehydration. The precipitate was then washed with absolute alcohol, and next kept over oil of vitriol for half a day. It was then dried at 100° fora short time, transferred to a flat platinum crucible, and dried for 4 hours at 110°, when its weight was found to be 121063 grms. This precipitate was put into a Geissler tube. A portion (A), amounting to 2°9398 grms., was however weighed out at once for analysis, and another (B) (2°9255 grms.), at the same time, for the determination of the water volatile at 150°, which, as we may state at once, amounted to 11°5 mgs., or 0°393 per cent. (after 23 hours’ drying). The reduction of “A” was conducted in the wet way with 200 c.c. of water, the platinum filtered off, and weighed; (it weighed 1:1768 grms.). Aliquot portions of the filtrate served for the determination of the fixed and total chlorine. The former was determined once; (found 0°42449 grms. per total solution) ; the latter twice ; found 127148, and 1:27122, mean = 1'27133 grms. adopted. Hence we had, by calculation, per 2K Cl parts— Platinum. Loose Chlorine. Myy0- Miso: 196°58 199496 x Cl, 491-087 489-166 “M,,.” stands for chloroplatinate dried for 4 hours at 110° C. ; 572 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON “M,,.” for the same, dried in addition for 23 hours at 150° C. The number for the loose chlorine suggests the presence of 0:00504 x (O = 16 parts of oxygen) in the PtX, part of the salt. Admitting this, we have, if Pt = 196°58, for the composition of M parts of this kind of chloroplatinate— Platinum, —. ; f § : : 19658 2KCl, . : } ; ; : 149'18 1:99496 x Cl,, t ‘ ; : , 3 14146 0:00504 x O, . ' ‘ f ; ‘ 0:08 487°30 Hence water, &c. in. Myo. and Mys0- Per M parts, .. : : Oate 1:87 Per cent., . , : : areal 0:381 The oxychloride oxygen amounts to very little, supposing it to be present at all. Taking the two determinations of the total chlorine as so many determinations of three times the fixed chlorine, and combining them with the direct determination of the latter, we had :— Mean value of fixed chlorine = 042401 ; and for 2KCl parts Platinum, . ; : ‘ ; : , 196°80 2KCl, . : : : : 5 : : 14918 Acie. ‘ ; : ; ‘ , : 141°82 PtCl,K,, : : ; : ‘ : : 487-80 and for the values M, M and for water, etc., in these, Per M parts, : ; : : 3°84 1:92 The results remain substantially the same. =491:64; Myo = 489-72. 110 Experiment VI. (Matthey’s Metal). At the time when this experiment was planned, we had arrived at the conviction that the atomic weight of platinum is very nearly, if not exactly, equal to 196, and the object of the experiment was to test this number in the most direct manner possible, namely, as follows :— 0°7548 grm. of chloride of potassium was dissolved in 20 c.c. of water, and the solution transferred to a tared glass-stoppered bottle of about 120 cc. capacity. 20°5 c.c. of standard platinum solution were then measured out, weighed exactly, and added to the chloride of potassium. From the weight and the known titre of the solution, it followed that the weight of platinum added amounted to 0:99086 grm. or to 195°9 per 2KCl. The idea was to let the chloroplatinate separate out as far as possible, to draw off some CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 573 of the clear liquor, and, by its analysis for platinum and fixed chlorine, prove the absence of excess, or determine such small excess as there might be, of either reagent. But very little chloroplatinate did settle out. We therefore added enough of absolute alcohol to nearly fill the bottle, shook up the contents, and allowed to stand, stopper on, over night. Next morning the whole was weighed, as much as possible of the clear liquor drawn off, weighed, (in a stoppered phial), and analysed for platinum, total chlorine, and fixed chlorine. For this purpose, the solution (after addition of water but without removing its alcohol), was kept at a temperature near to, but below, its boiling-point, within a conical flask, through which a very slow current of hydrogen was constantly passing. The out-going hydrogen was made to bubble through a quantity of water contained in a bulbed U-tube, to catch any hydrochloric acid that might go off. Let us at once state that no chlorine was found in this wash-water after the experiment. The platinum was filtered off and weighed ; the filtrate diluted to a known weight, and aliquot parts used for the determination of the fixed and total chlorine. As there was not enough of material for repeating either deter- mination, the chlorine, in each case, was precipitated with a known weight of standard nitrate of silver, the precipitated chloride of silver weighed, and the excess of silver in the filtrate determined by Volhard’s method, to check the gravimetric determination. From the weight of platinum obtained, the corresponding weight of chloroplatinate of potassium was calculated, the re- sulting number multiplied by the ratio of bottle contents to liquor analysed, and the result deducted from the total chloroplatinate (solid or dissolved) calculated from the total platinum used, to obtain a first approximation to the weight of solid chloroplatinate in the precipitation bottle, at the time when the liquor was withdrawn. The result, when subtracted from the weight of the total contents, gave an approximation to the weight of the total mother-liquor. From this weight a second approximation to the dissolved chloroplatinate was obtained, and thus a second approximation to the weight of the solid chloroplatinate, and consequently also to that of the mother-liquor. This second result served for the reduction of the chlorines formed in the drawn off part to the total liquor. We might have stated before that the platinum solution, before being used, was analysed to find the quantities of platinum, total chlorine, and fixed chlorine present in the weight of reagent employed. To enable the reader to form his own opinion on the probable uncertainty in the final results, we quote _ the following numbers :— Found, for the total liquor drawn off— Total Chlorine. Fixed Chlorine. Gravimetrically, . : : . 038263 0:02676 Titrimetrically, . : : . 038240 0:02675 Means (adopted), - —s = 038251 0:02675 574 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Calculated weight of total mother-liquor = 114-272 erms.; Weight of drawn-off liquor analysed, = 99718. Hence, Grms. of Chloride of Potassium. Loose Chlorine. Platinum. In reagents, 1:06538 099086 In mother-liquor, : : 0:06453 0:40768 0:07907 { In precipitate, . ! 069065 0:65770 OL TO Ratios, : ; . . 14918 : 142:063—: 196°95 The loose chlorine equals 4:00685 x Cl, which, considering the complexity of the method, may be accepted as a sufficient approximation to 40000. The value Pt, calculated from the loose chlorine, comes to 196°61; cal- culated from the total chlorine, to 196°72. This last number probably is the most exact of the three; but any are considerably above 196. The following experiment was made with the view of finding a lower limit for the value of the constant :— i Eaperiment VII. 9°3125 germs. of chloride of potassium were dissolved in 50 c.c. of water, and mixed with a measured volume of standard platinum solution, containing 2°45 erms. of metal, ¢.¢., only about one-fifth of what the chloride of potassium would demand for its conversion into chloroplatinate. Total volume of mixture =100c.c. about. After a night’s standing a large precipitate of chloroplatinate had settled out; it was washed with small successive instalments of water, until the excess of chloride of potassium, by calculation, was reduced to about 10 mgs. The washing was then continued with a cold-saturated solution of chloroplatinate of potassium. About 60 c.c. of this solution were used in all; after it followed two washings with 50 per cent., and at last a few with absolute, alcohol. The precipitate was dried, first for a night over vitriol, and then at 100° until constant within 4 to 5 mgs. It was then divided into two approxi- mately equal parts, and both analysed by reduction with hydrogen in the wet way. That combination of the gravimetric with the titrimetric modus, which had served in Experiment VI., was used again. Analysis of Part. A. B. Substance analysed, . . 3 3°1626 2°6760 grms. Platinum obtained, . ; : 1:2601 1:0672... ;} Total chlorine (grav.), ‘ 136772 Ls 5 07D os 3 (titr.), . , 136750 1:15750 ,, fs (mean), ; 5 1:36751 TIGT0°e ,, Fixed chlorine (grav.), 0:45921 038894. ,, y (titr.), 4 0:45894 0°38855 ,, ES (mean), 0°45908 038875 ,, Referring in both cases to 2K Cl parts, we have— Platinum. Total Chlorine. Substance = M. A, " : ; ; : 194-64 211:227 : B, , , ; ; 194°66 211:158 488:12 Means, . j : , ; 194°65 211°192 488°31 CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 575 Adopting the mean, it would follow that Pt = 19465, and that this Pt is combined with 211:192 = 5:95663 x Cl, supplemented by 004337 x (OH or 40). But we have no excuse for presuming that oxychloride-oxygen is present; on the other hand, we have good reason for suspecting the presence of surplus chloride of potassium. Referring the mean results to 4x Cl parts of loose chlorine, we have Platinum, . : , ; : : 3 196°78 1:01097 x K,Cl,, . ‘ J , . ; 150°82 4 x Cl of loose Cl, ‘ : : : ; 141°82 Total, . ; ; , ‘ ; : . 48942 And M, , ; ; 3 7 . = 49367 Water (7?) ; 3 ; , ; . 425 This, we believe, is the correct mode of interpreting the analysis. It does not follow that 196-78 is the true atomic weight of platinum. Experiment VIII. (Matthey’s Metal). All the experiments reported on so far were made with platinum solu- tions, made by means of the ordinary aqua regia process. We have reason to suspect that these solutions contained small quantities of the nitroso-com- pound PtCl,(NO),, which may have had an evil effect, although it is not quite easy to see how this can have been the case. However this may be, the following experiments (VIII. and IX.) are free of this flaw, because they were carried out with platinum solution made by the chlorine process (see page 565). Their most important feature, however, is that a very con- siderable excess of chloroplatinic acid was used in the preparation of the chloroplatinates. In Experiment VIII. 3:045 grms. of chloride of potassium were dissolved in 40 c.c. of water, and precipitated with 120 c.c. of. a platinum solution containing 6°0 grms. of platinum, so that the platinum added per 2KCI parts amounted to 2940 parts. The potassium salt was poured into the platinum solution, the mixture allowed to stand over night, the precipitate washed by decanting filtration, first with 5 c.c. of water (z.¢., chloroplatinic acid solution), and then with absolute alcohol; the aqueous and alcoholic washings were preserved separately. The precipitate was dried at 100° to 105° in a watch- glass for 44 hours, and then for another hour in a Geissler tube at the same temperature. The precipitate acquired a constant weight in a remarkably short time. After having been dried, it was divided into three parts, ab. ©, A = 33511 grms. was devoted to a direct determination of the water. Water obtained = 30°8 mgs. = 0°919 per cent. 576 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON B (= 2°8375) and C (2:8884 grms.) were dissolved separately in hot water and reduced by hydrogen, for the determination of the platinum, the fixed chlorine, and the total chlorine. The results were as follows :— Analysis of ‘“ B.”—Substance = 2°8375; platinum therefrom = 1:1312. Total chlorine [in fraction analysed, by titration = 0308683; gravimetrically = 0°308791: mean = 0°308737; whence, for the whole], 1:22855 germs. Fixed chlorine [by titration = 0°306219 ; gravimetrically = 0°306615 ; whence, for the whole |, 0409294 grms. Hence per 2K CI parts :— Platinum. Total Chlorine. Substance analysed. Dried at 100°C. Anhydrous, 195-98 212°85 = 6°0033 Cl. ' 491-60 487-08 Sum of components = 486°98, which leaves no room for oxychloride oxygen. : Analysis of “ C.”—Substance = 2°8884; platinum = 1:1518. Total chlorine [calculated from mean of 0°314118 and 0°314255], 1:25006. Fixed chlorine [calculated from mean of 0°312018 and 0°312574], 0°417213. Hence per 2KCl parts :— Platinum. Total Chlorine. Dictate ae ee ‘Anhydrous. 195°76 212°46—5:9924 x Cl. 490-92 486-40 Sum of components = 486°76, or adding in (0:0076 x 30 = 0-061) 486:82. But the small chlorine-deficit had better be viewed as an observational error, like the excess in the analysis of “B.” The filtrate from the chloroplatinate (A+B+C), which still contained some 2 grms. of platinum, was utilised for Experiment IX. thus. It was mixed with 1 grm. of chloride of potassium, 7.¢., two-thirds of what the 2 grms. of platinum required for their conversion into chloroplatinate and the mixture wrought in pretty much the way prescribed by Mr Tatiock in his form of the chloroplatinate process for determining potassium ; 7.¢., it was evaporated to near dryness on a water-bath, some added water evaporated over the residue to eliminate the free hydrochloric acid as fully as possible, and the residue, after cooling, digested in the cold with 10 ¢.c. of water, zz. virtually 10 c.c. of a 6 to 7 per cent.* platinum solution, for an hour. The liquor was then decanted off through a filter, and the precipitate washed, first twice with small added volumes of 5 per cent.* platinum solution, and at last with 95 per cent. (by weight) alcohol. It was dried at 100° in a Geissler tube (the weight became constant very soon) and weighed. It weighed 4:0487 grms. It was divided into three parts, A, B, and C. * Meaning a solution containing so many centigrammes of metallic platinum per c.c. ee i i i CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 9577 “A” was used for a direct determination of the water. 1°3573 grms. of substance gave 1'1 mgs., or 0081 per cent. “B” and “C” were analysed by wet-way reduction. Found for Gi) 3} etna Mean. Substance, ; ; i i 1:0990 15917 Platinum obtained, . , F 04416 0:6403 Substance per 1 grm. of platinum, 2:4887 24859 2°4873 The agreement being satisfactory, the filtrates from the platinum were mixed, and used for the determination of the fixed, and of the total, chlorine. Results.—Substance = 2°6912 (by direct weighing preceding the dividing into “B” and “C”). Total chlorine [calculated from the mean of 0:23554 and 023532] = 1:17466. Fixed chlorine [calculated from the mean of 0:23462 and 0:23501 | = 0°39223. ) Hence per 2K Cl parts :— Substance. Platinum. Total Chlorine, Dried at 100°, Anhydrous, M’. M”. 195-59 212°362 =5'9896 x Cl. 48653 486:14 Neglecting the small chlorine-deficit, we have for the sum of components 486°59, which, singularly, agrees better with M’ than with M”. Perhaps the 1:1 mgs. of water found in “ A” was the result of observational errors,—water absorbed after weighing of the substance, or water out of the joints, &c. In the preparation of the two chloroplatinates VIII. and IX. we were very much struck by the promptitude with which they acquired a constant weight in the drying chamber; all our previous chloroplatinates used to continue losing weight for hours and hours, and hardly ever really exhibited absolute constancy of weight. The most remarkable feature in these chloroplatinates, however, is, that although produced in the midst of a very large excess of chloroplatinic acid, they contained rather less platinum per 2KCl parts than we had found in the chloroplatinates previously produced in the presence of small excesses, or even negative excesses, of the platinic reagent. Yet it does not follow that even those quasi-exceptional chloroplatinates (of Experiments VIII. and IX.) were free of surplus platinum. At the time when Experiments VIII. and IX. were planned, this question had already been expiscated experimentally to some extent by special experiments on chloroplatinates V. and Va., which were done at the time when the analysis of V. and Va. were carried out, but which we prefer, in this memoir, to treat of separately in the following section, VOL. XXXIII. PART II. 4Q 578 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON II. ON THE CoMPOSITION OF CHLOROPLATINATE OF POTASSIUM. Second Series of Experiments. Chloroplatinate of potassium, as has been long known, is liable to contain water, and indeed in most cases does contain water, so intimately combined with the rest that it cannot be completely expelled at even 150°, That this water should all be present as such, as a mere enclosure within the crystals, is difficult to believe ; it is more likely to be present, at least partly, in the form of hydroxyl, functioning as part of the loose chlorine in the ideal substance. From certain observations of SEUBERT’S, indeed, it appears that whenever chloroplatinate of potassium is recrystallised from hot water, part of the chlorine passes into solution, and is replaced, of course, by its equivalent of hydroxyl or oxygen. That this exchange should take place only in hot, and not at all in cold, solutions, is by no means probable. Any chloroplatinate is liable to be thus contaminated, and as long as its purity is not proved, and quite apart from any free chloride of potassium, or surplus platinum in this form or that, which may adhere to it, must be looked upon as a (mixed) oxychloride of the general formula PtCl¢_.,O,.K,+2H,O, where y of course is a fractional number, and # may be greater than y, because the salt may contain combined water in addition to the water present as hydroxyl. To be able to make a direct and complete analysis of a “ chloroplatinate,” we must have methods for the direct determination of the water and of the oxychloride-oxygen. The determination of the water presents no difficulty; it indeed is so easy that we wonder that SEuBERT did not effect it with his chloroplatinate of potassium, and thus remove the cloud of uncertainty which hangs over those of his calculations of the atomic weight of platinum which are based on the ratio of platinum to non-platinum in the chloroplatinate. All that is required is, from a known weight of substance, contained in a porcelain boat standing in a combustion-tube, to expel what goes off at a dull red heat, to remove the liberated chlorine by passing the volatile products through a spiral of red-hot sheet silver, placed in the exit-end of the combustion-tube, to collect the thus purified water in a tared chloride of calcium tube, and weigh it. ‘The water determinations referred to in the above reports were carried out in this manner.* To avoid the uncertainties arising from the use of cork joints, the exit end of the combustion-tube was drawn out to quill size, and the entrance end of the chloride of calcium tube joined on by means of a piece of india- rubber, in such a manner that only a narrow line of the latter was exposed to the out-going vapours. This joint, during the whole of the operation, was * The water determination in Experiment III. forms an exception, in this sense, that the substance — was heated in a current of mitrogen, and the products filtered through a stopper of copper gauze (instead of metallic silver). CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 579 kept at 100° to 105° by means of a small chamber made of asbestos paste- board, which enclosed it ; and the first step in each analysis was to make this joint anhydrous by passing a current of chloride of calcium dry air through the (heated) tube, until the chloride of calcium tube attached to its end ceased to gain weight, taking care to have the silver spiral at a red heat during an earlier stage of this preliminary operation. The expulsion of the water of the substance was effected in a slow current of chloride of calcium dry air. The direct determination of the oxychloride-oxygen is a far more difficult problem, which we did not succeed in solving. To keep other chemists from wasting their time, we will shortly sketch out the two methods which we tried. First Method—A known weight of the chloroplatinate (contained in a boat standing within a combustion-tube, which has a spiral of red-hot sheet silver near its exit end) is heated to redness in a current of carbonic oxide, with the view of determining the water as such, and any surplus oxygen as carbonic acid. The former is collected in chloride of calcium, the latter in an evacuated flask containing a measured volume of standard baryta-water to be determined titrimetrically in the way which one of us employed successfully in the analysis of sea-water (see “Challenger Memoir”; also Dittmar’s Eve7~- cises in Quantitative Analysis, section on Sea-water Analysis, pages 227-230). To test the method, we applied it to a known weight of pure fused chloride of silver. The carbonic oxide was prepared from oxalic acid, and kept in a Pisani gas-holder over strongly alkaline pyrogallate of soda. Certain other precau- tions which were taken are too obvious to be stated; suffice it to say that we obtained 16 mgs. of carbonic acid from a substance containing no oxygen.* Second Method.—A known weight of the substance is heated in a current of pure hydrogen to obtain all the oxygen (present as water, or in other forms) as water, which is collected in a U-tube charged with pumice and oil of vitriol. By avoiding all cork-joints we had no difficulty in collecting our water in that U-tube, without any hydrochloric acid condensing along with it. To obtain air-free hydrogen, we employed a Kipp apparatus, charged with zinc and boiled out dilute sulphuric acid, and so communicating with a supplementary “ Kipp,” discharging hydrogen, as to maintain an atmosphere of hydrogen in the upper bulb of the other. The gas obtained was passed through a series of U-tubes charged with sulphuric acid and pumice, and from the last U-tube direct into a combustion-tube containing red-hot copper-gauze (and drawn out at both ends to avoid cork joints), and thence again through a U-tube containing _ sulphuric acid and pumice ; from this last tube the gas entered the combustion- tube to do duty. This method, like the first, was rehearsed with pure chloride of silver, and the result was that the chloride of silver yielded a very * No doubt some COCI, had been produced from the CO and AgCl, and not all been decomposed by the silver spiral, although we operated at the highest temperature permissible in a Bohemian glass tube. 580 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON appreciable proportion of oxygen! (2°03 grms. gave 15°6 mgs.). Where this oxygen came from we are unable to say; perhaps it came out of the glass of the combustion-tube, If ScHONBEIN were still alive he would perhaps say that the oxygen came out of the chloride of silver; we need not add that this is not our explanation of the phenomenon. While we are compiling this memoir, it strikes us that the oxygen in a chloro- platinate might perhaps be determined by heating it, behind red-hot silver, in a combustion-tube connected with, and previously evacuated by means of, a Sprengel pump. The chlorine should be retained as AgCl, and the oxygen go off as such (with the water), so that it could be collected, determined, and identified by the methods of gas analysis. Unfortunately the idea did not present itself to our minds at the time, and we had to rely on an obvious indirect method for determining the oxygen of a chloroplatinate. This is what we did in the following experiments, Supplement to Experiment V. of First Series. The principal chloroplatinate produced in this experiment,* while being dried at 130°, became slightly discoloured from some unexplained cause. Hence, instead of analysing it according to our general plan, we wrought it as follows :— The dried precipitate was dissolved in water in a large platinum basin, the solution filtered, and the filtrate allowed to cool with occasional agitation. The crystalline deposit was washed twice, each time with 10 c.c. of water. . It was then recrystallised from hot water. The crystals obtained were dried at 120° C., weighed, dissolved in water, and reduced in hydrogen in the wet-way, to determine the platinum and the two chlorines. The results were as follows :— Fixed Chlorine. Loose Chlorine. Platinum. Substance. Absolute weights, 0°52125 1:02916 14374 35704 grms. Relative 2 Hi Cl e100 140-005 19554 485-71 i =3°9488 x Cl The chlorine-deficit is too great to be explained by observational errors ; the salt must be assumed to have contained 0°0512 equivalents of oxygen, instead of chlorine, per molecule. Admitting this, we have, for the composition of 485:71 parts, Platinum, . A : i ; 2 : 19554 Chloride of potassium, , : : ‘ 5 14918 Other chlorine, , , ; , ; ; 140-00 Oxygen (as hydroxyl), . ; " . 0-41 Water (in the hydroxyl), , : ; 0°46 Total, t 485°59 instead of 485°71, which is very satisfactory. * We mean the bulk of the original precipitate; ¢.¢., that part of it which did mot pass on the filter, and which, as stated on page 570, amounted to 9°6244 grms, after drying at 130° C. A CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 581 The two mother-liquors derived from the recrystallisations were analysed (by wet-way reduction in hydrogen, &c.), so as to determine the absolute quantities of platinum, fixed chlorine, and loose chlorine contained in them. The jirst-mother liquor gave :— Platinum. Fixed Chlorine, Loose Chlorine, Absolute weights, . 07914 028214 0:59644 Relative - . 196°67 IO (2x< Cl 41798 x Cl 194:00 Sal x Cl 4124 xCl » ” The number 196°67 for Pt was chosen at the time, because it resulted from the synthesis of the original chloroplatinate *; we now utilise it as an upper limit, while 194 is used as a lower limit, for the unknown true “Pt.” Either mode of calculation leads to the conclusion that the liquor contained chloroplatinic acid and (really free) hydrochloric acid besides PtCl,H,. These analyses consequently prove, what SEUBERT only surmised, namely, that a chloroplatinate recrystallised from hot water contains oxygen in place of part of the chlorine of its PtCl,, and that the mother-liquor contains hydrochloric acid. Before inquiring into the origin of the chloroplatinic acid, let us give the results of the analysis of the second mother-liquor, It contained— Platinum. Fixed Chlorine. Loose Chlorine. Absolute weights, A 0°5249 019199 0:3609 Relative m . 196°67 2:0286 x Cl 3°8140 x Cl 194:00 20014 3°7629 x Cl Hither mode of calculation brings out oxychloride-oxygen, and with Pt = 19667 we obtain, moreover (for so much platinum), 0°0286 x KCl of surplus chloride of potassium. But the silver solution used for the determina- tion of the fixed chlorine (in the respective fraction of the filtrate from the platinum), amounted to only 27°3 grms. ; and either of the two numbers for the fixed chlorine (per Pt parts of platinum) must be considered uncertain by about + 0-002 of its value (because the weight of the platinum cannot be presumed to be free of error). Correcting it down by 0°002 of its amount, and reducing to Pt = 195°5 (which number we ultimately came to adopt as the most probable value, vide infra), we have— Platinum, Fixed Chlorine. Loose Chlorine. 195°50 2°0128 x Cl 3°7920 x Cl It really would appear that a little chloride of potassium has been eliminated in the second recrystallisation; but it is not permissible to draw such a weighty conclusion from a single analysis made on such a small scale. We prefer to look upon the fraction 0:0128 as resulting from observational errors. * And a small slip in a calculation ; the correctly calculated value is 196°64. 582 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Supplement to Experiment Va. of First Series. Of the 12 germs. of chloroplatinate produced in this experiment, the greater part was used for the following experiments :— 6:2356 grms. of chloroplatinate, dried at 110°, was dissolved in about 250 c.c. of boiling water in a platinum basin, and the solution allowed to stand (for about 40 hours), when a large crop of crystals was found to have separated out. These were washed thrice with small instalments of water and put aside as ‘‘ Crystals C.” The mother-liquor was distilled down, in a flask provided with a ground-in alembic and attached to a Liebig’s condenser, to about 25 ¢.c. The distillate contained a small quantity of hydrochloric acid, which was determined gravimetrically by nitrate of silver; its chlorine amounted to 0°87 mg. The contents of the flask, on cooling, deposited a crop of crystals which were collected and put aside as ‘“E,”’ the mother-liquor being labelled “ D.” Each of the products was analysed in toto (by reduction by hydrogen in the wet-way) so as to obtain the absolute weights of platinum, &c. Crystals “ C.” Substance unweighed. Platinum 1°9329. Total chlorine* 2:07648 and 207725 ; mean, 2:07686. Fixed chlorine 0:46467 and 0°46450; mean 0°464585 per fraction analysed. For the whole 0°701157. Hence per 2K Cl parts :— Platinum. Loose Chlorine. Oxygen (calculated). 195°48 139°129 or 39241 x Cl 0:0759x 40 About 2 per cent. of the loose chlorine is replaced by oxygen. Crystals “ E.” Fixed Chlorine. Loose Chlorine. Platinum. Oxygen. Absolute weights, 0:15260 0:29754 0°4195 Relative i} 70°91 138°26 194:93 =3°8996 x Cl 0:1004x40 Mother-liquor “ D.” Platinum. Fixed Chlorine. Loose Chlorine. t Absolute weights, . 0:1441 005044 0:11940 Relative M 195:00 ¢ 68°253 161'58 or 19250 x Cl or 45574 x Cl Hence, for the composition of the solution (if we take Pt = 195, as we * The AgCl in the titrations had a tinge of pink indicating platinum. + Including the 0°87 mg. from the distillate. t Instead of 194-93, as found for “ I.” - CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 583 did at the time; any other value which one could reasonably substitute would lead to, essentially, the same conclusion)— 1925 KCl PtCl \ *( 0-075 HCl \ 4 0-482 HCl. To check our work, we added up the several instalments of platinum, &c. found in “C,” “EK,” and “D,” and contrasted them with the quantities found in the original chloroplatinate. We had for Platinum, Fixed Chlorine. Loose Chlorine, | Dyes a aes SU Oar 2°4965 0-90419 1:79265 grms. Original precipitate, . 24961 090038 E625 *~ Considering the complexity of the operations involved in obtaining the upper set of numbers, the agreement is very satisfactory. From our analysis of chloroplatinates V. and Vw., and of their derivatives, we see that in both cases the original chloroplatinate, by being recrystallised, lost chlorine and platinum, with formation of free hydrochloric and chloroplatinic acids, which passed into solution, and the most plausible explanation of the result is to assume that the original chloroplatinates were mixtures of the constitution — PtCl,K,+«(Pt(OH),H, or PtO,H,). and that part of the hydrochloric acid formed in the substitution of oxygen or hydroxyl for PtCl,-chlorine served to dissolve away the #PtO,H, of oxyplatinate of hydrogen as chloroplatinic acid. Now, in the original chloroplatinate of Experiment V. the weight of platinum “ Pt” per 2K Cl parts was (by synthesis) =196'67 ; while in the twice recrystallised salt the corresponding quantity was = 195:54. In the case of chloroplatinate Va. we had “Pt”=196'58, and for its derivatives— Salts CO" tad bay ie Epes 194°93 The analysis “E” was carried out on a relatively small scale. The mean of the other two values is 195°51; and, supposing our theory to be correct, the true atomic weight of platinum should be either equal to or less than 195°51. Now the degree of completeness with which the surplus platinum is eliminated by recrystallisation should be the higher the purer the chloroplatinate started with, and of all our chloroplatinates those produced in Experiments VIII. and IX. by means of a large excess of chloroplatinic acid apparently come nearest to the ideal substance ; hence we thought the best thing we could do would be to prepare a large supply of such chloroplatinate, to analyse it, and then to see what value would come out for the weight of platinum per 2KCI parts, after recrystallisation from water and hydrochloric acid respectively, which latter solvent we hoped would eliminate the surplus platinum more completely, 584 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON and resubstitute chlorine for the oxychloride oxygen. This programme was carried out in the following experiments :— Experiment X. In each of two parallel experiments, I. and II.,* a known weight (about 3°8 grms.) of pure chloride of potassium was dissolved in 50 c.c. of water, and the solution mixed with 150 c.c. of chloroplatinic acid solution, made from Matthey’s metal by means of the chlorine process. This reagent contained 0:05 grm. of metal per c.c.; hence the excess of platinum used amounted to about 2°5 grms. The mixture was evaporated down on a water-bath, as far as possible, on stirring, the residue mixed with a little water, and re-evaporated. After cooling, 25 c.c. of water were added to produce a “ ten per cent” platinum solution, and the mixture allowed to stand cold, in the case of “I.” for an hour, in the case of “II.” for some 12 hours, with occasional stirring. The precipitate was then thrown on a, filter, the basin rinsed with 3 c.c. of the 5 per cent. platinum solution, and the precipitate, which was now all on the filter, washed with other 3 c.c. of the same reagent. After the liquor had drained off, the precipitate was washed exhaustively with alcohol of 95 per cent. (by weight). It was then dried in the filter at 100°, the bulk transferred to a tared glass- stoppered cylinder, and in it dried exhaustively at 100° C. The smallremnant on the filter was dissolved off with hot water, evaporated in a tared crucible, and weighed by itself. The aqueous and alcoholic washings (after removal of the alcohol by distillation from the latter) were reduced with hydrogen in the wet-way, the platinum was filtered off, and the filtrate evaporated to dryness to recover the potassium which had escaped precipitation. This potassium was determined as chloroplatinate (Fresenius’ modus). The results, so far, were as follows :— I II. Chloride of potassium taken, . 2 3 3°8062 38061 =A. 55 left unprecipitated, . 00319 0:0302 ‘i in the chloroplatinate, 3°7743 o Gio IA, Chloroplatinate obtained, . tenn . 12:3807 123691 =C, Hence A, : C, 0:30485 0°30525 which numbers seem to show that chloroplatinate “II.” having been lixiviated with chloroplatinic acid for 12 hours (instead of for one hour, like “I.”), was purer than No. “I.” Each of the two chloroplatinates was divided into three parts: one (A) for the determination of the platinum and total chlorine, another (B) for the determination of the water by the direct method, and a third (C) for * This second experiment was carried out for us by Mr Jamus Rogson. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 585 recrystallisation work. The water-determinations (carried out with 2:2 and 2:4 germs. of substance in I. and II. respectively) gave the following results :— Found in I. II. Per cents. of water, . ; : F : 0°353 0:269 The analyses were made with about 2:4 and 2°7 grms. of substance, dried at 100°. Their results, when combined with the synthetical results for the fixed chlorine, and reduced to Cl, = 7091 parts of the latter, were as follows :— Total Chlorine. Platinum. Spbetancs Anat sed. I. 4 : : 213:08=6:0099 x Cl 19601 489°35 IL. ; ‘ : 212'84=6:0032 x Cl 195°82 488°69 Neglecting the small excesses of chlorine found, we have for ie HI. Platinum, . : ; : ; : 19601 195°82 2KCl, : : : : : 5 149-18 149-18 4x Clof loose chlorine, . d : 141°82 141°82: Water (as above), E ; , 1°72 io Total, , , ; 488°73 48813 M., ‘ . » =489°35 48869 Unaccounted for, : ‘ i ; 0°62 0°56 These deficits might be allowed to pass as the cumulative effects of observa- tional errors; but possibly they may be owing to an element of uncertainty in the above numbers for the quantities of water. The samples “ B,” after having been weighed out for the determination of their water, were kept within their weighing-tubes,* over oil of vitriol, for a considerable time, before these determinations came to be carried out. On reweighing the weighing tubes and contents, they were found to have lost (‘1I”.) 1:3 and (“II.”) 2:1 mgs. Assuming these losses to have been suffered merely by the tubes and boats, the percentages of water come out as above reported. Assuming them to have been suffered by the chloroplatinates, their percentages of water rise to (“I.”) = 0-412, and (“II.”)= 0-356. The corresponding weights per M parts of chloroplatinate then are— I. IL. Water, ; ‘ S , : : 2:01 bee a c7id Total components, . : : ‘ 489-02 488-56 Deficits now, . : ? , : 0:33 0:13 _ which is more satisfactory. * Made each out of two lipless test-tubes sliding over one another. VOL. XXXIII. PART II. 4k 586 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Recrystallisations. The parts “C,” amounting conjointly to about 15:0 erms., were dissolved separately, each in 600 c.c. of boiling water, the solutions cooled down, finally in ice, the mother-liquors decanted and filtered off, and the crystals redissolved and recrystallised as before. The two twice-crystallised products were then united, dissolved in hot water, and the solution filtered, to make sure of the absence of every trace of dust in the final product. The filtered solution was evaporated over a water-bath, until crystals had abundantly separated out in the heat, then cooled down, finally in ice, the crystals collected, washed, first with a little cold water, and then exhaustively with absolute alcohol ; the latter washings being collected by themselves. The crystals were dried over oil of vitriol ; they amounted to 4°8 grms., showing that some 10 grms. of salt were contained in the several mother-liquors. The mother-liquors from the first and second crystallisations, when evaporated over a water-bath, gave off towards the end vapours of hydrochloric acid, showing that abundance of chlorine must have been eliminated, and its place taken by oxygen or hydroxyl. The whole of the liquors (minus a small quantity which had been used for these and other tests) were evaporated to dryness over a water-bath, and the residue washed with 91 per cent. (by weight) alcohol. The alcoholic liquors amounted to about 35 c.c.; they exhibited a faint yellow colour, indicating presence of chloroplatinic acid. 2°5 mgs. of platinum as chloroplatinic acid, when added to the same volume of alcohol, produced the same intensity of colour; hence (it would appear) very little platinum had been eliminated by the recrystallisations as chloroplatinic acid, or to speak more correctly, the PtCl,H, and the KCl in the (acid) mother-liquor were almost equally balanced, representing just so rauch PtCl,K, dissolved in hydrochloric acid. The chloroplatinate thus recovered was dried, recrystallised from water, the crystals mixed with the above 4°8 grms. of recrystallised salt, and the two again recrystallised conjointly. The crystals were washed, first with water, then exhaustively with 91 per cent. alcohol, dried over a water-bath, and preserved in a glass-stoppered bottle as “ ecrystallised precipitate.” Our mode of procedure may appear irrational to some of our readers, and so it is, inasense. Oar original programme was, starting from the chloroplatinate of Experiment X. to recrystallise it, and analyse both the crystals and the mother-liquor; then to recrystallise the crystals, and analyse the second mother-liquor; and so on, until the platinum per 2KCl would become constant, but we had not sufficient material for carrying out this scheme. The aqueous mother-liquors obtained were mixed with 10 c.c. of 20 per cent. hydrochloric acid, evaporated over a water-bath to about 70 c.c., the) liquor cooled down in ice, the erystals collected, washed first with + per cent. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 587 hydrochloric acid, then with water, and lastly and exhaustively with the 91 per cent. alcohol. They were dried, transferred to a glass-stoppered tube, and kept as “‘ Precipitate recrystallised from hydrochloric acid.” In order to see whether the chloroplatinic acid and the chloride of potassium were still balanced against each other in the hydrochloric mother-liquor, this liquor was evaporated down on a water-bath to about 10 c.c., and mixed, after cooling, with 71 c.c. of 91 per cent. alcohol, to bring down the chloroplatinate (and chloride of potassium, if present). The precipitate, amounting to about 0°66 erm., after having been washed with 91 per cent. alcohol, was extracted, very cautiously, with small successive instalments of ice-cold water. The aqueous washings were evaporated to dryness, the residues again taken up in a little water, filtered, and again evaporated to dryness. This last residue weighed 13 mgs., and consisted partly of free chloride of potassium; it gave a precipitate with added chloroplatinic. acid. Possibly the alcoholic liquors may have contained more free chloride of potassium, but we unfortunately forgot to examine them. XI. Analysis of the Chloroplatinate recrystallised from Water. This preparation was divided into two parts (amounting to about 2°8 and 3°4 erms.), and then analysed separately. Found per 2K Cl parts— ‘* Chloroplatinate.” Platinum. Loose Chlorine. Orie, é : 484°62 19550 138:55=3°9077 xCl ml ‘ : 484-91 195°69 138°66 =3°91077 x Cl Mean, : ‘ 484-77 19560 138°60=3°9092 xCl Substituting 0°0908x17 of hydroxyl for the chlorine deficit, we have (Mean of I. and II.)— Platinum, . : F : ; : ; 195-60 Chloride of potassium, . , 3 : 149°18 Loose Chlorine, . : . J , ‘ 138-60 Hydroxyl, . : : ' ‘ : : 154 484-92 Mrs I 104 : : 484-77 Excess, : ‘ 0:15 XII. Analysis of the Chloroplatinate recrystallised from Hydrochloric Acid (by Mr Robson). This preparation was also analysed twice (substance used = 1:9379 and 23525 germs.) ; but unfortunately, the determination of the fived chlorine in the 588 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON second analysis miscarried. Reducing to unit weight of chloroplatinate analysed we had— Platinum. Total Chlorine. Fixed Chlorine. i : ; , 5 0:40203 0:43639 0:14587 i : ' : ‘ 0:40179 0:43662 lost Mean,* 5 : : . 040191 043650 (0°14587) * Or reducing to 2KCI parts, Chloroplatinate. Platinum. Loose Chlorine. 486:10 195°37 141:28=3:9847 x Cl The water was determined directly in 3:2310 grms.: found 5:9 mgs., or 01826 per cent., or 0888 parts per M parts of chloroplatinate. Substituting 001534 x 17 of hydroxyl] for the chlorine deficit, we have— Platinum, ; ; ; 195°37 . i Ke ee ions 3°98466xCl, . } ; 141-28 0:01534x40). a) - 0-12 001534 x 4H,0, : ‘ 014 ‘ 0888 —0:138= Other water, by analysis, . 0-75 { 0-750 Total; ? \ . 486°84 1 = 486-10 Excess of analysis, = 0-74 which is not too much. It is surprising to see that the action of the hydrochloric acid did not resubstitute Cl for all the (OH) of the recrystallised salt. The following table summarises what we found regarding the quantity of platinum (“Pt”) present, per 2K CI parts, in our several chloroplatinates :— 1. Chloroplatinates prepared by Simple Precipitation ; Platinum moderately (if at all) IN excess. Experiments. Pe fea oat of KO) nse ue Pt. 1. 209 \ Mixture evaporated to small 196:28 >) TE 218 volume, then alcohol added, 195°67 IIL 206 32 GC. 19567 Mean IV 214 Ou. oz 195-79 P= V. 195 34 ,, 196°64 196°28 Va. 203 50M, 19658 VI. 196 (Precipitated by alcohol.) 196°95. 9) * «P” stands for weight of platinum used as chloroplatinic acid, per 2KCI parts. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 589 2. Chloroplatinate prepared by precipitating Chloroplatinie Acid with a large excess of Chloride of Potassvwm. Volume of mixture of Reagents Pt eeoament, be per 1 grm. of KCl used. 194:65 (196°78 per VII. 40 53°T c.0.* | 4Cl parts of loose (See context.) chlorine). 3. Chloroplatinate made by precipitating Chloride of Potassium with a large excess of Chloroplatinie Acid. A Volume of mixture of Reagents Experiment. = per 1 grm. of KCl used. ae ery: 294 160 c.c. 195°87 4. Chloroplatinates made from Chloride of Potassium, by evaporating with a large excess of Chloroplatinic Acid, washing with first 5 per cent. Platinum Solution, and lastly with Alcohol. Experiments. P. ee moc Pt. Tx. 294 (Evaporation) 195°59 ? M = X. (1) 294 i 196-01 } se X. (IL) 294 3 195°82 5. Recrystallised Chloroplatinates, Supplementary Experiment to V., . , ‘ : ; ; . 19554 . - to Von ; : : : . 195°48 ( Mean= Experiment XI, (Recrystallisation of X. from een : . 195°60 ( 195°50 ‘5 XII., (Recrystallisation from dilute hydrochloric pci) a Oboe, From our recrystallisation experiments (if our theory be correct), it follows that the true atomic weight of platinum (apart from observational errors) lies at, or is less than, 195°50. This number, 195:50, then, affords an upper limit to the unknown number Pt. SEuBERT’s experiments enable us to find a lower limit. SEUBERT, in his Memoir, gives the results of two analyses of one, and of six of another, preparation of chloroplatinate of potassium. The chloride of potassium and the platinum were determined in each case, the loose chlorine only three times. His mean results, referred to 100 parts of substance analysed, were as follows:—(Mean relative uncertainty means the mean deviation from the mean, measured by the mean itself; or the uncertainty per unit of the quantity determined). Per cents. Mean relative uncertainty. Platinum, . : : 4 40°110 + 0:0004 Chloride of nore i : : 30°685 + 00010 Loose chlorine, ; 4 P 29:144 + 00017 Error, water, &., . ‘ : , 0:061 100-000 * Per 1 grm. chloride of potassium as calculated from the platinum, assuming Pt to correspond to 2KCI. 590 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Reducing to 2KCl parts, we have 2KCl. , Platinum. Loose Chlorine. Chloroplatinate as analysed. 14918 195-00 + 27 141°688 48617 or 39963 x Cl+ 011 Cl But Sevusert’s chloroplatinate was prepared by precipitating a (rather dilute, ice-cold) solution of chloroplatinic acid, with, in two cases, 1°33 times, in the other six cases, twice, the calculated weight of chloride of potassium. His precipitate, therefore, in all probability, contained loosely combined (surplus) chloride of potassium. é Side by side with the potassium salt, SEUBERT prepared ammonium chloroplatinate by a closely similar process ; the latter he recrystallised to remove “edergerissenen salmiak”; the former he accepted as normal, although it also suffered a loss of alkyl chloride on recrystallisation; but in this case, it appears, SEUBERT assumes that the eliminated KCl came out of the chloro- platinate itself. For this, we submit, he had no excuse. Our view of the matter is that both his chloroplatinates contained niedergerissenes alkyl- chloride. In the case of his chloroplatinate of potassium, a very little surplus chloride of potassium was sufficient to make his value Pt by half a unit too low. In order to see by how much we would have to correct down his proportion of KCl in his chloroplatinate, let us refer his numbers to Pt = 19550; they then read as follows -— Chloride of Potassium. Platinum. Loose Chlorine. 149562 + 0:21 (195:50) (4:0065 + 00084) Cl for which we may substitute, without correcting by more than the mean errors 149:35 (195:50) 4:000 x Cl = 1:00114x K,Cl, Now SEvBERT’s chloroplatinate of potassium, from the way in which it was prepared, was bound to contain some “‘ niedergerissenes”” chloride of potassium; that its proportion should have amounted to less than 0:00114 of the chloride of potassium of the real chloroplatinate in his precipitate, is not at all likely ; hence we are justified in concluding that his analyses of chloroplatinate of potassium fall in better with our Pt = 195°5, than with his own 194°8. A critique of his analysis of the ammonium salt leads to a similar result. The results of these analyses may be summarised as follows :—Found for “Pt” (referred to O = 16). 1. By determining the weight-ratio of platinum to non-platinum, in a salt precipitated from sal-ammoniac solution, by an excess of ee acid (Darstellung I.), . ; , ; . 19576 2, By determining the same ratio in a salt Mined mie an excess of sal- ammoniac (and not recrystallised) (Darstellung IT.), . . ‘ } . 19453 CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 591 3. By determining the same ratio in a salt prepared from salt of Suid TT by recrystallisation, . . ; . 195-16 4. By determining the same ratio in a oat ened fee apr acid by addition of about 2°5 times the calculated weight of sal-ammoniac, washing, and recrystallising the precipitate (Darstellung [V.), . i 2) E9530 5. By determining the weight-ratio of platinum to total chlorine in “ae pre- parations, namely :— (a) One of Darstellung II. not recrystallised. (b) One of Darstellung IIL. (precipitation of chloroplatinic acid, prepared by means of chlorine gas and hydrochloric acid, with a somewhat considerable excess of sal-ammoniac). (c) One of Darstellung IV. The values Pt (calculated by us from his numbers for platinum and total chlorine) were as follows :— Salt, . : : . a, b, C, ee , : = 195:55 19578 196:10 Now, all these chloroplatinates were liable to contain water. Those which were not recrystallised,—excepting (1),—were almost bound to contain free sal-ammoniac ; those which were, probably contained OH instead of part of their Cl. In the case of determinations (5), (a), (0), (c), however, the presence of water does not tell upon the results; hence, if we had only these three analyses to go by, we should say, results () and (4) are probably too low, be- cause (free sal-ammoniac) chlorine was determined as chloroplatinate-chlorine ; (c) is probably too high, because the salt contained oxygen in place of part of its chlorine, and we should take the mean of the mean of (a) and (0d), and of (c) as the most probable value, and put down Pt = 195°89 + 0-22. Of result (1) it is difficult to say whether it is more likely to be too high or too low, because it may have contained surplus platinum ; we must accept the 195°18 as it stands. Result (2) is sure to be too low, because it must be presumed to have contained both water and surplus sal-ammoniac. Hence the value 194°53 is Jess than the true Pt. Preparations (3) and (4) were probably free of surplus sal-ammoniac or platinum; but they may have contained water, which would make the re- sulting Pt too low; and they probably contained hydroxyl in place of their chlorine, which would tend the opposite way. But result (3) is derived from only one analysis. Hence (as (2) is out of court), the most reasonable mode of utilising the determinations (1) to (4), is to take the mean of (1) and (4); or rather, as (1) included six and (4) included nine analyses, to take Pt = (6 x 195°18+9 x 19550) + 15 = 195°37, and assuming this to have, say, 10 times the “ weight” of the result deduced from analyses (5), we have, finally, Pt =195-42, which number falls in well enough with SruBERT’s analyses of 592 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON chloroplatinate of potassium, and may be adopted as being virtually his final result. Combining it with our own number 195'50, we have Pt = 195°46, or rather 195°5 (because the uncertainity on either side is more than + 0:1), as being at present the most probable value of the constant. The true number, we mean a number ranking in probable precision with, say, Marienac’s number for chlorine, will, we hope, be determined one day, but if so, it must be derived from other experiments than analyses of chloro- platinates, which are clearly unfit for the purpose. If, instead of searching for the true atomic weight, we want the quasi “constant,” which tells us how much platinum is associated with 2KCl parts of chloride of potassium, in a chloroplatinate produced in the ordinary methods of analysis, even our value is too low by about half a unit. So at least we must conclude from our experiments on FINKENER’s and on TatTiock’s form of the chloroplatinate process for the determination of potassium. These experiments are detailed in the next following section. II. FINKENER’s AND Tattock’s Metuops oF PoTAsH DETERMINATION. Finkener’s Method. This method is not so widely known in this country as it ought to be, we therefore begin by shortly explaining it. Assuming, for the sake of greater de- finiteness, that the substance to be analysed is a mixture of chlorides and sulphates of potassium, sodium, and magnesium, it is dissolved in water, and the solution mixed with a quantity of sulphuric acid sure to be sufficient for converting all the foreign oxides into sulphates; a quantity of platinum solution, a little more than equivalent to the potassium to be determined is now added, and, if necessary, so much water that the chloroplatinate precipitate produced is dis- solved in the next operation, which is to heat the mixture on a boiling water bath. The solution produced is evaporated on a water-bath to the consistence (after cooling) of amagma. This is allowed to cool, mixed with ether-alcohol,* and allowed to stand, well covered, until the precipitate has settled completely. The precipitate,—a mixture of chloroplatinate of potassium and the sul- phates of the foreign metals,—is washed with ether-alcohol, to be worked up in one or other of the following two ways :—(A) The precipitate is heated in hydrogen gas to dull redness, so as to reduce the chloroplatinate to Pt+2KCl, the product treated with water (then with hydrochloric acid, if necessary, and again with water); the residual platinum ignited and weighed, to be calculated into potassium. (B) The precipitate is lixiviated, as quickly as possible, with concentrated (cold) solution of sal-ammoniac, until the filtrate is * “ Ether alcohol,” in connection with Frnkener’s method, always means 1 volume of anhydrous ether and 2 volumes of absolute alcohol. te PERS = « CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 593 free of sulphate ; the residue (chloroplatinate of potassium plus sal-ammoniac) is dried, ignited in hydrogen, the platinum collected as before, and weighed. In this case, of course we have the option of determining the chloride of potassium extracted by water from the ignited residue, either quite directly by evaporation, &c., or indirectly by determining its chlorine. When one of us, some years ago, was commissioned by the Challenger Authorities to carry out exact analyses of a large number of samples of ocean- water, he inquired critically into the several known methods which might have been used for the determination of the small proportion, of potassium present in sea-water salts, and found that FINKENER’s was the only one which afforded a fair approximation to the truth. To render his results susceptible of subsequent correction, he brought the FINKENER method for his purpose into a definite form, regarding which it may suffice here to state the main points as follows :—* 100 c.c. of sea-water are weighed, evaporated to near dryness, and the salts made into normal sulphates; these are dissolved in water, any sulphate of lime, &c., is filtered off, and in the filtrate, the potassium is determined by FINKENER’S method (form A) by means of a quantity of chloroplatinic acid containing 200 mgs. of platinum, 7.¢., about twice the calculated quantity. A number of analyses of synthetically prepared mixtures showed that the platinum-weight, when multiplied by K,O + Pt = 94+ 198 = 04747, gave results about 1 per cent. short of the potash used (as chloride of potassium), When the factor, calculated from SEvBERT’s atomic weight for platinum (Pt = 194°8) and Sras’ value for K,O was used (7.¢., the factor 0°48386)t the results were about by 0:01 of their value too high. We may state, in passing, that it was this observation which gave the start to the experiments reported above as Nos. I. to VII. in the First Series. After the “ Chalienger” analyses had been reported, we again determined, for our own satisfaction, what degree of exactitude would have been attained if we had separated out the potassium and sodium as chlorides (free of calcium and magnesium), and then applied what we are in the habit of calling “ Fresenius’ Method,” because it is the one recommended for mixtures of the two alkyl-chlorides in his handbook of analysis. Leaving the errors involved in the elimination of the lime and magnesia on one side, we prepared a mixture containing very nearly the weights of chloride of potassium and sodium present in 100 c.c. of ocean-water (that of the former, of course, adjusted exactly), and analysed it by means of the following method :— * For details, see ‘Challenger Memoirs,” Physics and Chemistry, vol. i. (appendix), p. 233 et segq.; see also body of Memoir, pp. 12 e¢ segq. } New factor _ 1.9399, Old factor VOL. XXXIII. PART II. As 594 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON The mixture is dissolved in a little water, mixed with more of chloro- platinic acid than necessary for the conversion of both metals into chloro- platinates, and evaporated to a magma on a water-bath. The residue, after | cooling, is digested in 30 c.c. of alcohol of 80 per cent. by volume, the liquid decanted off through a small filter, and the precipitate washed with the same alcohol, until the last runnings give only an opalescence with nitrate of silver. The washed precipitate is dried on the filter, dissolved off with boiling water, the solution evaporated in a tared crucible to dryness, dried exhaustively at, first 105° C., then at 130°, and weighed. From the weights, the potassium is calculated (as K,O) in order to see what the exact, but unreasoning, applica- tion of the method would lead to. But the precipitate can neither be presumed to contain the whole of the potassium, nor to be pure; hence— . Firstly, the crude precipitate is reduced in hydrogen, the platinum weighed, and from the aqueous solution of the chloride of potassium, the latter recovered by a renewed application of FRESENIUS’ method. Secondly, the filtrate from the original chloroplatinate (which contains the sodium and excess of chloroplatinic acid), is freed from its alcohol, by evaporation to dryness, the residue reduced in hydrogen, the alkalies are extracted with water, and made into normal sulphates. In these, the potassium which escaped precipitation is recovered by FINKENER’s method (sal-ammoniac form). The platinum obtained in the reduction of this small quantity of chloroplatinate of potassium is weighed, and the chloride of potassium reconverted into chloroplatinate by Fresenius’ process, to be weighed as such. The pure chloride of potassium used for preparing the mixtures was made from recrystallised chlorate.* For the preparation of potash-free chloride of sodium, we used two methods, of which the following gave the best results :— Ordinary “pure” sulphate of soda is dissolved in water, and the solution saturated with hydrochloric acid gas, to precipitate about one-half of the alkali metal as chloride, which is collected on a funnel stopped up with around glass bead fitting pretty closely into the neck of the funnel, and washed with fuming hydrochloric acid. The salt is then dried, redissolved in water, and reprecipitated by hydrochloric acid gas. The dried product contained a mere trace of sulphate, which was neglected. To test the salt for potassium, a large quantity of it was made into sulphate, and 23 grms. of this subjected to the FINKENER process, sal-ammoniac form. The potassium extracted was determined in the way just described, and its chloroplatinate identified by microscopic inspection. It amounted to 0:38 mg. of K,O calculated from the platinum, and to 0°43 mg. calculated from the chloroplatinate. From the mean, 0°40 mg., we * We subsequently came to prefer the perchlorate for obvious reasons. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 595 calculated that 100 grms. of the original chloride of sodium contained 2:1 mgs. of potassium calculated as K,O. This potassium was allowed for in the test experiments. Two such experiments were made, and both are reported in the “ Challenger Memoir,” along with some further details which are here omitted for brevity’s sake. We satisfy ourselves with quoting one of the two reports. Chloride of sodium used, ; ‘ i ; 2°9 germs. Potassium present, elodlaied as K Oi in ee by Old Atomic New Atomic Weights.* Weights. t Chloride of potassium taken, . ‘ , ; 50:04 50°12 . in the chloride of coin, : 0:06 0:06 50°10 50:18 Analysis,—Potassiwm found, calculated as K,O in milligrammes. I. In the crude chloroplatinate by calculation, ; 47°90 48°35 Ia. By calculation from the metallic Pt obtained therefrom, . : ; : ; 47-62 48°53 II. In the pure peercnlauetes : j oat OG 47°50 IIa. Calculating from the metallic Pt obtained, : 46°62 47°51 III. In filtrate from crude chloroplatinate, ; 4 2°64 2°68 IV. In filtrate from pure chloroplatinate, . ; ; 0°85 0:87 Sum of IL, III., and IV.,. 7 ; , : 50°55 51:05 Excess over synthesis, . 5 _ ; < 0°45 0:87 Sum of IIa., III, and IV., : ; : : 50°11 51:06 Excess over synthesis, . : 5 0-01 0°88 From the numbers given under I. and the ee we see that FRESENIUS’ method, if used as it stands, would have given a loss of potash amounting to. : ; 2:20 1:83 Or, per 100 of K,O to be determined, to ; : 4:4 ~ at Note—The chloroplatinates were weighed after abn dried at 130°. The weights, after drying at 105°, were only about 0-001 more per unit weight of precipitate. For the analysis of salt mixtures poor in potassium FINKENER’s is the only method that works at all; for the analysis of potassiferous substances generally, it of course only competes with other methods, but over any of these it offers the great advantage of being in a high degree independent of the - nature of the foreign bases present. It must be admitted, however, that the FINKENER method, in the form in which it came out of the inventor’s hands, owes what there is in it of precision, to some extent, to compensation of errors. FINKENER’S own test-analyses prove this: they gave very exact results, because FINKENER, in reducing his platinum-weights to potash, used the old atomic *K=39; Cl=355; PE=198. + K=39'13 ; Cl=35°454; Pt=194'8. 596 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON weight of platinum (Pt = 198); if he had used SeuBERt’s value, which no doubt comes nearer to the truth, his results would have been by about 2 per cent. of their value higher (in calculating for K,O). | We thought it worth while to try and ascertain the several errors involved in the method, with the view of either evading them, or finding a formula for their correction, There is, however, one other form of the platinum process which, although of less general applicability, as far as it goes, would appear to be as inde- pendent of the nature of the impurities in the substance analysed as FINKENER’S. We refer to Mr Tattocr’s method, which, with anything that falls within the denomination of “ potash salt ” (pure or impure) is well known to give, to say the least of it, fair results. We accordingly decided to test both these methods, as far as they compete side by side with each other. Passing over a deal of pioneering work, which, instructive as it was to ourselves, would probably not interest the majority of our readers, we will begin with our experiments on FINKENER’S method, and in the first instance detail certain experiments made for determining its value for the determination of relatively minute quantities of potassium salts, diffused throughout a substance consisting chiefly of sulphates and chlorides of sodium and magnesium. Let us state at once that we always worked with mixtures of this kind, because other acid-tests (than SO, or Cl,) are of rare occurrence, and other bases than MgO and Na,O are easily removed by analytical processes. From what we are going to say, every chemist will easily see to what extent the method is more widely applicable,—as a matter of probability at least. Blank Experiments with Unmixed Sulphate of Soda. The sulphate was made from the chloride of sodium which had served for the experiments detailed above, pages 594 and 595. It contained potassium = 1:93 mgs. of chloride in 71:08 grms. Experiment I.—1°5 grms. of the sulphate of soda (anhydrous) were dissolved in 30 c.c. of water, the solution mixed with chloroplatinic acid, containing 50 mgs. of metallic platinum, and evaporated toa magma. To it, alcohol (1 vol.) was added, and then (4 vol. of) ether, to dissolve out the chloroplatinic acid, which was washed away as far as possible with ether-alcohol. The remaining — salt was almost, but not quite white. This was no more than we had expected, having previously found that the mixture obtained in FINKENER’s process is liable to contain an excess of platinum. To remove this admixture, the salt, after removal of the ether and alcohol by drying at a gentle heat, was re- dissolved in water, and “re-Finkenerised ” without platinum; 7.¢., again evapo- rated to a magma, and the latter washed with ether-alcohol. The platinum thus CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 597 extracted was determined, and found to amount to 0°38 mg. The recrystallised * sulphate, when heated in hydrogen gas under a funnel, and (the product) treated with water, Jeft 1:2 mgs. of platinum, corresponding to 0°91 mg. of chloride of potassium. The 0:04 mg. (which had been present in the sulphate used) are not deducted. Experiment I1.—A repetition of I., except that 100 mgs. of H,SO, in the form of standard acid, were added in the recrystallisation, with the view of decomposing the chloroplatinate of sodium suspected to be present. Platinum obtained from the recrystallised salt, = 0°5 mg. = 0°34 mg. of chloride of potassium (corrected for the potassium in the sulphate of soda used). Eaperiment \11.—Three grammes of sulphate of soda used, and the sulphate obtained recrystallised twice, without addition of sulphuric acid. Residual platinum, = 0-4 mg. = 0°30 mg. of chloride of potassium, or 0°22 mg. after deducting the 0°08 mg. really present. Eaperiment 1V.—A repetition of Experiment III., except that 200 mgs. of H.SO, were added in each recrystallisation. Residual platinum exactly the same ; 7.¢., a quantity indicating 0°22 mg. of adventitious chloride of potassium. Experiments III. and IV. were made side by side of each other. Eaperiment V.—To study the effect of the added sulphuric acid on chloro- platinate of potassium, 0°2006 grm. of pure chloroplatinate of potassium was dissolved in 40 ¢.c. of boiling water, containing 25 mgs. of H,SO,, the mixture evaporated to a magma, and Finkenerised. The resulting salt was “ recrystal- lised ” again with 25 mgs. of H,SO,. Both mother-liquors were yellow. They contained, that from the first precipitation 0'9 mg., that from the second 0°6 mg. of platinum, equal to 0°68 and 0°46 mg. of chloride of potassium respectively. Seeing that sulphuric acid, under the circumstances, decomposes chloro- platinate of potassium appreciably, we tried, in Experiment VI., the effect of sulphate of Mthium on chloroplatinate of potassium, because, supposing it to prove inert towards chloroplatinate of potassium, it would have afforded an admirable reagent for the elimination of foreign chloroplatinates from a FINKENER residue. 0°5 grm. of chloroplatinate of potassium when Finkenerised with addition of 0:1 grm. of pure sulphate of lithia, yielded a filtrate containing 3°8 mgs. of platinum, = 2°9 mgs. of chloride of potassium. On “ recrystallising” with 50 mgs. of the lithia salt, the liquor contained 4:0 mgs. of platinum, equal to 3:0 mgs. of chloride of potassium. This shows that sulphate of lithia is not available for the purpose aimed at. Experiments with Sulphate of Soda, containing added Potassium. I. A quantity of sulphate of soda equivalent to 8°73 grms. of chloride (NaCl), and a known weight of a standard solution of chloride of potassium * We adopted this word for designatimg the operation described. U 598 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON containing 235°5 mgs. of KCl, were dissolved in hot water, and Finkenerised with a quantity of platinum solution containing 600 mgs. of platinum, 2.¢. about twice the quantity demanded by the potassium present. The magma obtained on evaporation was mixed with 50 c.c. of absolute alcohol, and allowed to stand for three quarters of an hour ; 25 c.c. of ether were then added, the precipitate well pounded up with a pestle, and the whole allowed to stand under a small bell-jar for two hours. The mother-liquor was then decanted off through a filter, and the precipitate washed with ether-alcohol (2 volumes of alcohol and 1 volume of ether), until the last runnings gave only an opalescence with nitrate of silver. The alcohol-ether filtrate was kept as “I.” From the precipitate the sulphates were extracted by lixiviation with (140 grammes of) saturated sal-ammoniac solution. The last runnings gave only a slight turbidity with chloride of barium. - The solution obtained was put aside as “ IT.” The remaining chloroplatinate was dissolved in hot water and in the solution, reduced by hydrogen. The platinum obtained was filtered off and weighed; it amounted to 311°0 mgs. The filtrate from the metallic platmum was evaporated to dryness, the residue ignited to drive off the ammonia-salt, and the resulting crude chloride of potassium weighed; it amounted to 241°5 mgs. This salt was dissolved in water, and its potassium eliminated and weighed as chloroplatinate, by a method which was essentially TaTiock’s, (extraction of foreign matter from the evaporated residue by 5 per cent.* chloride of platinum solution, and washing finally with strong alcohol). From the alcoholic washings the alcohol was distilled off, the residue united with the aqueous washings and filtrates, and from the mixture the platinum eliminated by hydrogen in the wet way. The solution was weighed and fractionated for the determination of the sulphuric acid, and of the total bases as sulphates. The following is a summary of the results, stated in milligrammes :— Chloride of potassium, used as such, . 5 ; ; ; : 235°52 is a in the sulphate of soda used, . : ; 0°29 Total, : A ‘ 23581 Chloride of Potassiwm obtained. A. (Apparently) in the erude chloroplatinate, as calculated from the Pt (factor = (6117, 4 ' ‘ : : ' . ‘ ‘ . f . 236°72 Excess found, . : 4 : : : ; ’ : ; ; 0:29 B. In the alcohol-ether filtrate and washings; determined as PtC],K,, Weight calculated from this precipitate=0°61 mg.; from the platinum eliminated from it by H,=0°84 mg; mean, Une. 1 Ohaeagy 4) YY op 2 aie * Meaning a solution containing 5 centigrms. of platinum metal perc.c. The phrases “ 5 per cent.,” “10 per cent.,” &c., platinum solution must be read in this sense. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 599 C. From the sal-ammoniac liquor (“II.”); determined in ignited residue as PtCl],K,, and as Pt eliminated from the same. Weight calculated from the PtCl, K, was 1:22; from the Pt it was 1°60 mgs.; mean, 1°41; corrected for the KCI contained in the sal-ammoniac used,* . ‘ 1:13 D. The crude chloride of potassium, extracted from the prude: Eiloreplnnieet weighed 241°5 mgs. According to the analysis of the filtrate from the PtCl,K, obtained from it, it contained 6°57 mgs. of Na,SO,; deducting this we have— F. Pure chloride of potassium, . ; . 234°93 The pure chloroplatinate of seioseatee ppiaredt aon the ane eee of potassium was weighed ; its platinum extracted by H, in the wet-way, and weighed likewise. D,. The KCl, calculated From the PtCl,K,, was 231:48 \ 4 Pt 93185 mean, . ; ; ; ; ; . 231°67 This D, then was the weight of chloride of potassium which the FINKENER process had actually separated out; the close correctness of the number “A” was owing to the presence of foreign chloroplatinates in what was nominally PtCl,K,. E. In the analysis of the crude chloride of potassium, by means of the platinum process, the little potassium which escaped precipitation was determined as PtCl,K,, and calculated as KCl; found equal to : : : ; 3°46 To check the work, let us add up the various instalments of chloride of potassium found. D,+B+C+E, . ; F 5 : ; ; ; 237°0 Excess found, . ‘ : ? 1:2 F+B+C, . ; : , : : : s : 2368 Excess found, . : : ; 10 II. A Second Experiment differed from the (“I.”) just detailed in the following points :—(To the same quantity of sulphate of soda as had been used in “I.”) only 23°86 mgs. of chloride of potassium were added, and the mixed solution Finkenerised with jive times the calculated quantity of platinum solution. The salt mixture (vNa,SO,+ PtCl,K,), after a final wash with pure ether, was allowed to dry, and “recrystallised” without any addition (of H,SO,, &c.). In the treatment of the “recrystallised” salt mixture (7Na,SO,+ PtCl,K,) with sal-ammoniac, an exhaustive extraction of the sulphate was not insisted upon, for not decomposing too much of the chloroplatinate of potassium. The impure residue obtained after treatment with 90 grms. of sal-ammoniac solution was dissolved in hot water, the platinum reduced out by H, and weighed. The sal-ammoniac liquor was preserved. The results, so far, were as follows (in milligrammes) :— * Determined by a blank carried out on a large scale. 600 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Chloride of potassium used, including 0°29 mg., from The Na,SO,, t ; : . ; 5 ; ; : ; . 24-15 — Found. (Apparently) in the ee Giliadi as calculated from the platinum extracted, . : 4 . . ; . : ; . 1918 In the second ether-alcohol elbrlte! 5: 8 mgs. Of these 5°8 mgs. of platinum, however, 0°8 was present as PtCl,K,, hence the re- crystallisation process, besides removing 5°8 — 0°3=5-0 mgs. of platinum present as chloroplatinate, removed also 0°8 mg. of platinum which ought to have remained. Hence, if recrystallisation had not been resorted to, the chloride of potassium found would have been. : » 2000 Frrors in the actual and imagined gus witli =-4 97 anil = 0: 56 rel ney: Experiment Ila.—The residue (yNa,SO,+ PtCl,K,+2NH,Cl) obtained in the inexhaustive treatment with sal-ammoniac was, as stated, reduced with hydrogen in the wet-way, to weigh the platinum present in it. The filtrate from the platinum was evaporated to dryness, the residue made into sulphates, and thus the original potassium—apart from what had gone into the two ether-alcohol filtrates, and into the sal-ammoniac extract—recovered, as part of a mixture of alkali-sulphates. This mixture was dissolved, Finkenerised with only a small excess of platinum (50 mgs. of Pt instead of about 32 as calculated), and no “ recrystallisation” effected. The washed chloroplatinate plus sulphate was treated exhaustively with sal-ammoniac (25:3 grms. solution) ; the sal-ammoniac liquors being put aside. The chloroplatinate (+xNH,Cl) was dissolved in water, and the platinum reduced out ; it weighed 24:2 mgs., equal by calculation to 18°42 mgs. of chloride of potassium. The chloride of potassium was recovered from the filtrate: it weighed 184 mgs., including, however, 0°2 mg. of insoluble matter, hence corrected weight = 18'°2 mgs. It was dissolved in water, the SO, precipitated by nitrate of baryta (precipitate calculated as Na,SO,= 0:29 mg.), and the chlorine of the filtrate determined by nitrate of silver. Chloride of potassium calculated from the weight of the chloride of silver = 18°11 mgs. . To be able to check our work, we determined the potassium in the two ether-alcohol filtrates of Experiment II., and in the two sal-ammoniac liquors, obtained in Experiments II. and IIa. respectively. Summary of Results. Mgs. Chloride of potassium taken, ; ; : ; ; ‘ , . 2415 KCl present in the chloroplatinate of Ilias galbalated from the Pt (24:2 mgs.) is, 1842 The crude KCl from the same, etukerted by deduct- ing SO, as Na,SO,, . : 5 Se Lge Calculated trom the chlorine of the orn clilbyids of } 18:01 potassium, . ; ; , ; : : . aon Aj=mean of 1842 and1811, . . . .». ‘ « « « «6 ain CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 601 B. KCl obtained from first ether-alcohol filtrates of Experiment II, . ; es ONG B’. KCl obtained from second ether-alcohol filtrates of Experiment II., ; a , Was C. KCl from the two sal-ammoniac liquors, worked conjointly :— Calculated from PtCl,K,, . 481) mean=5-03, deducting 0°23 present i ‘3 Ptsie . 521 sal-ammoniac, . sit) 4:80 X. Ether-alcohol filtrates of Experiment IIa. were not worked ae | they con- =} tained, let us say, the mean of B and B, : = 0°65 Total chloride of potassium found = A,+B+B’ 404X, = 25:03 Excess of KCl anal ‘ =" 088 The relative exactitude in Experiments II. and IIa., it is true, is not very high; but the absolute precision of the results is high, considering that the substance analysed was, virtually, a mixture of the two chlorides, NaCl and KCI, which contained only 0°2766 per cent. of the latter. Hence the FINKENER process, whatever it might be otherwise, is invaluable as a means for the determination of small quantities of potassium which escaped the meshes of other analytical methods, and in this sense, amongst others, we have used it largely. Experiments on synthetically prepared Mixtures containing relatively large quantities of Potassium. We could not detail all our experiments of this kind without filling a great many pages. We prefer to give the conclusions which we drew from a considerable experience concerning the FINKENER process, and then pass on to reporting, mainly, on our final series, which was carried out by what we at the end came to recognise as the best form of the process for general purposes. The conclusions referred to are these,— 1. In the analysis of a mixture of chlorides and sulphates of the bases K,0, Na,O, MgO for K,O, it is not necessary to begin by converting the bases into normal sulphates;* it suffices to add a sufficient quantity of sulphuric acid, equivalent, by calculation, to the chlorine present. In the test analyses to be reported on, we added measured volumes of standard sulphuric acid calculated from our knowledge of the weight of chlorine present, In actual practice, a preliminary determination of the latter by, say, Mour’s method, would give the necessary guidance. 2. In the treatment of the residue (of chloroplatinate of potassium plus sul- phates) it is expedient to add, first a sufficiency of absolute alcohol, say 10 c.c., to allow to stand for some time, and then to add the necessary (5 c.c, of) ether, and allow to stand longer, but under a small bell-jar on a glass plate. For a long time it was our rule to let the alcohol act for half an hour, and the alcohol and ether for other two hours; but we subsequently found that five * FINKENER does not say it is; but for a time it was owr method. VOL, XXXIII. PART II. z 47 602 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON minutes for the alcohol and then twenty-five minutes for the alcohol and ether suffice. In two test analyses (of a “95 per cent.” salt, vide infra) we used — this shorter mode, and found that the errors in the chloride of potassium amounted to only +909 and +0°94 mer., for about 753 mgs. to be deter- mined. The mixture is filtered, and the precipitate washed with ether-alcohol. To prepare it for “recrystallisation,” it is washed finally with plain ether, and allowed to dry. 3. The “ recrystallisation” of the mixture (PtCl,K,+#R,SO,) obtained in the FINKENER process is necessary, in general, for the removal of chloro- platinates, but addition of sulphuric acid in this subsidiary operation does no good. 4. The alcohol and the ether must be absolutely free of ammonia; we always distilled them with a little phosphoric acid before use. ‘5. With the generality of substances, the sal-ammoniac form of the process offers no advantages over the straight-forward determination of the ae in the “recrystallised” precipitate (of sulphates and PtCl,K,). 6. The determination of the platinum is best effected by reduction with hydrogen in the wet way. In our final series, the FINKENER and the TATLock processes were worked side by side of one another, in this sense, that for every mixture analysed by means of one of the processes, a substantially identical mixture was analysed by the other. Yet we prefer, meanwhile, to detail our test experiments on the FINKENER process first, and by themselves. For the preparation of the mixtures to be analysed, we used the following materials:— (1) Standard solutions of chloride of potassium made from perchlorate. The solutions were standardised synthetically by weight, and, immediately after their preparation, quantities containing the desired amounts of salt were weighed out into so many bottles, which were marked, and kept ready for use. (2) Chloride of sodium solution, prepared from potassium-free salt, and standardised volumetrically. 1 c.c. contained 8°194 mgs. of dry salt. (3) A sulphate of magnesia solution, prepared from pure (alkali-free) magnesia (MgO)* by solution in a very slight excess of standard sulphuric acid, and diluting to a definite volume. 1 ¢.c. contained 1539 mgs. of MgO. (4) A standard solution of sulphuric acid, made from distilled acid. 1 ¢.c.= 47°87 mgs. H,SO,. For the preparation of a mixture for analysis, one of the portions of chloride of potassium solution (see (1)), was mixed with measured volumes of solutions (2) and (3), and in general (4). The resulting solution was then Finkenerised * For mode of preparation, see “‘ Challenger Memoir,” p. 16. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 603 with a measured volume of a standard solution of chloroplatinic acid prepared from chemically pure metal by (in most cases) the chlorine process (see page 564). Our final experiments were made in sets of, in general, four analyses of the same kind of mixture. Before passing to these sets, we will shortly report on a single experiment, made with the view of seeing how the FINKENER process works with relatively pure potassium salts. Analysis of a Mixture of Sodium and Potassium Chlorides, containing 98°4 per cent of KCl. The solution analysed contained 7:1 mgs. of Na,SO,,=5°9 mgs. of NaCl, and exactly 0°3701 grms. of KCl. It was mixed with sulphuric acid equal to 49 mgs. of H,SO, (to give the sodium a better chance of separating out as sulphate), and then Finkenerised with 11 ¢c.c. of a platinum solution, of which 10 c.c. would have sufficed by calculation.* Salt-mixture (vR,SO,+ PtCl,K,) “recrystallised.” Platinum obtained from the final chloroplatinate (by reduction in the wet way) = 0°4855 erm. = 03696 erm. of KCl: error, = —0°5 mg. The second ether-alcohol filtrate was worked up for potassium. It amounted to 03 mg. of KCl. Assuming the jirst ether alcohol filtrate (which was not analysed) to have contained the same quantity, we have— Total KCl recovered, = 369'6+0°6 = 0°3702, instead of 03701 grm. The weight of platinum eliminated by “recrystallisation” was 3:1 mgs. Hence, if this operation had been omitted, we should have had 0:4855 +0°0031 = 0°4886 erm. of platinum, equal to 0:3719 of KCl; #2, a positive error = 1°8 mgs. _ We now pass to the series of trials referred to.t I. Set of Experiments. Subject :—a “95 per cent.” salt.{ Chloride of potassium operated upon in each analysis, about 0°75 grm. Sulphuric acid required to replace every Cl of the mixture by 48O,, = 10°9 cc. ; actually added,= 12°0 c.c. The whole was evaporated to dryness, and the residue ignited to expel the chlorine. Platinum solution required by calculation, = 20 c.c., we added 21°5 c.c., 7e., a very small excess, which, as we zow know, was a mistake. The mixture of PtCl,K, and * See last line of this page. +The designations of the following sets of experiments do not in general indicate the order in which they were carried out. {Meaning a mixture containing 95 parts of chloride of potassium in 100 of total anhydrous - salts. 604 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON sulphates obtained was “recrystallised,” the crystals obtained dissolved, the platinum reduced out by hydrogen and weighed.* Results in Grammes. Experiment. 1, 2. 3. 4, 5. I. KCl taken, . ; " ‘75750 75984 “75791 76042 75982 If. Platinum obtained, . ‘9959 "9952 wold, 9945 "9991 PE KOM it=0-76F17 x11... 75805 "75752 75485 ‘75698 "76048 Error, 2.¢., LIT.—-L, . +°55 — 2°32 — 3:06 —344 —0-66 mgs. The first alcohol and ether washings from all the five analyses were worked up for potassium by FINKENER’s method, sal-ammoniac form. Ultimately — the KCl was determined in FRESENIUS’s way, as PtCl,K,. Found for the five analyses, 1°37 mgs. of KCL The second alcohol and ether washings contained 39°1 mgs. of platinum, equal to 29:76 of chloride of potassium, or 5°95 mgs. per analysis. So much more would have been found (than quantities III.) if the recrystallisation had been omitted. But a determination of the potassium (as Pt) showed that 0:26 mg. of KCl per analysis was present in the ether-alcohol liquor. Viewing the five analyses as one, we have — A. Total chloride of potassium taken, . ; : : . 3°7955 grms. a. KCl lost in the ether-alcohol liquors, ; . . WOO2T. : A,=A-—a,. . ; ; : E 4 ; , S925) | p. Platinum obtaniteal!. , ; s : i ‘ — (40764); A :p = 0°76270 Ay : p = 0°76216 (p: Ay) x (14918 = K 20h) = quasi PiL=195-73 ,, (pr X 3 Fg t= 195-60 II. Set of Experiments. This set was carried out before we had come to adopt the recrystallisation modus for purifying the FINKENER product. It seemed to us at the time that the most exact method of potassium determination would be to eliminate the bulk of the potassium as chloro- platinate by precipitation from purely aqueous solutions, and to utilise the FINKENER process only for the recovery of the unprecipitated remnant. The substance worked upon was the 95 per cent. salt used in the I. Set. A known weight of chloride of potassium (amounting to about 0°76 grm.) was dissolved with the necessary impurities, and the solution next evaporated to about 5 c.c. About 1°5 times the calculated minimum of platinum solution was now added to produce some 35 ¢.c. of mixture, which was allowed to stand over night. The mother-liquor was then decanted off through a small filter, — *Tn only one case, No. (4), did the filtrate contain a trace of platinum. It was recovered by H,§, the PtS, made into Pt, and its weight (0:4 mg.) added on. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 605 the precipitate washed with small instalments of water until pure by calculation (18 c.c. of water were used in all), then with dilute, and finally with absolute alcohol. The precipitate was dried at 110° and weighed. It was then dissolved in water, the platinum reduced out by hydrogen, and weighed likewise. The alcoholic washings were evaporated to dryness, the aqueous liquors added, and after addition of enough of standard sulphuric acid for replacing the Cl of the chlorides present by }SO,, the whole evaporated, Finkenerised, and the sulphates removed by sal-ammoniac solution. The resulting chloro- platinate was dissolved in hot water, reduced by hydrogen, the platinum filtered off, and the filtrate next evaporated to dryness, and the residue ignited, to drive off the ammonia-salt. In the ignited salt, the potassium was determined by means of the Fresenius’ form of the chloroplatinate process. In some cases the chloroplatinate obtained was reduced in hydrogen (wet way), and the platinum weighed. The factors used were — For reducing PtCl,K, to K,Cl,, . : : : : ‘ . 030435 sy Pt zh : ; : : : : . 0°76117 In Experiments IT. to V. chloroplatinic acid made by the chlorine process was used; in Experiment I. a reagent made with aqua regia. Results. Experiment. 1. 2. 3. 4, 5. Chloride of potassium used, . : . 75380 75308 ‘75497 “75467 75390 grm. TI. Bulk of chloroplatinate, . eeoold “<2aviG gicoo2st Zao2S 23569 , II. K,PtCl, from liquors, . : , 138" * “0887 0926 "1202 190" 5 III. Total PtCl,K,, . : : . 24717 24663 24750 24730 24759 ,, IV. KCleorresponding, . ’ 3752260 5375062,.4 ipe26 =. “75266... "75354 V. Platinum from I, : ; « » 9456 . :9529 "9549 9430 9423 —C, VI. e some ; : . 0454 ee Sie 0487 0484 _—C, wi —V.+VI.,. : j » Joe oe ise ily S907 VSS, VIII. KCl corresponding to v,, FISICA (2502) A 268k OLS) 71725. ,, IX. ‘5 VI. i 11), 03456 (02700) (02818) ‘03707 03684 _,, X. =VIIL. +IX, . : 75432 (75232) (75502) “75485 75409 ,, Error in IV. (in day ‘ . —154 -—246 -—-171 -201 -036 mg. X. ; : . +052 -—076 +005 +4018 +019 ,, SO, found i in I, 5 a 0°51 0°75 cae Total chloroplatinate eadaeedl 1.0. rae = aE) as , : : . 12°3619 grms. Total chloride of potassium employed, =, di (04, 4; Hence chloride of acon per unit of piibeepiatinate aed at LO? C., : : ‘ p . 0°30500 ,, Total “iste feted in Sracenagu 1, 4, aad 5y. : : . =29734 =p. Total chloride of potassium for experiments 1,4,and 5, . ; . =22624 =A. Hence A: p = 0°76087 and (p: A) x K,Cl, = 196-06. 606 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON III. Set of Experiments. Subject, the same “95 per cent.” salt as was used in Sets I. and II. Method the same as that used in Set L; except that the salts were made into sul- phates only virtually, by addition of the calculated amount of standard sulphuric acid. Platinum added on Finkenerising, = 25 c.c. or 1°25 times the calculated minimum. Product (PtCl,K, and 2 R,SO,) “ recrystallised.” Results. Experiment. "iB 2. 3. 4, I. Chloride of potassium taken, 75330 "75253 75327 75426 grm. “II. Platinum obtained, . : 9900 9880 9902 0907 a TE, KC) found af ="76lt7 AL, "75356 75204 “75371 75409 Error=1I1.—I, . : + 0°26 — 0°49 + 0°44 —017 mg. The ether-alcohol liquors not analysed. Uniting the four analyses into one, we have— Chloride of potassium taken, . ; ; : . =A=3-0134 grms. Platinum obtained, . : ; . : : . =p=s0580 %, Hence A : p.=0°76116. (op: A) X KOCE =" Pi2— 195-99; IV. Set of Experiments. The solutions analysed were such as to represent very nearly a salt consist- ing of 82 per cent. of chloride of potassium, 15 of chloride of sodium, and 3 of sulphate of magnesia. The quantity of chloride of potassium used was about 0°656 grm. per analysis. | | The solution to be analysed was mixed with the quantity of standard sulphuric acid equivalent to the chlorides present, and 1:25 times the calculated — volume of platinum solution, and Finkenerised ; the resulting product being “recrystallised.” All else as in III. Results. Experiment. Ie 2. 3. 4, I. Chloride of potassium taken, 65390 65294 65032 65046 grm. II. Platinum obtained, . ? 8597 8553 8543 8547" ,, III. KCl, corresponding . =II. x 0°76117, 65428 65103 65027 65057 Error, III.—I.,. : . + 0°48 —1:91 —0°05 +011 mg. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 607 Uniting the four analyses into one, we have— Total chloride of potassium used, .. : : . =2°6076 grms.= A. , platinum obtained, . : : : ~ Soe =p: Hence A : p=0°76157 and (p : A) x K,Cl,=195°89. V. Set of Experiments. Salt analysed consisted of 33:3 per cent. of chloride of potassium, 33°3 of chloride of sodium, and 33°3 of sulphate of magnesium. Method exactly as in IV. Set. Results. Experiment. at 2. 3. 4, I. Chloride of potassium taken, *26603 26606 -26490 “26490 orm. II. Platinum obtained, . z “3501 3500 3484 oy) ae IIL. KCl, corresponding, 2.¢., : 0-76117 x IL, i 26649 26641 26519 26572, Error, III. — [., - +0°46 +0°35 +0°29 +0°82 mg. A =Total alae of SES used, =1:0619 grms. _ p= _.,, platinum obtained, =1:3976 , Hence A : p=0°75980 and (p : A) x K,Cl,=19634. Tatlock’s Method. This method was invented expressly for the assaying of commercial potash- salts, @.¢., of salt mixtures similar in constitution to those which we employed for our test-analyses by FINKENER’S method. Mr Tartocx’s method, according to his own description,* is as follows:—Assuming the substance to be analysed to have been converted into a standard solution, a quantity equal to (10 grains” =) 0°6 to 0°7 grm. of dry salt is measured off, to be analysed as follows :—For every one gramme of salt the solution is diluted to about 40 c.c. ; it is then acidified with a few drops of hydrochloric acid, and mixed with 50 c.c. of a “5 per cent.” chloride of platinum solution, meaning a solution contain- 5 centgrms. of metal per c.c. The mixture is evaporated to near dryness over a water-bath, and the residue re-evaporated with addition of a little water, to more fully eliminate the free hydrochloric acid. [Observe that this large proportion of platinum is prescribed for a// kinds of salts, rich or poor. Now 1 grm. each of the anhydrous salts, NaCl, MgCl,, MgSO,, demands only 1°69, 2°08, 1°65 grms. of platinum, assuming Pt to be equal to 198; hence Mr TaTLock’s intention apparently is to have sufficient chloroplatinic acid present for converting all the metals into chloroplatinates, and, in addition thereto, some * As communicated by him to a Committee of the British Association, and published by them in a Report presented to the Meeting at Glasgow, in 1876. 608 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON 0'8 grm. (we calculate from the NaCl number) as surplus chloroplatinic acid.] Some 5 c.c. more of the choride of platinum solution are mixed with the residue (which, in the case of NaCl, will produce some 6 c.c. of a 17 per cent. solution) ; the whole is stirred well, and set aside in a cool place for at least an hour, with occasional stirring. The precipitate is then thrown on a very small filter, the basin rinsed out with about 15 drops of the platinum solution, and the precipitate on the filter washed with 16-24 drops more. The basin, and filter and contents, are then washed with the smallest possible quantity of alcohol of 95 per cent. (by weight or volume ?; we used 95 per cent. by weight), and dried at 100° C. The precipitate is transferred as far as possible to a tared capsule, and further dried until it assumes a distinct orange colour. The filter with the remnant of precipitate adhering to it is incinerated, and the residue calculated as Pt+K,Cl,. The weight of the chloroplatinate of potassium, multiplied by 0°3056, gives the weight of chloride of potassium to be determined. [The factor is calculated from Cl=35°457; K =39:137; Pt=197°19. | Whenever in the following we state that an analysis was executed according o “ Tatlock’s directions,” these directions were followed closely as above given, except that we allowed ourselves to recover the small quantity of chloro- platinate sticking to the filter, by dissolving it off in hot water, and evaporating the solution to dryness in the tared crucible intended to receive the main quantity, and that we continued the drying process at 100° until the weight became constant. In the case of a substance rich in sulphates, TATLock recommends to add a quantity of pure chloride of sodium. This rule, however, is obviously based on the misapprehension that “platinum solution” is one of PtCl,* while it really is one of PtCl,H,. Yet the chloride of sodium may do good by substituting acid sulphate of soda for the H,SO, liberated, and_ besides, by displacing some of the HCl in the surplus PIC, lel Of the various sources of error involved in Datos method, the most. obvious is the appreciable solubility of chloroplatinate of potassium in water, and aqueous liquids generally. We therefore, at-an early stage of our investigation, determined the Solubility of the Chloroplatinate in the following reagents :—Five small flasks were tared, each charged with 0:2 grm. of chloroplatinate of potassium, and a convenient volume of the respective * We used to be under this erroneous impression ourselves until some three years ago, when we analysed a carefully prepared platinum solution (which had been specially freed from extra hydro- chloric acid) for chlorine and platinum, It contained very nearly 6 x Cl for 1 x Pt, which, by the way, is in accordance with an old analysis of “ chloride of platinum,” quoted in Gmetins’s handbook as having been made by VAUQUELIN. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 609 solvent, the flasks stopped up, and allowed to stand with occasional agitation. Whenever the precipitate threatened to dissolve completely, an additional weighed instalment of chloroplatinate was added. After six days the contents of the flasks were weighed, filtered, and the filtrates analysed for the dissolved chloroplatinate. The temperature during those six days varied from about 13° in the mornings, to about 16°°5 in the evenings. The results were as follows :— 100 parts by weight of solvent dissolve g parts of PtCl,K,. Solvent. q. A. Water, . : ‘ : : j : . 0628 B. a ene hata of 5 per a ; ; : : ; : . 0°662 C. “5 per cent.” chloride of platinum solution, . 0:233 D. Solution of chloride of platinum, containing 0:05 grm. of eae Gey HCl, and 0:05 grm. of platinum, per c.c., ‘ : : sy 0-168 E. Sulphuric acid, containing 4SO,=40 grms. per foe ‘ . 0900 Methods of Analysis: A. and B.—¥vaporation to dryness, and weighing of residue, dried at 150° C., as PtCl,K,. C. and D.—Evaporation to dryness on a water-bath, treatment with abso- lute alcohol, and weighing of the washed precipitate after drying at 150° as PtCl,K,. £.—The liquid almost neutralised with pure (potassium-free) caustic soda, the platinum reduced out by hydrogen, and weighed. We regret now not to have determined the action of stronger solutions of chloroplatinic acid, because Mr Tattockx virtually begins by washing his chloroplatinate with a “17 per cent.” solution of the reagent. In now passing to our analyses, we begin with a series in which we deliberately departed from certain of Mr Tattock’s rules, in order to bring the errors into greater prominence, and also on the chance of being able to rectify these by suitable modifications of the process. Let us at once confess that our success in the latter direction amounted to very little, if anything. Preliminary Trials. Experiment I.—The solution analysed contained exactly 0°3471 germ. of chloride of potassium, and about 100 mgs. Na,SO,, and 150 mgs. of MgSO, ; it consequently represented a salt of “58 per cent.” Evaporated down with platinum solution equal to 515 mgs. of platinum, or 50 mgs. more than demanded by the potassium. Residual magma washed five times, each time with 0°5 c.c. of water, then exhaustively with absolute alcohol. The chloro- platinate dried at 150°, and weighed ; then dissolved in water, the platinum reduced out, and weighed likewise. VOL. XXXII. PART II. 4U 610 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON The aqueous washings were evaporated with 1 c.c. of normal sulphuric acid (49 mgs. of H,SO,) to a magma and Finkenerised ; no recrystallisation ; sal-ammoniac form applied. Resulting chloroplatinate dissolved, platinum reduced out and weighed. Alcoholic Washings.— After removal of the alcohol by distillation, the platinum was removed by hydrogen, the filtrate evaporated to dryness, residual salts made into neutral sulphates and Finkenerised; sal-ammoniac process. The results were as follows (mgs.) :— I. Chloride of potassium taken, . : : , : . + S471 a II. First chloroplatinate precipitate, : ; ; : . 10265 =C, III. KCl corresponding (=II. x fal es ; : 4 . 313°63 IV. Platinum from II, . - } : . 4113 =p. Ve RCI] Ghincsp dialing (IV. x 0: 76117), ‘ : é 2 BLGOF VI. KCl in aqueous filtrates, . § 5 i 33°11 VII. , alcoholic washings, ee as 5 PLC] Ky : ; 1:12 I.—(VIL+VI), . , ‘ . 9312°88=A,, RP C_020d8 A, p= 07007; Ex)>P 196-11. Conclusion.—It is quite possible, by operating as described, 7.¢., without wasting so much platinum as Tatiock does, to obtain a chloroplatinate fit for the balance; but the chloroplatinate includes only about 90 per cent. of the chloride of potassium. Experiment 11.—The solution analysed represented 1:04 grms. of a “67 per cent.” salt, including 0°24 grm. of Na,SO,, 0°20 of NaCl, and 0°10 of MgCl,. Platinum used, 2°6 grms., or 2°49 per grm. of salt analysed. TatTLock’s directions followed, except that the washing with 5 per cent. platinum solu- tion was continued until the last runnings contained only a trace of SO,R,. Washing completed with 95 per cent. alcohol. Precipitate weighed, after drying at 100°, and after further drying at 150°. A =0°7018 grm. Chloro- platinate obtained, dried at 100° = C’ = 2:2745: same dried at 150° = C” = 22737. Platinum from C, by wet-way reduction,= 0:9132 grm.= p. Potassium in filtrates collected by FivKener’s (sal-ammoniac) process, and weighed as PtCl,K, .— KCl thus found = 0:00819. C’ x 030435 * = 0°69224; px 0°76117 = 0°69510. Mean = 0°69367 ; loss = 0°00813,= 1°16 per cent. of the KCl taken. KCl in precipitate by synthesis = 0°69361 = A,. Ay : C’= 030500; A, : p = 0°75954. ECL ; Pr = 196-41. 0 Conclusion.—It will not do to “improve” upon Tatiock’s method by washing with platinum solution, until the SO, is proved to be away. * These factors were calculated from results for “‘M” in the first series of potassium experiments. We did not consider it necessary to recalculate the analyses with our present factors. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 611 ‘Further Trials. These were carried out with solutions representing 95 per cent., 82 per cent., 33°3 per cent. salts, which were prepared from the same materials as those used for the corresponding trials with FinKENER’s method. In the case of each kind of salt, indeed, the two methods were worked side. by side of each other, so as to give no advantage to either. General Method.—A solution representing 0°648 grm. of a “95 per cent.” salt mixed with a few drops of hydrochloric acid, and 32°4 c.c. of 5 per cent. chloride of platinum ; mixture evaporated to a magma, mixed with 2 to 3 c.c. of water, and evaporated again. After cooling, 3°25 c.c. of platinum solution added, and allowed to stand for an hour. So far, all the four analyses conducted in the same way. In (1) and (2).—Washing with chloroplatinic acid continued until the impurities by calculation were reduced to about 0:06 mg. and the SO, to 0:012 (but a direct test with BaCl, showed that there must have been more). The washing then completed with 95 per cent. alcohol. In (3).—After decanting off the mother-liquor, the precipitate was washed once, with 1 c.c. of platinum solution. The precipitate was then dissolved in hot water, and the solution, after addition of 2 c.c. of platinum solution, re-evaporated as far as possible on a water-bath. 2 c.c. of water were then added, and the whole allowed to stand for an hour, The precipitate was then filtered off, washed with two successive cubic centimetres of platinum solution (when as a matter of calculation, the impurities should have been reduced to 0°25 mg.; the SO, to 0:06 mg., yet a drop tested with BaCl, gave a precipitate), and lastly with 95 per cent. alcohol. In (4), TatiLocr’s directions were strictly obeyed. Chloroplatinates dried at 100° to 105°, weighed, and reduced with hydrogen, (wet way), to determine their platinum, The results are stated in the following table (in grammes) :— Experiment, 1, 2. 3. 4, I. Chloride of potassium taken, . 3 61075 61033 60992 60917 II. Chloroplatinate obtained, . + toned 19696 19755 1:9860 III, Platinum from “II.”, . ] : 7927 ‘7918 “7947 7964. IV. II. x 0:30435, ‘ ; 3 ‘ 60021 59945 60124 60444 Error = IV. minus 1.; mgs, . -—10°54 —10°88 — 868 —4:73 Ve. TT x O76R Neon o. : : é 60338 60269 ‘60490 60620 Error = V. minus 1.; mgs. . —737 —7'64 — 502 —2:97 Filtrates—from (1), (2), and (3),* freed from alcohol by distillation: their potassium recovered by FINKENER’s (sal-ammoniac) process, and determined * Those of (4) were lost by a disaster in the laboratory. 612 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON as PtCl,K,. The KCl recovered amounted to 22°55 mgs. or 7°52 mgs. per analysis. . Uniting analyses (1), (2), and (3) into one, using the same symbols as before, we have— . A, : C='30563; A): p="76011; (p: Ay) K,Cl,=196-26. For experiment (4) : by our analysis A : C = 0°30673. The Committee’s (TATLOcK’s) factor is 0°30560. Final Experiments. These were all carried out strictly according to TaT1Lock’s directions. I. Series. ‘In it, a 95 per cent. salt, virtually the same as that used for Set I. of the FINKENER analyses, was used. Chloride of potassium taken per analysis, *6104 to 6124 grm. The chloroplatinates dried at 100°, and their weights multiplied by the Committee’s factor, 0°30560, to find the chloride of potassium. The errors were, in Analysis, (1) (2) (8) (4) —1°60 — 1:60 +0:91 —1-06 mgs. The platinum of each precipitate was determined as usual; calculating from the weight of the platinum, by multiplying with 0°76117, the errors were— (1) (2) (3) (4) — 2°66 —3°43 — 0°30 —158 mgs, The volume of the platinic washings = 4°6 c.c. ; that of the alcoholic, about 19 c.c. per analysis. Total KCl recovered from united washings (by FINKENER’S sal-ammoniac process), determined as PtCl,K, = 11:11 mgs., or 2°78 mgs. per analysis. Uniting the four analyses into one, we have— A=2'44447 grms, — Ay =2'43336 C=7:9880 Hence KCl by Committee’s factor, = 2°44112, = “T.” T—A, = +7°76 mgs.; T—A = —3°35 mgs., showing that the smallness of the latter difference is owing to impurities in the chloroplatinate. Calculating constants from the slumped analysis, we have— A :C=030602 (Committee’s factor =0°30560). Ay : C=0°30463 Ay : p=0°76019 (p : Ay) K,Cl,=196'24 Let us note down before passing on, that the KCl found in the filtrates amounted to 11:11+18°6, = 0597 mg. per c.c. of aqueous platinic washings. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 613 Il. Series: with 82 per cent. Salt. Comprising again 4 analyses, each made with about 0:52 grm. of chloride of potassium. Taking “T” as symbol for the weight of chloride of potassium found in an analysis, from the chloroplatinate by the Committee’s factor, and A as designating the corresponding weight of chloride of potassium taken, we had :— ‘ Analysis, (1) (2) (3) (4) T—Ainmgs., . ; : —1-49 — 0:90 —1-21 —1:22 Volume of platinic filtrates, 48 48 4:8 5:0 c.c. These, united with the alcoholic washings, contained in all 9°71 mgs. of KCl, or 2°43 per analysis, or 0°500 mg. per c.c. of aqueous platinic filtrates. Uniting the four analyses into one, we had— A=2-10646; A,=209675; C=68771; T=2:10164; T-A= —482 mgs.; T-A,= +489 mgs. A : C=0°30630; A, : C=0°30489. From (1), (8) and (4). Total platinum=p=2'0689 ; (platinum from (2) lost). Aj=1°57274. A, : p=0°76018; (p : Ay) K.Cl,=196:24. III. Series : 33:3 per cent. Salt. Four analyses, each with about 0-215 grm. of chloride of potassium. Analysis, (1) (2) (8) (4) T—A (in mgs.), . ; : — 0°66 — 0-48 — 0°88 — 0-44 Volume of platinic filtrates, 5:8 59 53 53 CC. From ail the filtrates, including alcoholic, KCl recovered as usual, and found, = 9:10 mgs., or 2°28 mgs. per analysis, or 0°408 mg. per c.c. of aqueous platinic filtrates. Uniting the four analyses into one, we had— A=0°86132; A,=0°85222 ; C=2°8104; p=1:1259; T=0°85886 ; T—A= 9-46 mgs.; T— — +664 mgs. A 1g = 030648: A, : C=0:30324; Ay: =0°75692; (p : A) K, Cl,=197-09. The precipitate obviously was impure, and included foreign chloroplatinates. Before passing on, let. us summarise the principal result of our test analyses by the two methods. ; A. Finkener’s. Each set of analyses united into one. Column I. refers to the respective page of this memoir; Column II. specifies the set of analyses referred to ; Column III. the percentage of KCl in the salt analysed. 614 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Page| Re De neat Arp. (p : A) K,Cly. (p : A,)K,Cl,=195°78 (Set L.), | 604 if 95 (‘76270) 195-60 Mean (p: A,) K,Cl, may 605 TE 95 (‘76087) 196-06 perhaps be put down at 606 III. 95 ‘76116 195-99 195:98 +0°13=196:11. 606 IV. 82 “76157 195°89 607 vi 333 75980 19634 Means, . . 76084 195-98 Calculated from III., IV., & V. All. Set I. Salts made into normal sulphates; platinum used = 1°075 times the calculated minimum. Set II. A combination of TATLock’s and FINKENER’S methods. Set III. Only sulphuric acid added ; no evaporation: platinum used, = 1:25 times the calculated minimum. Only sets III, [V., and V. correspond to owr present method: hence the mean factor A : p was calculated from only these 3 sets. We have recalculated the 12 analyses of sets III., [V., and V., with the new factor 0°76084, and found the “errors ” of the recalculated numbers. Set, III. IV. v. A, about 0°75 0°65 0:266 gm. Analysis, (1) (2) (8) 4) yn) (2) (3) @> |) @)" 2) x6) (4) Error, -"07 -°82 +411 -'50/+°20 -219 --33 -'17 14°34 4:23 4:18 +°'70mg. Mean error = + 0°49 mg,, or, excluding No. 2 in IV., it is + 0°33. B. Taitlock’s. In the following table, the first column refers to the respective page of this memoir ; the third gives the percentage of KCl in the set of salts analysed ; the figures in the second are reference marks. Me, cao fac A,:0. (C : A,)K,Cl,. Aoi p. (p : A,)K,Cly. 612 (1) 95 30675 * 30563 48811 ‘76011 196:26 612 (2) 95 30602 30463 489°71 ‘76019 196°24 613 (3) 82 30630 30489 489°29 ‘76018 196°24 613 (4) 333 -30648 (30324) 491-96 (75692) (19709) Means, . 30627 30505 489:04 ‘76016 196:25 Calculated from (2), (3), &(4) (1), (2), & (3) (1), (2), & (3) (1), (2), & (3) (1), (2), & (8) Recalculating the 12 analyses we made (by Tattock’s exact method) with the factor 0°30627, we arrive at the following errors for the individual results :— * From the one analysis of this set in which Tatlock’s directions were strictly obeyed. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 615 (2) (3) (4) Salt of, 95 per cent. 82 per cent. 33°3 per cent. A about, 610 520 ‘215 grm. Analysis, (1) (2) (3) (4) (1) (2) (3) (4) @, @) @ . Error, —°26 -—°26 42°25 +:28 |--34 +°25 -06 --07 |--19--01 —-41 +-03 mg. Mean error, = + 0°37, or excluding No (3) under (2),=+ 0:20 mg. We see that the degree of precision afforded by the two methods is about the same, and is amply sufficient for all practical purposes. But it still remains to be seen how far the TatLocx method is available for the analysis of salts which are relatively poor in potash. Analyses made with the view of deciding this question will be submitted presently ; but we prefer to interpolate a duplicate analysis, by both methods, of an imitation of the double salt MgK,S,O, + 6H,O. 0°5370 grm. of chloride of potassium was converted into neutral sulphate (obtained 0°6273 grm. equivalent, by calculation, to 0°5368 of KCl); this was dissolved in water, and diluted to a known weight. Two portions of the solution were weighed out, each mixed with the calculated weights of a standard solution of MgSO,, prepared from pure oxide by solution in dilute sulphuric acid, and analysed, one by the TatTLock method, the other by our form of the FINKENER process. Found by the method of Finkener. Tatlock. I. Chloride of potassium used, . wth Ju! ‘ pee Tel ‘23929 prm. II. Chloroplatinate obtained, . : : : ' 2 FIGD Ne es IIL. II. x 30627 (see page 614) = : ‘ ae 23834 __,, IV. Platinum from IL., : : : : 1 “OOLS ca Ot ihe ae V. IV. x 0°76084 (see page 614)= _ . : : ay pubs Q3ial« 3 “Excess” of KCl found by III, . j ; aa — 0°95 mg. fs, rs Wists Na? ae . +0°01 =1'°98 ,, The TatLock method, as we see, gave a deficit of about 1 mg. of KCI; but this, after all, is only =4,th of the quantity to be determined, which suffices for all purposes ; or, in other words, the TatLock method works well enough even with unmixed sulphates. The substance analysed contained the equivalent of about 50°6 per cent. of KCl. We now pass to A Set of Three Analyses of a “10 per cent.” Salt. A standard solution was prepared, which represented a mixture containing (about) 79 of NaCl, 6 of Na,SO,, and 5 of MgSO, in 90 parts. The 10 per cent. of KCl were weighed out specially for each analysis, as a standard solution. Analysis (1)—Solution Finkenerised with twice the calculated weight of 616 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON platinum, and enough (by calculation) of standard sulphuric acid to displace the chlorine of the chlorides. Mixture (#R,SO,+PtCl,K,) “ recrystallised,” | platinum reduced out in the wet-way, and weighed. Analysis (2)—Solution Finkenerised exactly as in (1); but recrystallisation omitted. From mixture #R,SO,+ PtCl,K,, the sulphates extracted by sal- ammoniac ; the residual chloroplatinate dissolved, the platinum reduced out, and weighed. The KCl contained in the filtrate beside NH,Cl recovered by evaporation and ignition, and weighed; then dissolved in water, wrought with PtCl,H, (in FREsENIvus’ way), and the chloroplatinate weighed. In Analysis (3), Tattock’s method was applied in all strictness, except that the chloroplatinate received an extra washing with 6 drops of platinum solution. Analysis (1)\—Finkener’s Method; Recrystallisation. A = 77-22 mgs.; p = 1022; px 076084 = 77-76; Error = + 0°54 mg., or 0°7 per cent of A. Analysis (2)—Finkener’s Method ; Sal-Ammoniac Form. A = 162°44 mgs.; p = 217°4; px 0°76084 =165-41 ; Excess over A = 2°97 mgs.‘ Crude KCl from filtrate (from p) = 166°9; Excess over’ A = 45 mgs. Chloroplatinate from the crude chloride of potassium = 527°6 mgs.; whence by ‘imultiplication with 0°30627 = 161°59 mgs. of chloride of potassium; deficit against A = 0°85 mg., or 0°52 per cent. Platinum out of the last chloroplatinate = p’ = 212°0; p’ x 0°7608 = 161°30, which is less than A by 1:14 mgs., or 0°70 per cent. Analysis (3)—Tatlock’s Method. A = 65-44 mgs.; C = 2092; Cx 0:30627 = 64:07 T; T—A = —1°37, or 21 per cent of A. Platinum from C = 840=p. px0°7608 = 63:91 =T” T’—A = — 1°53 mgs., or 2°3 per cent. of A. Analysis (2) did not do justice to its method, through unobserved causes, it is true. Yet the error in (1) or (2) did not rise beyond 0°7 per cent. of the | small quantity to be determined. The Tattock method loses 2 per cent. of — the chloride of potassium to be determined ; 7.¢., it would report 9°8 instead of 10 per cent. We believe the line of the applicability of TarLocx’s process must be drawn at about the “10 per cent.” salt. Three Analyses of synthetically prepared Sea- Water Salts. (Average ocean-water salts contain 2°11 per cent. of potassium calculated as KCI.) | CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 617 A kind of potash-free sea-water was made from pure materials : chloride of sodium, magnesia, and standard solutions of sulphuric and hydrochloric acids. For each analysis a volume corresponding to so and so much average ocean- water was measured out, and the exact weight of a standard solution of chloride of potassium added. The methods were exactly the same as those used in the preceding set. Analysis (1)—Finkener’s Method ; “ Recrystallisation.” Chloride of potassium used as such, . , ‘ : . . =148-69 mgs. a es present in the chloride of sodium used, . = 0°19 _,, Total chloride of potassium, . : : é : : 5 BARS be aA, p = 198°3; px 0°76084, = 150°87; excess over A = 1:99 mgs., or 1°3 per cent. of A :—partly through compensation of errors. The ether-alcohol washings, when worked up for potassium as usual, gave, “ctrl ieee ETE This analysis was made by means of our present form of the FINKENER process ; the foreign bases were converted into sulphates only virtually, z.e¢., by adding the calculated volume of standard sulphuric acid, &c. In the “ Challenger” analyses, the first step always was to actually convert all the bases into neutral sulphates, which probably ensures greater constancy in the results. Analysis (2)—Finkener’s Method ; Sal-Ammoniac Form. Total chloride of potassium operated upon (including that of the NaCl), = A = 148°59 mgs. Platinum from the chloroplatinate (+2NH,Cl), = 2021=p. px 0°76084, = 153°77; excess over A = 5°18 mgs., or 3°3 per cent of A. Crude KCl (from filtrate from platinum), = 1460. Chloroplatinate from the same = 0°4642 erm. = 142:17 =“ a’ mgs. of KCl; this is less than A by 6°42 mgs., or 4:3 per cent. of A. KCl recovered (as ultimately PtCl,K,) from sal-ammoniac liquors, =6'54 mgs. = A; from ether-alcohol washings=0'46 mg.=6. By addition, a+4=14871 mgs. (a+ A)—A = +012 mg,, or 0-08 per cent. of A. Here again the sal-ammoniac process failed to do justice to itself; in many similar cases we obtained better results, in the sense that far less potassium passed into the sal-ammoniac. The method, unfortunately, is somewhat capricious ; the chloroplatinate does not always stand the sal-ammoniac treatment equally well. To determine small quantities of potassium correctly, the sal-ammoniac liquors must be worked up; and in no case dares the crude chloride of VOL. XXXIII. PART II. 4x 618 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON potassium be accepted as pure KCl. Yet the sal-ammoniac form of the FINKENER method is invaluable, being the only method for extracting a small proportion of potassium from a mass of, eg., soda-salts; only it must be wrought with great circumspection, and in its original form be only used as a mode of extracting the potassium, not as a method for its determination. Analysis (3)—Tatlock’s Method. Chloride of potassium used = A = 14:22 mgs. (We could not have used as much as in (1) or (2) without wasting an unreasonable amount of platinum solution.) C =41:0; C x 030627 = 12°56 =T. T—A = —1°66 mgs., or 11°7 per cent. of A. Platinum from C =16'‘9mgs. = 12°86 mgs. of KCl. Chloride of potassium recovered from the washings, and weighed, ultimately as PtC]l,K, = 2°92 mgs. ; hence A, = 11°30. T—A, =+1°26 mgs., or 11°1 per cent. of Ay. With salt-mixtures like sea-water salts the Tartock method obviously loses its applicability. Nor was it ever intended for such mixtures. IV. EXPERIMENTS ON CHLOROPLATINATE OF RUBIDIUM. These experiments were planned at a very early stage of our investigation. They were suggested by the obvious consideration that for the synthetical determination ‘of the weight-ratio Pt:2Cl between the platinum and fixed chlorine in chloroplatinates, chloride of rubidium should be better adapted than the potassium-salt, because, while itself soluble in alcohol, its chloroplat- inate is less soluble in water than chloroplatinate of potassium. For a similar reason, chloride of caesium should be preferable to chloride of rubidium ; but we shrank from the great expense which would have been involved in procuring the necessary supply of the rarer of the two rare alkalies. Our raw material for the preparation of chloride of rubidum was a supply of “pubidium alum” from Trommsdorff in Erfurt. The alum was dissolved in hot water, and its rubidium precipitated by addition of chloroplatinic acid, the precipitate washed, reduced in the dry way with hydrogen, and the chloride of rubidium extracted with water. As it turned out to contain a very appreciable quantity of sulphate, it was redissolved, reconverted into chloroplatinate, and recovered from the latter by means of hydrogen. The salt thus obtained was contaminated with sulphuric acid, and two or three repetitions of the cycle of operations failed to eliminate this impurity quite completely. Going by the aspect of the chloride of barium precipitate, the last precipitation, indeed, seemed to have effected no improvement ; we therefore evaporated the whole of our (last) chloride of rubidium to dryness, and thus obtained about 32 grms. CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 619 of a salt, which, apart from that trace of sulphuric acid, seemed to be very pure. In its aqueous solution, sulphuretted hydrogen produced no change; sul- phide of ammonium had no immediate effect, and, on long standing, only an almost invisible precipitate settled out. 05 germ. of the salt, when dissolved and mixed with iodide of potassium, gave no precipitate (absence of thallium). The spectrum-apparatus revealed no trace of potassium. In order to, at the same time, determine the sulphuric acid, and rehearse a method for its removal, 5 grms. of the salt were dissolved, chloride of barium added, the precipitate allowed to settle, and weighed. It amounted to 65 mgs., indicating 0'045 per cent. of SO, in the preparation. From the (concentrated) filtrate, the rubidium was precipitated by addition of a slight excess of chloroplatinic acid, the chloroplatinate allowed to settle, and washed by decanting filtration, first with water, then with 50 per cent., and lastly with absolute, alcohol. The weight of the chloroplatinate, after a preliminary drying at 120°, amounted to 11°187 grms. 1:0094 germs. of this chloroplatinate were dissolved in hot water, in an Erlenmeyer flask, the platinum was reduced out by hydrogen, and filtered off, and the filtrate evaporated to dryness over a water-bath. The residue was dissolved in 50 c.c. of water, and separate portions examined. 10 c.c. when mixed with sulphide of ammonium, gave a mere trace of (FeS and A1,O,%) ; 10 c.c. when Nesslerised, gave 0°08 mg. of NH,, corresponding to 0-4 mg. per 1 orm. of salt; 10 c.c. when mixed with chloride of barium, and other 10 cc. when mixed with sulphuric acid, gave both slight clouds of sulphate of baryta. From the last two tests, it was clear that the application of chloride of barium to our stock of salt would have done little good, and we accordingly decided upon using the preparation as it was. A portion of it was dehydrated in a platinum crucible without fusion, then fused very cautiously, and poured out into a platinum basin. 18°1321 grms. of such salt were dissolved in water to 300 c.c., and thus converted into 313°522 germs. of a standard solution, of which every 1 grm. contained 57°834 mgs. of salt. Three titrimetric determinations of the chlorine gave the follow- ing results :— Analysis, i 2. 3. Approximate weight of solution taken, . 104 20°8 20°8 grms. Chlorine per gramme of solution, . : . 16°8253 16°8200 16°8226 mgs. Mean = 16°8226, corresponding to 29:087 per cent. of chlorine in the original salt. Taking Rb = 85-4, and Cs = 133°0, the precentages of chlorine in the chlorides of the two metals are— In RbCl. CsCl. Equal to, : 4 : : . 29°3387 21-047 620 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON Hence our “ chloride of rubidium ” contained Real chloride of rubidium, : : ; 5 . 96°985 Chloride of czesium, ; ; : ’ f L MUS OLS 100-000 These 3 per cent. of czsium chloride, although inconvenient, did not unfit the salt for our purpose; all we had to do was to base our calculations of “ Pt” not upon the weight of alkyl-chloride present in the respective chloro- platinate, but upon the weight of the chlorine in that chloride. The reason why we took such pains in standardising our rubidium solution, of course was, that we intended to rely chiefly on the synthetical data of our experiments ; but unfortunately, these became almost valueless through a change in the strength of the rubidium solution, which was observed only after the greater part of the work had been completed; yet they were, and still are, of great use to ourselves as affording checks for the respective analyses. Only the latter are reported on in the following paragraphs :— To begin with a case where no quantitative synthesis was attempted, let us give the results of an analysis of the chloroplatinate of rubidium referred to above, as having been obtained incidentally from about 5 grms. of salt. 3°2495 grms. of this preparation (weighed after a preliminary drying at 120° C.) were placed in a “Geissler tube,” and dried systematically, first at 120°, then at 130°, and lastly at 150° C. The weights recorded were as follows :— After 2 hours at 120°, . ; . : : : . 382457 ? a 1307" =, : , 5 ‘ : » 32432 gh © 150%" : : : 4 H . 32342 Even then the weight was not constant ; yet the drying process was stopped, and 3°1970 grms. of the salt transferred to an Erlenmeyer flask, to be reduced with hydrogen in the wet way, and analysed in the way we had before applied to many specimens of the potassium salt. The reduction set in very readily, but was very slow in coming to an end, It took in all about six days for its completion, although every morning the precipitate was broken up with a glass rod, to bring the hidden chloroplatinate to the surface. When at last the reduction seemed to be completed, the platinum was filtered off, and the filtrate divided gravimetrically for the deter- mination of the fixed and of the total chlorine. Suspecting that the platinum might include some undecomposed chloroplatinate, the greater part was removed from the filter, and next dehydrated at a dull-red heat. It weighed 1:0454 grms.; after subsequent strong ignition this weight was 1:0452 grms. and after a succeeding strong ignition in hydrogen it was 1:0449 grms. Hence it appears that the proportion of undecomposed chloroplatinate CRITICAL EXPERIMENTS FOR DETERMINATION OF POTASSIUM, ETC. 621 in the “platinum” was at the worst extremely small. The total weight of platinum obtained amounted to 1:0923 grms. For the isolation of the fixed chlorine, the respective portion of the solu- tion was evaporated to dryness, the residue dried at 130° for 2 hours, then dis- solved in water, the solution re-evaporated, and the residue again dried at 130° for15hours. The solution of the thus dried salt was absolutely neutral to litmus. The determinations of the chlorine were effected by the “Challenger ” method as usual. Summary of Results. Found per 2C1=70°91 parts of fixed chlorine. RCL peadines i Platinum. Loose Chlorine. Substance. weighing, 24433 203°39 144-71 595-29 =4:0816 x Cl. These results at the time surprised us very much ; but we have no difficulty now (after our later experience with the potassium salt) in explaining them. Part of the 203°39 parts of platinum must be assumed to be present as PtX,H,; the X, including the 00816 x Cl of chlorine, besides the necessary amount of oxygen or hydroxyl. Taking Pt as 195°5 as it follows from our potassium experiments, we have for the composition of 595:29 of the chloro- platinate :— 2RCL-~ . ; ; ‘ : ‘ . 24433 1:04036 x Pt, . ‘ 5 ; : . 203°39 Loose chlorine, F ; ; : . 144°71 Hydroxyl, . : : : : . 273 = 01606 x OH. Hydrogen, . : : : " . 0:08 595°24 which agrees very well with the above 595:29 of substance analysed ; only the closeness of the agreement is probably accidental, as the direct determination of the “ RCl” was made only on a very small scale.* We now pass to those synthetical experiments with standardised solutions, of which, unfortunately, only the analytical parts are worth publishing. Experiment I. 40 ¢.e. of the rubidium solution were evaporated to 15 c.c., poured into 40-4 c.c. of standard platinum solution (1 c.c. = 49°5 mgs. of metal), and the * A weighed portion of the filtrate from the platinum evaporated to dryness, and made neutral, as above explained, and weighed. Actual weight=-2213 grm. for the fraction analysed. The salt was analysed, and found to contain 29:09 per cent. of chlorine, z.e., almost exactly as much as the original chloride of rubidium. 622 PROFESSOR DITTMAR AND MR JOHN M‘ARTHUR ON mixture was allowed to stand overnight. Next morning the liquor was decanted through a small filter, the precipitate washed four times with water (10 c.c. — each time), and then twice with absolute alcohol. Absolute weight of platinum used = 1:9976 grms.; platinum per Cl, = 70°91 parts of fixed chlorine = “P” = 201°8 (nearly, the exact data need not be reproduced here). Total chloro- platinate produced = 5°78 grms. It was dried at 130° C. for 5 hours, and then divided into two parts, A and B. & i au | P}) J Vol XXXII, Plate XXXVIII. M‘Farlame & Erskine, Lith®® Bain is. Hoy. Soc. Edin™ Vol. XXXII, Platte XXXIX, tr bl ' bl ér ; ty PROLLY ven? ei feet ae eee Bie eoeetes rao a=). 's AZ este ae es. ES McFarlane & Erskine, Lith®? Edin® Roy. Soc. Edin? Vol. XXXII, Plate XL. e eet. i) che MSParlane & Erskine, Lith®* Edin™ Vol, XXXII, Plate XLI. ins. Roy. Soc. Edin® M‘Farlane & Erskine, Lith? Edin™ Vol. XXXII, Plate XLII. 's. Roy. Soc. Edin? MFarlane & Erskine, Lith? Edin® SROs £c RRRRES WN SOAs ai ; NOISES SINUSES S/n ¥ ees rad ns. Roy. Soc. Edin™ ‘Vol. XXXIII, Plate XLIII. M‘Farlane & Erskine, Lith'? Edin® & Erskine, Lith? Edin™ M‘Parlane Vol. XXXII, Plate XLIV. ogy oy oe fg ae a ete rE Naame eet sae ii i: Hees PEP YD) al.C SS IT is. Roy. Soc. Edin? Vol. XXXII, Plate XLV. ims. Roy. Soc. Edin? SCs ee Edin® M‘Ferlans & Erskine, Lath! by . ye of kk ny . < SS =p ame x, SE j CAN J p Sar ae S94) yy ® 1s. Roy. Soc. Edin? Vol. XXXIII, Plate XLVI. M°Farlane & Erskine, Lith? Edin™ . Roy. Soc. Edin? Vol. XXXITI, Plate XLVII. - eM NOU UU M'Farlane & Erskine, Lith™® Edin® XP PEN De xX: TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. VOL. XXXiI, PART 11, 5H CONTENTS. THE COUNCIL OF THE SOCIETY, ALPHABETICAL LIST OF THE ORDINARY FELLOWS, LIST OF HONORARY FELLOWS, LIST OF ORDINARY FELLOWS ELECTED DURING SESSIONS 1886-87, LAWS OF THE SOCIETY, THE KEITH, BRISBANE, NEILL, AND VICTORIA JUBILEE PRIZES, AWARDS OF THE KEITH, MAKDOUGALL-BRISBANE, AND NEILL PRIZES, FROM 1827 TO 1886, AND OF THE VICTORIA JUBILEE PRIZE, IN 1887, PROCEEDINGS OF THE STATUTORY GENERAL MEETING, LIST OF PUBLIC INSTITUTIONS AND IN DIVIDUALS ENTITLED TO RECEIVE COPIES OF THE TRANSACTIONS AND PROCEEDINGS OF THE ROYAL SOCIETY, : : INDEX, PAGE 688, 689 702 704 707 714 rf kes 721 i 725 731 LIST OF MEMBERS. COUNCIL, ALPHABETICAL LIST OF ORDINARY FELLOWS, AND LIST OF HONORARY FELLOWS. At January 1888. THE COUN. a. OF THE ROYAL SOCIETY OF EDINBURGH, NOVEMBER 1887. PRESIDENT. Str WILLIAM THOMSON, LL.D., D.C.L., F.R.S., Foreign Associate of the Institute of France, Regius Professor of Natural Philosophy in the University of Glasgow. HONORARY VICE-PRESIDENTS, HAVING FILLED THE OFFICE OF PRESIDENT. His Grace THE DUKE or ARGYLI, K.G., K.T., D.C.L. Oxon., F.R.S., F.G.S. Tae Ricut Hon. Lorp MONCREIFF, LL.D., Lorp Justics-Cierx. VICE-P RESIDENTS. JOHN MURRAY, Ph.D., Director of the Challenger Expedition Commission. D. MILNE HOME of Milne-Graden, LL.D. Str DOUGLAS MACLAGAN, M.D., President of the Royal College of Physicians, Edin., F.R.C.S.E., and Professor of Medical Jurisprudence in the University of Edinburgh. Tue Hon. Lorp MACLAREN, LL.D. Edin. and Glas., F.R.A.S., one of the Senators of the College of Justice. Tue Rev. Proressor FLINT, D.D., Corresponding Member of the Institute of France. GEORGE CHRYSTAL, M.A., LL.D., Professor of Mathematics in the University of Edinburgh, GENERAL SECRETARY. P. GUTHRIE TAIT, M.A., Professor of Natural Philosophy in the University of Edinburgh. SECRETARIES TO ORDINARY MEETINGS. Sr WILLIAM TURNER, M.B., F.R.C.S.E., F.R.S., Professor of Anatomy in the University of Edinburgh. ALEXANDER CRUM BROWN, ML.D., D.Sc, F.R.C.P.E., F.R.S., Professor of Chemistry in the University of Edinburgh. TREASURER. ADAM GILLIES SMITH, Esq., C.A. CURATOR OF LIBRARY AND MUSEUM, ALEXANDER BUCHAN, Esq., M.A., LL.D., Secretary to the Scottish Meteorological Society. COUNCILLORS. S. H. BUTCHER, M.A., LL.D., Professor of Greek in the University of Edinburgh. JOHN G. M‘KENDRICK, M.D., F.R.C.P.E., F.R.S., Professor of the Institutes of Medicine in the University of Glasgow. THOMAS MUIR, M.A., LL.D., Mathematical Master in the High School of Glasgow. WILLIAM CARMICHAEL M‘INTOSH, M.D., LL.D., F.R.S., F.L.8., Professor of Natural History in the University of St Andrews. Sirk ARTHUR MITCHELL, K.C.B., M.A., M.D., LL.D., Commissioner in Lunacy. STAIR A. AGNEW, Esgq., C.B., M.A., Advocate, Registrar-General. A. FORBES IRVINE, Esq. of Drum, LL.D. ROBERT M. FERGUSON, Esq., Ph.D. J. BATTY TUKE, M.D., F.R.C.P.E. FREDERICK 0. BOWER, M.A., F.L.S., Regius Professor of Botany in the University of Glasgow. GERMAN SIMS WOODHEAD, M.D., F.R.C.P.E. ROBERT COX, Esq. of Gorgie, M.A. Date of Election. 1879 1871 1881 1878 1875 1878 1856 1886 1874 1883 1883 1881 1867 1883 1886 1849 1887 1885 1879 1875 1843 1879 1877 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY, B. K. N. Weds 1k KP. CORRECTED TO JANUARY 1888. N.B.—Those murked * ave Annual Contributors. prefixed to a name indicates that the Fellow has received a Makdougall-Brisbane Medal. » 3 16 Keith Medal, Neill Medal. the Victoria Jubilee Prize. 3 ” », contributed one or more Papers to the TRANSACTIONS. Abernethy, Jas., Memb, Inst. C.E., Prince of Wales Terrace, Kensington * Aonew, Stair A., C.B., M.A., Advocate, Registrar-General, 22 Buckingham Terrace Aitchison, James Edward Tierney, C.I.E., M.D., F.R.S., F.LS., Brigade-Surgeon, Secretary to the Surgeon-General, H.M.F. Bengal, and Naturalist with the Afghan Delimitation Commission, H.M. Bengal Army, North Bank, Simla, Punjab, India, 55 Parliament Street, London, S.W. * Aitken, Andrew Peebles, M.A., Sc.D., F.I.C., 18 Dublin Street * Aitken, John, Darroch, Falkirk 5 Allchin, W. H., M.B. (Lond.), F.R.C.P., Physician to the Westminster Hospital, 5 Chandos Street, Cavendish Square, London Allman, George J., M.D., F.R.S., M.R.LA., F.L.S., Emeritus Professor of Natural History, University of Edinburgh, Ardmore, Parkstone, Dorset * Anderson, Arthur, M.D., C.B., Ex-Inspector-General of Hospitals, Pitlochry Anderson, John, M.D., LL.D., F.R.S., Superintendent of the Indian Museum, and Professor of Comparative Anatomy in the Medical College, Calcutta, 71 Harrington Gardens, Lond. * Anderson, Robert Rowand, LL.D., 19 St Andrew Square 10 Andrews, Thomas, F.R.S., F.C.S., Memb. Inst. C.E., Ravencrag, Wortley, near Sheffield Anglin, A. Hallam, M.A., LL.D., M.R.1.A., Professor of Mathematics, Queen’s College, Cork, Brighton Villas, Western Road, Cork * Annandale, Thomas, M.D., F.R.C.S.E., Professor of Clinical Surgery in the University of Edinburgh, 34 Charlotte Square Archibald, John, M.B., C.M., Lynton House, Brixton Rise, London * Armstrong, George Frederick, Professor of Engineering in the University of Edinburgh 15 Argyll, His Grace the Duke of, K.T., D.C.L., F.R.S. (Hon. Vicz-Pres.), Inveraray Castle * Ashdown, Herbert H., M.B., 49 Upper Bedford Place, Russell Square, London * Baildon, H. Bellyse, B.A., Duncliffe, Murrayfield, Edinburgh * Bailey, James Lambert, Royal Bank of Scotland, Ardrossan * Bain, Sir James, 3 Park Terrace, Glasgow 20 Balfour, Colonel David, of Balfour and Trenabie, Balfour Castle, Kirkwall * Balfour, George .W., M.D., LL.D., F.R.C.P.E., 7 Walker Street - * Balfour, I. Bayley, Sc.D., M.D., C.M., F.R.S., Sherardian Prof. of Botany in the Univ. of Oxford 690 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Date of | Election 1870 1886 1872 1883 1887 1882 1874 1887 1878 1857 1880 1882 1887 1886 1874 1876 1887 1875 1881 1880 1884 1850 1863 1862 1878 1884 1872 1869 1886 1884 1871 1873 1886 1886 1877 1887 1864 1881 1883 | _* Balfour, Thomas A. G., M.D., F.R.C.P.E., 51 George Square * Barclay, A. J. G., M.A., 5 Ethel Terrace 25 * Barclay, George, M.A., 17 Coates Crescent * Barclay, G. W. W., M.A., 40 Princes Street Barlow, W. H., Memb. Inst. C.E., High Combe, Old Charlton, Kent Barnes, Henry, M.D., 6 Portland Square, Carlisle Barrett, William F., M.R.I.A., Professor of Physics, Royal College of Science, Dublin 30 * Bartholomew, J. G., 32 Royal Terrace Bateman, John Frederic La Trobe, Memb. Inst. C.E., F.R.S., F.G.S., F.R.G.S., 18 Abing- don Street, Westminster Batten, Edmund Chisholm, of Aigas, M.A., 16 Pelham Crescent, South Kensington, London * Bayly, General John, C.B., R.E., 58 Palmerston Place Beddard, Frank E., M.A. Oxon., Prosector to the Zoological Society of London, Zoological Society’s Gardens, Regent’s Park, London 35 * Begg, Ferdinand Faithful, 6 Draper’s Gardens, London * Bell, A. Beatson, Chairman of Prison Commissioners, 130 George Street * Bell, Joseph, M.D., F.R.C.S.E., 2 Melville Crescent * Belcombe, Rev. F. E., 14 Merchiston Avenue * Bernard, J. Mackay, 25 Chester Street 40 Bernstein, Ludwik, M.D., Lismore, New South Wales * Berry, Walter, Danish Consul-General, 11 Atholl Crescent * Birch, De Burgh, M.D., Professor of Physiology, University College, Leeds, 16 De Grey Terrace, Leeds * Black, Rev. John S., 6 Oxford Terrace Blackburn, Hugh, M.A., LL.D., Emeritus Professor of Mathematics in the University of Glasgow, Roshven, Ardgour 45 Blackie, John S., Emeritus Prof. of Greek in the University of Edin., 9 Douglas Crescent Blaikie, The Rev. W. Garden, M.A., D.D., LL.D., Protessor of Apologetics and Pastoral Theology, New College, Edinburgh, 9 Palmerston Road * Blyth, James, M.A., Professor of Natural Philosophy in Anderson’s College, Glasgow Bond, Francis T., M.D., B.A., M.R.C.8., 1 Beaufort Buildings, Spa, Gloucester * Bottomley, J. Thomson, M.A., Lecturer on Nat. Philosophy in the Univ. of Glasgow 50 * Bow, Robert Henry, C.E., 7 South Gray Street * Bower, Frederick O., M. 4 F.L.S., Regius Professor of Botany in the University of Glas- gow, 45 Kerrsland anaes Hillhead, Glasgow Bowman, Frederick Hungerford, D.Sc., F.R.A.S., F.C.S., F.L.S., F.G.S., West Mount, Halifax, Yorkshire * Boyd, Sir Thomas J., Chairman of the Scottish Fishery Board, 41 Moray Place * Boyd, William, M.A., Peterhead 55 * Bramwell, Byrom, M.D., F.R.C.P.E., 23 Drumsheugh Gardens Brittle, John Richard, Memb. Inst. C.E., Vanbrugh Hill, Blackheath, Kent Broadrick, George, Memb. Inst. C.E., The Hall Cross, Doncaster * Brown, A. B., C.E., 19 Douglas Crescent .|* Brown, Alex. Crum, M.D., D.Sc., F.R.C.P.E., F.R.S. (Secretary), Professor of Chemistry in the University of Edinburgh, 8 Belgrave Crescent 60 * Brown, J. A. Harvie, of Quarter, Dunipace House, Larbert, Stirlingshire * Brown, J. Graham, M.D., C.M., F.R.C.P.E., 16 Ainslie Place ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 691 Date of Election. 1885 * Brown, J. Macdonald, M.B., F.R.C.S.E., 6 Atholl Place 1861 | P. Brown, Rey. Thomas, 16 Carlton Street 1870 Browne, Sir Jas. Crichton, M.D., LL.D., 7 Cumberland Ter., Regent’s Park, Lond. 65 1883 * Bruce, Alexander, M.A., M.B., M.R.C.P.E., 16 Alva Street 1878 Brunlees, Sir James, Memb. Inst. C.E., 5 Victoria Street, Westminster 1867 | * Bryce, A. H., D.C.L., LL.D., 42 Moray Place 1869 |B. P.|* Buchan, Alexander, M.A., LL.D., Secretary to the Scottish Meteorological Society (Curator oF Lisrary), 72 Northumberland Street 1870 | P. |* Buchanan, John Young, M.A., F.R.S., 10 Moray Place 70 1882 * Buchanan, T. Ryburn, M.A., M.P. for the City of Edinburgh, 10 Moray Place 1887 * Buist, J.B., M.D., F.R.C.P.E., 1 Clifton Terrace 1887 * Burnet, John James, Architect, 1 Granby Place, Hillhead, Glasgow 1887 * Burton, Cosmo Innes, B.Sc., F.C.S., 6 Montpellier, Viewforth, Edinburgh 1883 * Butcher, S. H., M.A., LL.D., Professor of Greek in the University of Edinburgh, 27 Palmerston Place 75 1887 * Cadell, H. M., B.Sc., H.M. Geological Survey, 13 Douglas Crescent 1869 * Calderwood, Rev. H., LL.D., Professor of Moral Philosophy in the University of Edin. burgh, Napier Road, Merchiston 1879 * Calderwood, John, F.I.C., Belmont Works, Battersea, London 1878 Campbell, John Archibald, M.D., Garland’s Asylum, Carlisle 1887 * Capstick, J. W., Lecturer in Mathematics and Physics, University College, Dundee 80 1874 Carrington, Benjamin, M.D., Eccles, Lancashire 1882 * Cay, W. Dyce, Memb. Inst. C.E., 1074 Princes Street 1876 * Cazenove, The Rev. John Gibson, M.A., D.D., 22 Alva Street, Chancellor of St Mary’s Cathedral 1885 * Chambers, Robert, 10 Claremont Crescent 1866 * Chalmers, David, Redhall, Slateford 85 1874 * Chiene, John, M.D., F.R.C.S.E., Professor of Surgery in the University of Edinburgh, 26 Charlotte Square 1875 * Christie, John, 19 Buckingham Terrace 1872 Christie, Thomas B., M.D., F.R.C.P.E., Royal India Asylum, Ealing, London 1880 | K. P.|* Chrystal, George, M.A., LL.D., Prof. of Mathematics in the Univ. of Edin., 5 Belgrave Crescent 1875 * Clark, Robert, 7 Learmonth Terrace 90 1886 | P. | * Clark, The Right Hon. Sir Thomas, Bart., Lord Provost of Edinburgh, 11 Melville Crescent 1863 Cleghorn, Hugh F. C., of Stravithie, M.D., LL.D., F.L.S., St Andrews, United Service Club, 14 Queen Street 1875 * Clouston, T. 8., M.D., F.R.C.P.E., Tipperlin House, Morningside 1882 * Coats, Sir Peter, of Auchendrane, President of the Glasgow and West of Scotland Horti cultural Society, Auchendrane, Ayr 1887 * Cockburn, John, 6 Atholl Crescent 95 1887 * Coleman, Joseph James, Ardarroch, Bearsden, Glasgow 1886 Connan, Daniel M., M.A., Education Department, Cape of Good Hope 1872 * Constable, Archibald, 11 Thistle Street 1863 Cowan, Charles, of Westerlea, Murrayfield 1879 * Cox, Robert, of Gorgie, M.A., 34 Drumsheugh Gardens 100 692 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Election. 1875 |* Craig, William, M.D., F.R.C.S.E., 7 Bruntsfield Place 1886 * Croom, John Halliday, M.D., 25 Charlotte Square 1887 * Crawford, William Caldwell, Lockharton Gardens, Slateford, Edinburgh 1887 * Cumming, A. 8., M.D., 18 Ainslie Place 1878 * Cunningham, Daniel John, M.D., Professor of Anatomy in Trinity College, 69 Har- court Street, Dublin 105 1886 * Cunningham, David, Memb. Inst. C.E., Harbour Chambers, Dock Street, Dundee 1877 * Cunningham, George Miller, 2 Ainslie Place 1884 | P. |* Cunningham, J. T., B.A., Marine Biological Laboratory, Plymouth 1871 * Cunynghame, R. J. Blair, M.D., 6 Walker Street 1841 | Pp. Dalmahoy, James, 9 Forres Street 110 1878 * Dalziel, John Grahame, 2 Melville Terrace, Pollokshields, Glasgow 1885 * Daniell, Alfred, M.A., LL.B., D.Sc., Advocate, 3 Great King Street 1867 * Davidson, David, Somerset Lodge, Wimbledon Common, Wimbledon 1848 Davidson, Henry, Muirhouse, Davidson’s Mains 1884 Davy, Richard, M.B., F.R.C.S., Surgeon to the Westminster Hospital, 33 Welbeck Street, Cavendish Square, London 115 1870 * Day, St John Vincent, C.E., 115 St Vincent Street, Glasgow, and 12 Rothesay Place, Edin. 1876 * Denny, Peter, Memb. Inst. C.E., Dumbarton 1869 | P. | * Dewar, James, M.A., F.R.S., Jacksonian Professor of Natural and Experimental Philosophy in the University of Cambridge, and Fullerian Professor of Chemistry at the Royal Institution of Great Britain, London 1869 | P. |* Dickson, Alexander, M.D., Professor of Botany in the University of Edinburgh, 11 Royal Circus 1884 * Dickson, Charles Scott, Advocate, 59 Northumberland Street 120 1876 | P. |* Dickson, J. D. Hamilton, M.A., Fellow and Tutor, St Peter’s College, Cambridge 1869 * Dickson, William, 38 York Place 1863 | P. Dittmar, W., LL.D., F.R.S., Professor of Chemistry, Anderson’s College, Glasgow 1885 Dixon, J. M., M.A., Professor of English Literature in the University of Tokio, Japan 1881 * Dobbin, Leonard, Ph.D., 16 Kilmaurs Road 125 1867 | P. |* Donaldson, J., M.A., LL. D. , Principal of the United College of St Salvador and St Leonard, St Andrews 1882 * Dott, D. B., Memb. Pharm. Soc., 7 Victoria Terrace, ar 1866 * Douglas, David, 22 Drummond Place 1878 Drew, Samuel, M.D., D.Se., Chapelton, near Sheffield 1880 * Drummond, Henry, F.G.S., Prof. of Nat. History in the Free Church College, Glasgow 130 1860 Dudgeon, Patrick, of Cargen, Dumfries 1863 | P. Duncan, J. Matthews, M.A., M.D., F.R.C.P.E., LL.D., F.R.S., 71 Brook Street, London 1870 * Duncan, John, M.D,, F.R.C.P.E., F.R.C.S.E., 8 Ainslie Place 1876 * Duncan, James, of Benmore, Kilmun, 9 Mincing Lane, London 1878 * Duncanson, J. J. Kirk, M.D., F.R.C.P.E., 22 Drumsheugh Gardens 135 1859 Duns, Rev. Professor, D.D., New College, Edinburgh, 14 Greenhill Place 1874 * Durham, William, Seaforth House, Portobello 1869 * Elder, George, Knock Castle, Wemyss Bay, Greenock 1885 * Elgar, Francis, LL.D., The Admiralty, London 1875 Elliot, Daniel G., New York 140 Date of Election. 1880 1855 1884 1863 1879 1878 1875 1866 1859 1883 1868 1874 1886 1852 1876 1880 1872 1859 1828 1887 1858 1867 1885 1867 1880 1861 1871 1881 1877 1885 1887 | 1879 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 693 BSE. 12 B. P. * Elliot, T. Armstrong, M.A., 6 Sanderson Road, Newcastle-on-Tyne Etheridge, Robert, F.R.S., Assistant-Keeper of the Geological Department at the British Museum of Natural History, 14 Carlyle Square, Chelsea, London * Evans, William, F.F.A., 18a Morningside Park, Edinburgh Everett, J. D., M.A., D.C.L., F.R.S., Professor of Natural Philosophy, Queen’s College, Belfast * Ewart, James Cossar, M.D., F.R.C.S.E., Professor of Natural History, University of Edin- burgh, 3 Great Stuart Street 145 * Ewing, James Alfred, B.Sc., F.R.S., Professor of Engineering and Drawing in University College, Dundee Fairley, Thomas, Lecturer on Chemistry, 8 Newton Grove, Leeds * Falshaw, Sir James, Bart., Assoc. Inst. C.E., 14 Belgrave Crescent Fayrer, Sir Joseph, K.C.S.1., M.D., F.R.C.P.L., F.R.C.S.L. and E., LL.D., F.R.S., Honorary Physician to the Queen, 53 Wimpole Street, London * Felkin, Robert W., M.D., F.R.G.S., Fellow of the Anthropological Society of Berlin, 20 Alva Street, Edinburgh 150 * Ferguson, Robert M., Ph.D., 12 Moray Place * Ferguson, William, of Kinmundy, F.L.S., F.G.S., 21 Manor Place, Edinburgh, and Kinmundy House, Mintlaw Field, C. Leopold, F.C.S., Upper Marsh, Lambeth, London Fleming, Andrew, M.D., Deputy Surgeon-General, 3 Napier Road * Fleming, J. S., 16 Grosvenor Crescent 155 * Flint, Robert, D.D., Corresponding Member of the Institute of France, Professor of Divinity in the University of Edinburgh (Vicz-PResiDEnT), Johnstone Lodge, 54 Craigmillar Park * Forbes, G., Professor, M.A., F.R.S., F.R.A.S., M.S.T.E. and E., 34 Great George Street, Westminster Forlong, Major-Gen. J. G., F.R.G.S., R.A.S., Assoc. C.E., &., 11 Douglas Crescent Foster, John, Liverpool Fowler, Sir John, Memb. Inst. C.E., Thornwood Lodge, Kensington, London 160 Fraser, A. Campbell, M.A., LL.D., D.C.L., Professor of Logic and Metaphysics in the University of Edinburgh, 20 Chester Street * Fraser, Thomas R., M.D., F.R.C.P.E., F.R.S., Professor of Materia Medica in the University of Edinburgh, 13 Drumsheugh Gardens * Fraser, A. Y., M.A., Secretary to the Mathematical Society of Edinburgh, 4 Mayfield Road Gayner, Charles, M.D., Oxford * Geddes, Patrick, Assistant to the Professor of Botany in the University of Edinburgh, and Lecturer on Zoology, 6 James’ Court, Lawnmarket 165 Geikie, Archibald, LL.D., F.R.S., F.G.S., Director of the Geological Surveys of Great Britain, and Head of the Geological Museum, 28 Jermyn Street, London B. P.| * Geikie, James, LL.D., F.R.S., F.G.S., Professor of Geology in the University of Edinburgh, 31 Merchiston Avenue * Gibson, G. A., D.Sc., M.D., F.R.C.P.E., 17 Alva Street * Gibson, John, Ph.D., 29 Greenhill Gardens * Gibson, R. J. Harvey, M.A., Lecturer on Botany, Victoria University, 98 Coltart Road, Prince’s Park, Liverpool 170 * Gilmour, William, 10 Elm Row * Gilray, Thomas, M.A., Professor of English Language and Literature and Modern History in University College, Dundee VOL. XXXIII. PART III. ak 694 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Date of Election. 1880 * Gilruth, George Ritchie, Surgeon, 48 Northumberland Street 1850 Gosset, Major-General W. D., R.E., 70 Edith Road, West Kensington, London 1867 * Graham, Andrew, M.D., R.N., Army and Navy Club, 36 Pall Mall, London 175 1880 * Graham, James, 195 Bath Street, Glasgow 1851 Grant, The Rev. James, D.D., D.C.L., 15 Palmerston Place 1883 * Gray, Andrew, M.A., Professor of Physics in University College, Bangor, North Wales 1880 | P. Gray, Thomas, B.Sc., 17 Broomhill Avenue, Partick 1886 * Greenfield, W. S., M.D., Professor of General Pathology in the University of Edinburgh, 7 Heriot Row 180 1872 * Grieve, David, 19 Abercorn Terrace, Portobello | 1884 * Grieve, John, M.A., M.D., F.L.S., 212 St Vincent Street, Glasgow 1886 * Griffiths, Arthur Bower, Ph.D., Principal and Lecturer on Chemistry in the School of Science of the City and County of Lincoln, 15 Broadgate, Lincoln 1883 Gunning, R. H., M.D., LL.D., 12 Addison Crescent, Kensington 1886 . Haddington, The Right Hon. the Earl of, Tyninghame House, Haddington 185 1867 * Hallen, James H. B., F.R.C.S.E., F.R.P.S.E., Inspecting Veterinary Surgeon in H.M. Indian Army, Pebworth, near Stratford-on-Avon 1881 | Pp. |* Hamilton, D. J., M.B., F.R.C.S.E., Professor of Pathological Anatomy in the University of Aberdeen, 1a Albyn Place, Aberdeen 1876 | Pp. |* Hannay, J. Ballantyne, Cove Castle, Loch Long 1886 * Hare, Arthur W., M.B., F.R.C.E., Professor of Surgery, Owens College, Victoria Univer- ’ sity, 3 Adelphi Terrace, Salford, Manchester 1869 Hartley, Sir Charles A., K.C.M.G., Memb. Inst. C.E., 26 Pall Mall, London 190 1877 Hartley, Walter Noel, F.R.S., Professor of Chemistry, Royal College of Science for Ireland, Dublin 1870 * Harvey, Thomas, M.A., LL.D., Rector of the Edinburgh Academy, 32 George Square 1880 | P. |* Hayceraft, J. Berry, M.B., B.Sc., Lecturer on Physiology in the University of Edinburgh 1875 Hawkshaw, Sir John, Memb. Inst. C.E., F.R.S., F.G.S., 33 Great George St., Westminster 1870 Heathfield, W. E., F.C.S., 1 Powis Grove, Brighton 195 1862 Hector, Sir James, C.M.G., M.D., F.R.S., Director of the Geological Survey, Wellington, New Zealand 1876 \K, P. | * Heddle, M. Forster, M.D., Emeritus Professor of Chemistry in the University of St Andrews 1884 * Henderson, John, jun., 4 Crown Terrace, Dowanhill, Glasgow 1881 \N. P.|} * Herdman, W. A., D.Sc., Professor of Natural History in University College, Liverpool 1871 Higgins, Charles Hayes, LL.D., Alfred House, Birkenhead 200 1859 Hills, John, Lieut.-Colonel, Bombay Engineers, C.B., United Service Club, London 1879 Hislop, John, Secretary to the Department of Education, Wellington, New Zealand 1885 Hodgkinson, W. R., Ph.D., F.LC., F.C.S., Professor of Chemistry and Physics at the Royal Military Academy and Royal Artillery College, Woolwich, 75 Vanbrugh Park, Black heath, London 1828 | P. Home, David Milne, of Milne-Graden, LL.D., F.G.S. (Vicu-Preswwent), 10 York Place 1879 * Hood, Thomas H. Cockburn, F.G.S., Walton Hall, Kelso 205 1881 | P. | * Horne, John, F.G.S., Geological Survey of Scotland, 41 Southside Road, Inverness 1883 | P. | * Hoyle, William Evans, M.A., M.R.C.S., Office of Challenger Commission, 32 Queen Street 1886 Hunt, Rev. H. G. Bonavia, Mus. D. Dublin, Mus. B. Oxon., F.L.S., La BelleSauvage, London 1872 * Hunter, Major Charles, Plis Coch, Llanfair, Anglesea, and 17 St George’s Sq., London 1887 * Hunter, James, F.R.C.S.E., F.R.A.S., 20 Craigmillar Park 210 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 695 Date of Election. 1887 1864 1855 1882 1874 1886 1875 1882 1860 1880 1865 1869 1867 1874 1877 1866 1886 1877 1880 1883 1878 1875 1880 1875 1886 1878 1885 1870 1881 1872 1872 1882 1883 1863 1858 1874 1870 Bers * Hunter, William, M.D., 8 West Maitland Street * Hutchison, Robert (Carlowrie Castle), and 29 Chester Street Inglis, Right Hon. John, LL.D., D.C.L., Lord Justice-General of Scotland, and Chancellor of the University of Edinburgh, 30 Abercromby Place * Inglis, J. W., Memb. Inst. C.E., Myrtle Bank, Trinity * Irvine, Alex. Forbes, of Drum, LL.D., Advocate, Sheriff of Argyll (Vicn-PREsIDENT), 25 Castle Terrace 215 * Irvine, Robert, Royston, Granton, Edinburgh Jack, William, M.A., Professor of Mathematics in the University of Glasgow * Jamieson, A., Prof., Memb. Inst. C.E., Principal, the Glasgow and West of Scotland Technical College, Glasgow Jamieson, George A., 24 St Andrew Square Japp, A. H., LL.D., The Limes, Elmstead, near Colchester 220 * Jenner, Charles, Easter Duddingston Lodge Johnston, John Wilson, M.D., Surgeon-Major, 11 Windsor Street * Johnston, T. B., F.R.G.S., Geographer to the Queen, 9 Claremont Crescent Jones, Francis, Lecturer on Chemistry, Monton Place, Manchester * Jolly, William, H.M. Inspector of Schools, F.G.S., Ardgowan, Pollokshields 225 * Keiller, Alexander, M.D., F,R.C.P.E., LL.D., 21 Queen Street * Kidston, Robert, F.G.S., 24 Victoria Place, Stirling * King, The Right Hon. Sir James, of Campsie, LL. D., Lord Provost of Glasgow, 12 Claremont Terrace, Glasgow * King, W. F., Lonend, Russell Place, Trinity * Kinnear, The Hon. Lord, one of the Senators of the College of Justice, 2 Moray Place 230 * Kintore, The R. H. the Earl of, M.A. Cantab., Keith Hall, Inglismaldie Castle, Laurencekirk * Kirkwood, Anderson, LL.D., 7 Melville Terrace, Stirling * Knott, C. G., D.Se., Prof. of Natural Philosophy in the Imperial University of Tokio, Japan * L’Amy, John Ramsay, of Dunkenny, Forfarshire, 107 Cromwell Road, London * Laing, Rev. George, 17 Buckingham Terrace 235 * Lang, P. R. Scott, M.A., B.Sc., Professor of Mathematics in the University of St Andrews \* Laurie, A. P., B.A., B.Sc., Nairne Lodge, Duddingstone, Edinburgh * Laurie, Simon S8., M.A., Professor of Education in the University of Edinburgh, Nairne Lodge, Duddingstone * Lawson, Robert, M.D., Deputy-Commissioner in Lunacy, 24 Mayfield Terrace * Lee, Alexander H., C.E., Blairhoyle, Stirling 240 * Lee, The Hon. Lord, one of the Senators of the College of Justice, Duddingstone House, Edinburgh * Leslie, Alexander, Memb. Inst. C.E., 12 Greenhill Terrace * Leslie, George, M.B., C.M., Old Manse, Falkirk Leslie, Hon. G. Waldegrave, Leslie House, Leslie Leslie, James, Memb. Inst. C.E., 2 Charlotte Square 245 * Letts, E. A., Ph.D., F.LC., F.C.S., Professor of Chemistry, Queen’s College, Belfast * Lister, Sir Joseph, Bart., M.D., F.R.C.8.L., F.R.CS.E., LL.D., D.C.L, F.R.S., Professor of Clinical Surgery, King’s College, Surgeon Extraordinary to the Queen, 12 Park Crescent, Portland Place, London 696 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Bleetion. 1882 * Livingston, Josiah, 4 Minto Street 1861 Lorimer, James, M.A., Advocate, Professor of Public Law in the University of Edinburgh, | 1 Bruntsfield Crescent 1884 * Low, George M., Actuary, 19 Learmonth Terrace 250 1849 Lowe, W. H., M.D., F.R.C.P.E., Woodcote, Inner Park, Wimbledon 1886 Lyster, George Fosbery, Memb. Inst. C.E., Gisburn House, Liverpool 1855 Macadam, Stevenson, Ph.D., Lecturer on Chemistry, Surgeons’ Hall, Edinburgh, 11 East Brighton Crescent, Portobello 1887 M‘Aldowie, Alexander M., M.D., Brook Street, Stoke-on-Trent 1885 * M‘Bride, Charles, M.D., Wigtown 255 1883 * M‘Bride, P., M.D., F.R.C.P.E., 16 Chester Street 1867 * M‘Candlish, John M., W.S., 27 Drumsheugh Gardens 1886 * Macdonald, The Right Hon. J. H. A., C.B., Q.C., M.P., LL.D., M.S.T.E. and E., Lord Advocate of Scotland, 15 Abercromby Place 1886 * Macdonald, William J., M.A., 6 Lockharton Terrace 1847 Macdonald, W. Macdonald, of St Martin’s, Perth 260 1878 * MacDougall, Alan, Memb. I.C.E., Mail Building, 52 King Street West, Toronto, Canada 1878 | P. Macfarlane, Alex., M.A., D.Sc., LL.D., Professor of Physics in the University of the State of Texas, Austin, Texas 1885 | P. |* Macfarlane, J. M., D.Sc., 15 Scotland Street 1877 * Macfie, Robert A., Dreghorn Castle, Colinton 1878 + M‘Gowan, George, F.I.C., Ph.D., University College of North Wales, Bangor 265 1886 * MacGregor, Rev. J., D.D., 11 Cumin Place, Grange 1330) P: MacGregor, J. Gordon, M.A., D.Sc., Professor of Physics in Dalhousie College, Halifax, Nova Scotia 1879 \* M‘Grigor, Alexander Bennett, LL.D., 19 Woodside Terrace, Glasgow 1869 | N. P.|* M‘Intosh, William Carmichael, M.D., LL.D., F.R.S., F.L.S., Professor of Natural History in the University of St Andrews, 2 Abbotsford Crescent, St Andrews 1882 * Mackay, John Sturgeon, M.A., LL.D., Mathematical Master in the Edinburgh Academy, 69 Northumberland Street 270 1873 | P. | * M‘Kendrick, John G., M.D., F.R.C.P.E., F.R.S., Professor of the Institutes of Medicine in the University of Glasgow 1840 Mackenzie, John, New Club, Princes Street 1848 Py Maclagan, Sir Douglas, M.D., President of the Royal College of Physicians, Edinburgh, and F.R.C.S.E., Professor of Medical Jurisprudence in the University of Edinburgh (Vice-PresipEnT), 28 Heriot Row 1853 Maclagan, General R., Royal Engineers, 4 West Cromwell Road, 8S. Kensington, London,S. W. 1869 * Maclagan, R. Craig, M.D., 5 Coates Crescent 6 1864 * M‘Lagan, Peter, of Pumpherston, M.P., Clifton Hall, Ratho 1869 * M‘Laren, The Hon. Lord, LL.D. Edin. and Glasg., F.R.A.S., one of the Senators of the College of Justice (Vicn-PresipENT), 46 Moray Place 1870 * Macleod, Sir George H.B., M.D., F.R.C.S.E., Regius Prof. of Surgery in the University of Glas- gow, and Surgeon in Ordinary to the Queen in Scotland, 10 Woodside Crescent, Glasgow 1876 * Macleod, Rev. Norman, D.D., 7 Royal Circus 1883 * Macleod, W. Bowman, L.D.S., 16 George Square 280 1872 * Macmillan, Rev. Hugh, D.D., LL.D., Seafield, Greenock 1876 * Macmillan, John, M.A., B.Se., Mathematical Master, Perth Academy ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 697 Date of Election. 1884 1883 1858 1880 1882 1869 1864 1866 1885 1883 1885 1886 1852 1885 1833 1886 1866 1865 1870 1871 1868 1887 | 1887 1877 1873 1874 1877 1870 1857 1877 1887 1884 1877 1887 1883 1884 K.P, * Macpherson, Rev. J. Gordon, M.A., D.Sc., Ruthven Manse, Meigle * M‘Roberts, George, F.C.S., Ardeer, Stevenston, Ayrshire Malcolm, R. B., M.D., F.R.C.P.E., 126 George Street 285 Marsden, R. Sydney, M.B., C.M., D.Sc., F.1.C., F.C.S., Pembroke House, King Street, Stockton-on-Trent Marshall, D. H., M.A., Professor of Physics in Queen’s University and College, Kingston, Ontario, Canada Marshall, Henry, M.D., Clifton, Bristol * Marwick, Sir James David, LL.D., Town-Clerk, Glasgow * Masson, David, LL.D., Professor of Rhetoric and English Literature in the University of Edinburgh, 58 Great King Street 290 * Masson, Orme, D.Sc., Professor of Chemistry in the University of Melbourne * Matthews, James Duncan, Springhill, Aberdeen * Mill, Hugh Robt., D.Se., F.C.S., Scot. Marine Station, Granton, 3 Glenorchy Terrace, Edin. * Miller, Hugh, H.M. Geological Survey Office, George IV. Bridge Miller, Thomas, M.A., LL. D., Emeritus Rectorof Perth Academy, Inchbank House, Perth 295 * Miller, William, S.S.C., 59 George Square Milne, Admiral Sir Alexander, Bart., G.C.B., Inveresk * Milne, William, M.A., B.Sc., Mathematical and Science Teacher, High School, Glasgow * Mitchell, Sir Arthur, K.C.B.,M.A., M.D., LL.D., Commissioner in Lunacy, 34 Drummond PI. * Moir, John J. A., M.D., F.R.C.P.E., 52 Castle Street 300 * Moncreiff, The Right Hon. Lord, of Tullibole, Lord Justice-Clerk, LL.D. (Honorary Vicr- PRESIDENT), 15 Great Stuart Street * Moncrieff, Rev. Canon William Scott, of Fossaway, Christ’s Church Vicarage, Bishop-Wear- mouth, Sunderland * Montgomery, Very Rev. Dean, M.A., D.D., 17 Atholl Crescent Moos, Nanabhay A. F., L.C.E., B.Sc., Assistant Professor of Engineering, College of Science, Bombay More, Alexander Goodman, M.R.1I.A., F.L.S., 77 Leinster Road, Dublin 305 * Morrison, Robert Milner, D.Sc., F.I.C. * Muir, M. M. Pattison, Prelector on Chemistry, Caius College, Cambridge * Muir, Thomas, M.A., LL.D., Mathematical Master, High School, Glasgow, Beechcroft, Bothwell, Glasgow Mukhopadhyay, Asftosh, M.A., F.R.A.S., Examiner in Mathematics in the University of Calcutta, Professor of Mathematics at the Indian Association for the Cultivation of Science, 77 Russa Road North, Bhowanipore, Calcutta * Munn, David, M.A., 2 Ramsay Gardens 310 Murray, John Ivor, M.D., F.R.C.S.E., M.R.C.P.E., 24 Huntriss Row, Scarborough .|* Murray, John, LL.D., Ph.D., Director of the Challenger Expedition Commission, 32 Queen Street, and United Service Club (Vicz-PREsIDENT) Muter, John, M.A., F.C.S., Winchester House, 397 Kennington Road, London Mylne, R. W., C.E., F.R.S., 7 Whitehall Place, London * Napier, John C., Audley Mansions, Grosvenor Square, London 315 * Nasmyth, T. Goodall, M.B., C.M., D.Sc., Foulford, Cowdenbeath, Fife * Nelson, Thomas, St Leonard’s, Dalkeith Road * Newcombe, Henry, F.R.C.S.E., 5 Dalrymple Crescent, Edinburgh 698 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Date of | Election. 1866 * Nicholson, J. Shield, Professor of Political Economy in the University of Edinburgh, Eden Lodge, Eden Lane, Newbattle Terrace 1880 | P. |* Nicol, W. W. J., M.A., D.Se., Lecturer on Chemistry, Mason College, Birmingham 320 1878 Norris, Richard, M.D., Professor of Physiology, Queen’s College, Birmingham 1886 Oliver, James, M.I)., C.M., M.R.C.P., Assistant Physician, Hospital for Women, 18 Gordon Square, London 1884 * Omond, Robert Traill, Superintendent of Ben Nevis Observatory, Fort-William, Inverness 1877 Panton, George A., 95 Colmore Row, Birmingham 1886 * Paton, D. Noel, M.D., B.Sc., 4 Walker Street 325 1881 |N.P.| Peach, B. N., F.G.8., Acting Paleontologist of the Geol. Sury. of Scot., 13 Dalrymple Cres. 1863 Peddie, ee M.D., F.R.C.P.E., 15 Rutland Street 1887 * Peddie, William, hecetaah to the Professor of Natural Philosophy, Edinburgh Universe 1886 * Peebles, D. Bruce, Tay House, Bonnington, Edinburgh 1869 | Pender, John, 18 Arlington Street, Piccadilly, London 330 1883 Phillips, Charles D. F., M.D., 10 Henrietta Street, Cavendish Square, London, W. 1859 | Pp, Playfair, The Right Hon. Sir Lyon, K.C.B., M.P.,LL.D., F.R.S., 68 Onslow Gardens, London 1877 Pole, William, Memb. Inst. C.E., Mus. Doc., F.R.S., 31 Parliament Street, Westminster — 1886 * Pollock, Charles Frederick, M.D., F.R.C.S.E., 1 Buckingham Terrace, Hillhead, Glasgow 1874 Powell, Baden Henry Baden-, Forest Department, India 335 1852 Powell, Eyre B., C.S.1., M.A., 28 Park Road, Haverstock Hill, Hampstead, London 1880 * Prentice, Charles, C.A., Actuary, 8 St Bernard’s Crescent 1875 Prevost, E. W., Ph.D., The Poplars, Shuttington, Tamworth 1849 Primrose, Hon. B. F., C.B., 22 Moray Place 1882 * Pryde, David, M.A., LL.D., Head Master of the Ladies’ College, 10 Fettes Row, Edinb, 340 1885 * Pullar, J. F., Rosebank, Perth. 1880 * Pullar, Robert, Tayside, Perth 1884 Ramsay, E. Peirson, M.R.LA., F.L.S., C.M.Z.S., F.R.G.S., F.G.S., Fel. of the Imperial and Royal Zool. and Bot. Soc. of Vienna, Curator of Australian Museum, Sydney, N.S.W. 1882 * Rattray, James Clerk, M.D., 61 Grange Loan 1885 | P. |* Rattray, John, M.A., B.Sc., Natural History Department, British Museum, London 345 1869 Raven, Rev. Thomas Milville, M.A., The Vicarage, Crakehall, Bedale 1883 * Readman, J. B., 9 Moray Place 1875 * Richardson, Ralph, W.S., 10 Magdala Place 1872 Ricarde-Seaver, Major F. Ignacio, Conservative Club, St James’ Street, London and 2 Rue Lafitte, Boulevard des Italiens, Paris 1883 * Ritchie, R. Peel, M.D., F.R.C.P.E., 1 Melville Crescent 350 1877 * Roberton, James, LL.D., Professor of Conveyancing in the University of Glasgow, 1 Park Terrace East. Glasgow 1880 Roberts, D. Lloyd, M.D., F.R.C.P.L., 23 St John Street, Manchester 1872 * Robertson, D. M. C. L. Argyll, M.D., F.R.C.S.E., Surgeon Oculist to the Queen for Scot- land, and President of the Royal College of Surgeons, 18 Charlotte Square 1859 Robertson, George, Memb. Inst. C.E., Athenzeum Club, Pall Mall, London 1886 * Robertson, J. P. B., Q.C., M.P., 19 riknchlonsh Gardens 355 1877 | P. |* Robinson, George Dass EUCS staves on Chemistry in the Royal Institution, Hull 1881 * Rogerson, John Johnston, B.A., LL.B., Merchiston Castle Academy ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 699 Date of Election. 1862 1881 1880 1880 1869 iB. EP. Ronalds, Edmund, LL.D., Bonnington House, Bonnington Road Rosebery, The Right Hon. the Earl of, LL.D., Dalmeny Rowland, L. L., M.A., M.D., President of the Oregon State Medical Society, and Professor of Physiology and Microscopy in Williamette University, Salem, Oregon 360 * Russell, J. A., M.A., B.Sc., M.B., F.R.C.P.E., Woodville, Canaan Lane * Rutherford, Wm., M.D., F.R.C.P.E., F.R.S., Professor of the Institutes of Medicine in the University of Edinburgh, 14 Douglas Crescent Sanderson, James, Deputy Inspector-General of Hospitals, F.R.C.S.E., 8 Manor Place Sandford, The Right Rev. D. F., LL.D., Bishop of Tasmania Sang, Edward, C.E., LL.D., Sec. to the Royal Scottish Society of Arts, 31 Mayfield Road 365 Schmitz, Leonard, LL.D., 81 Linden Gardens, London * Schulze, Adolf P., 2 Doune Gardens, Kelvinside, Glasgow Scott, Alexander, M.A., D.Sc., 4 North Bailey, Durham Scott, J. H., M.B., C.M., M.R.C.S., Prof. of Anatomy in the Univ. of Otago, N. Z. Scott, Michael, Memb. Inst. C.E., 35 Dudley Road, Tunbridge Wells 370 * Sellar, W. Y., M.A., LL.D., Professor of Humanity in the University of Edinburgh, 15 Buckingham Terrace * Seton, George, M.A., Advocate, 42 Greenhill Gardens * Sexton, A. H., F.C.S., Professor of Chemistry, College of Science and Arts, Glasgow * Sibbald, John, M.D., Commissioner in Lunacy, 3 St Margaret’s Road, Whitehouse Loan * Sime, James, M.A., South Park, Fountainhall Road 375 * Simpson, A. R., M.D., F.R.C.P.E., Professor of Midwifery in the University of Edinburgh, 52 Queen Street Skene, Wm. F., W.S., LL.D., D.C.L., Historiographer-Royal for Scotland, 27 Inverleith Row * Skinner, William, W.S., Town-Clerk of Edinburgh, 35 George Square * Smith, Adam Gillies, C.A. (TREasuRER), 64 Princes Street Smith, C. Michie, B.Sc., Professor of Physical Science, Christian College, Madras, India 380 * Smith, George, F.C.S8., Polmont Station Smith, James Greig, M.A., M.B., 16 Victoria Square, Clifton * Smith, John, M.D., LL.D., F.R.C.S.E., President of the Medico-Chirurgical Society, 11 Wemyss Place Smith, Robert Mackay, 4 Bellevue Crescent * Smith, Major-General Sir R. Murdoch, K.C.M.G., R.E., Director of Museum of Science and Art, Edinburgh 385 * Smith, Rev. W. Robertson, M.A., LL.D., Librarian to the University of Cambridge Smith, William Robert, M.D., D.Sc., 74 Great Russell Street, Bloomsbury Square, London Smyth, Piazzi, Professor of Practical Astronomy in the University of Edinburgh, and Astronomer-Royal for Scotland, 15 Royal Terrace Sollas, W. J., M.A., D.Sc., late Fellow of St John’s College, Cambridge, and Professor of Geology and Mineralogy in the University of Dublin, 4 Clyde Road, Dublin * Sorley, James, F.F.A., C.A., 2 Dean Park Crescent 390 * Sprague, T. B., M.A., 29 Buckingham Terrace Stark, James, M.D., F.R.C.P.E., of Huntfield, Underwood, Bridge of Allan * Stegeall, J. E. A., Prof. of Mathematics and Natural Philosophy in University Coll., Dundee * Stevenson, C. A., B.Sc., C.E., 45 Melville Street * Stevenson, David Alan, B.Sc., C.E., 45 Melville Street 395 * Stevenson, James, F.R.G.S., 4 Woodside Crescent, Glasgow 700 ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. Date of Election. 1868 1868 1878 1866 1873 1848 1877 1823 1870 1848 1844 1875 1885 1872 1861 1870 1872 1873 1885 1884 1870 1887 1875 1887 1880 1863 1870 1847 1882 1870 1876 1878 1874 1874 1879 K-P. K.P, Vids W.P. Stevenson, John J., 18 Queen’s Road, Bayswater, London Stewart, Major J. H. M. Shaw, Royal Engineers, 61 Lancaster Gate, London, W. * Stewart, James R., M.A., 10 Salisbury Road * Stewart, T. Grainger, M.D., F.R.C.P.E., Professor of the Practice of Physic in the University of Edinburgh, 19 Charlotte Square 400 * Stewart, Walter, 22 Torphichen Street Stirling, Patrick J., LL.D., Kippendavie House, Dunblane * Stirling, William, D.Sc., M.D., Brackenbury Professor of Physiology and Histology in Owens College and Victoria University, Manchester Stuart, Captain T. D., H.M.LS. * Swan, Patrick Don, Provost of Kirkcaldy 405 Swan, Wm., LL.D., Emeritus Professor of Natural Philosophy in the University of St Andrews, President of the Royal Scottish Society of Arts, Ardchapel, Helensburgh Swinton, A. Campbell, of Kimmerghame, LL.D., Duns Syme, James, 9 Drumsheugh Gardens * Symington, Johnson, M.D., F.R.C.S.E., 2 Greenhill Park Tait, the Very Rev. A., D.D., LL.D., Provost of Tuam, Moylough Rectory, County Galway, Treland 410 Tait, P. Guthrie, M.A., Professor of Natural Philosophy in the University of Edinburgh (GENERAL SEcRETARY), 38 George Square * Tatlock, Robert R., City Analyst’s Office, 138 Bath Street, Glasgow * Teape, Rev. Charles R., M.A., Ph.D., 15 Findhorn Place * Tennent, Robert, 23 Buckingham Terrace * Thompson, D’Arcy W., Professor of Natural History in University College, Dundee 415 * Thoms, George Hunter, of Aberlemno, Advocate, Sheriff of the Counties of Orkney and Zetland, 13 Charlotte Square Thomson, Rev. Andrew, D.D., 63 Northumberland Street * Thomson, Andrew, M.A., D.Sc., Assistant to the Professor of Chemistry in the University College, Dundee, 1 Blackness Crescent, Dundee * Thomson, James, LL.D., F.R.S., Professor of Engineering in the University of Glasgow, 2 Florentine Gardens, Hillhead, Glasgow * Thomson, J. Arthur, M.A., Lect. on Zoology, School of Medicine, Edin., 10 Kilmaurs Rd. 420 Thomson, John Millar, King’s College, London Thomson, Murray, M.D., Professor of Chemistry, Thomason College, Roorkee, India * Thomson, Spencer C., Avtuary, 10 Eglinton Crescent Thomson, Sir William, LL.D., D.C.L., F.R.S. (Presipent), Regius Professor of Natural Philosophy in the University of Glasgow, Foreign Associate of Institute of France, aud Member of the Prussian Order Pour le Mérite Thomson, Willliam, M.A., B.Sc., Professor of Mathematics, Victoria College, Stellenbosch, Cape Colony 425 * Thomson, Wm. Burns, F.R.C.P.E., F.R.C.S.E., 110 Newington Green Road, London, N. Thomson, William, Royal Institution, Manchester Thorburn, Robert Macfie, Uddevalla, Sweden * Traquair, R. H., M.D., F.R.S., F.G.S., Keeper of the Natural History Collections in the Museum of Science and Art, Edinburgh, 8 Dean Park Crescent Tuke, J. Batty, M.D., F.R.C.P.E., 20 Charlotte Square 430 Turnbull, John, of Abbey St Bathans, W.S., 49 George Square ALPHABETICAL LIST OF THE ORDINARY FELLOWS OF THE SOCIETY. 701 Date of Election. 1861 | N.P.| Turner, Sir William, M.B., F.R.C.S.E., F.R.S., Professor of Anatomy in the University of Edinburgh, and President of the Royal Physical Society (Szcrerary), 6 Eton Terrace 1877 * Underhill, Charles E., B.A., M.B., F.R.C.P.E., F.R.C.S.E., 8 Coates Crescent 1875 Vincent, Charles Wilson, Royal Institution, Albemarle Street, London 1867 * Waddell, Peter, 5 Claremont Park, Leith 435 1873 * Walker, Robert, M.A., University, Aberdeen 1886 * Wallace, Robert, Professor of Agriculture and Rural Economy in the University of Edinburgh 1864 * Wallace, William, Ph.D., City Analyst’s Office, 138 Bath Street, Glasgow 1883 * Watson, Charles, Redhall, Slateford 1870 * Watson, James, C.A., 45 Charlotte Square 440 1866 * Watson, John K., 14 Blackford Road 1866 * Watson, Patrick Heron, M.D., F.R.C.P.E., F.R.C.S.E., LL.D., 16 Charlotte Square 1862 | P. Watson, Rev. Robert Boog, B,A., Free Church Manse, Cardross, Dumbartonshire 1887 * Webster, H. A., Librarian to the Univ. of Edinb., 7 Duddingstone Park, Portobello 1873 Welsh, David, Major-General, R.A., 1 Barton Terrace, Dawlish 445 1840 Welwood, Allan A. Maconochie, LL.D., of Meadowbank and Garvoch, Kirknewton 1882 * Wenley, James A., Treasurer of the Bank of Scotland, 5 Drumsheugh Gardens 1887 * White, Arthur Silva, Secretary to the Scottish Geographical Society, 22 Duke Street 1881 Whitehead, Walter, F.R.C.S.E., 202 Oxford Road, Manchester 1883 Wickham, R. H. B., M.D., F.R.C.S.E., Medical Superintendent, City and County Lunatic Asylum, Newcastle-on-Tyne 450 1887 * Wieland, G. B., Whitehill, Rosewell, Mid-Lothian 1879 * Will, John Charles Ogilvie, M.D., 305 Union Street, Aberdeen 1868 * Williams, W., Principal and Professor of Veterinary Medicine and Surgery, New Veterinary College, Leith Walk 1879 * Wilson, Andrew, Ph.D., Lecturer on Zoology and Comparative Anatomy in the Edinburgh Medical School, 118 Gilmore Place 1877 * Wilson, Charles E., M.A., LL.D., H.M. Senior Inspector of Schools for Scotland, 19 Palmerston Place 455 1878 * Wilson, Rev. John, M.A., 27 Buccleuch Place 1875 Wilson, Daniel, LL.D., President of the University of Toronto, and Professor of English Literature in that University 1882 Wilson, George, M.A., M.D., 23 Claremont Road, Leamington 1834 Wilson, Isaac, M.D. 1847 Wilson, John, LL.D., Emeritus Professor of Agriculture in the University of Edinburgh, Sandfield, Tunbridge Wells 460 1870 Winzer, John, Chief Surveyor, Civil Service, Ceylon, 7 Dryden Place, Newington 1880 * Wise, Thos. Alex., M.D., F.R.C.P.E., F.R.A.S., Thornton, the Beulah, Upper Norwood 1886 * Woodhead, German Sims, M.D., 6 Marchhall Crescent 1884 Woods, G. A., M.R.C.S., Carlton House, 57 Houghton Street, Southport 1864 * Wyld, Robert S., LL.D., 19 Inverleith Row 465 1887 * Yeo, John S., Carrington House, Fettes College 1882 * Young, Andrew, 22 Elm Row 1882 * Young, Frank W., F.C.S., Lecturer on Natural Science, High School, Dundee, Woodmuir Park, West Newport, Fife 1882 * Young, Thomas Graham, Durris, Aberdeenshire 469 VOL. XXXIII. PART III. 3K 702 APPENDIX—LIST OF HONORARY FELLOWS. LIST’ Of BONOERARY FELLOW ss AT JANUARY 1888. His Royal Highness The Princk or WALES. FOREIGNERS (LIMITED TO THIRTY-SIX BY LAW X.). Elected. 1884 Pierre J. van Beneden, Louvain. 1864 Robert Wilhelm Bunsen, Heidelberg. 1867 Michel Eugéne Chevreul, Paris. 1888 Rudolph Julius Emmanuel Clausius, Bonn. 1877 Alphonse de Candolle, Geneva. 1883 Luigi Cremona, Rome. 1858 James D. Dana, New Haven, Conn. 1879 Franz Cornelius Donders, Utrecht. 1877 Carl Gegenbaur, Heidelberg. 1879 Asa Gray, Harvard University. 1888 Ernest Haeckel, Jena. 1883 Julius Hann, Vienna. 1884 Charles Hermite, Paris. 1864 Hermann Ludwig Ferdinand von Helmholtz, Berlin. 1879 Jules Janssen, Paris. 1875 August Kekulé, Bonn. 1864 Albert Kolliker, Wirzburg. 1875 Ernst Eduard Kummer, Berlin. 1876 Ferdinand de Lesseps, Paris. 1864 Rudolph Leuckart, Leipzig. 1881 Sven Lovén, Stockholm. 1876 Carl Ludwig, Leipzig. 1878 J. N. Madvig, Copenhagen. 1888 Demetrius Ivanovich Mendeléef, St Petersburg. 1886 Alphonse Milne-Edwards, Paris. 1864 Theodore Mommsen, Berlin. 1881 Simon Newcomb, Washington. 1886 H. A. Newton, Yale College. 1874 Louis Pasteur, Paris. 1886 L’Abbé Renard, Louvain. 1881 Johannes Iapetus Smith Steenstrup, Copenhagen. 1878 Otto Wilhelm Struve, Pulkowa. 1886 Tobias Robert Thalen, Upsala. 1874 Otto Torell, Lund. 1868 Rudolph Virchow, Berlin. 1874 Wilhelm Eduard Weber, Gottingen. Total, 36. APPENDIX—LIST OF HONORARY FELLOWS. 703 BRITISH SUBJECTS (LIMITED TO TWENTY BY LAW X.). Elected. 1849 John Couch Adams, LL.D., F.R.S., Corresp. Mem. Inst. France, 1835 Sir George Biddell Airy, K.C.B., M.A., LL.D., D.C.L., F.R.S., Foreign Associate Inst. France, 1864 Arthur Cayley, LL.D., D.C.L., F.R.S., Corresp. Mem. Inst, France, 1884 Edward Frankland, D.C.L., LL.D., F.R.S8., 1874 John Anthony Froude, LL.D., 1881 The Hon. Justice Sir William Robert Grove, M.A., LL.D., DC. F.RS;, 1883 Sir Joseph Dalton Hooker, K.C.S.I., M.D., LL.D., D.C.L., F.B.S., F.G.S., Corresp. Mem. Inst. France, 1884 William Huggins, LL.D., D.C.L., F.R.S., 1876 Thomas Henry Huxley, LL.D., D.C.L., F.R.S., F.LS., F.Z.S., F.G.S., Corresp. Mem. Inst. France, 1867 James Prescott Joule, LL.D., D.C.L., F.R.S., Corresp. Mem. Inst. France, 1845 Sir Richard Owen, K.C.B., M.D., LL.D., D.C.L, F.R.S., Foreign Associate Inst. France, 1886 The Lord Rayleigh, D.C.L., LL.D., Sec. R.S., 1881 The Rev. George Salmon, D.D., LL.D., D.C.L., F.RB.S., Foreign Associate of the Institute of France, 1884 J. 8. Burdon Sanderson, M.D., LL.D., F.R.S., 1864 George Gabriel Stokes, M.A., LL.D., D.C.L., Pres. R.S., Corresp. Mem. Inst. France, 1874 James Joseph Sylvester, M.A., LL.D., F.R.S., Corresp. Mem. Inst. France, 1864 The Right Hon. Lord Tennyson, D.C.L., LL.D., F.R.S., Poet Laureate, 1883 Alexander William Williamson, LL.D., F.R.S., V.P.C.S., Corresp. Mem. Inst. France, 1883 Colonel Henry Yule, C.B., LL.D., Member of the Council of India, Total, 19. Cambridge. Greenwich. Cambridge. London. London. London. London. London. London. Manchester. London. London. Dublin. Oxford. Cambridge. Oxford. Isle of Wight. London. London. 704 APPENDIX—LIST OF MEMBERS ELECTED, ORDINARY FELLOWS ELECTED Durine Session 1886-87, ARRANGED ACCORDING TO THE DATE OF THEIR ELECTION. 6th December 1886. AsttosH Muxuopipuyay, M.A, JOSEPH JAMES COLEMAN. F.R.A.S. Joun James BuRNeET. 3rd January 1887. Nanabnay A. F. Moos, L.C.E., B.Se. 7th February 1887. J. B. Burst, M.D. J. G. BARTHOLOMEW. A. B. Brown, C.E. Wititram Hunter, M.D. FERDINAND Faiturut Bree. THomas GoopaLtt Nasmytu, D.Sc. J. ArtHur THomson, M.A. Sir Jonn Fowterr, M. Inst. C.E. Anprew Tomson, M.A., D.Sc. W. H. Bartow, M. Inst. C.E. WIt1am CaLDWELL CRAWFORD. JouN Murer, M.A., F.C.S. 7th March 1887. ARTHUR Sitva WHITE. H. M. Canetti, B.Sc. Witiram Peppie, D.Sc. G. B. Wiexanp. A. H. Srxtoyn, F.C.S. 4th April 1887. J. Mackay Brernarp. James Hunter, F.R.A.S, Hersert H. Asupown, M.D. Arexanprer M. M‘Atpowin, M.D. Witi1am Ginmoor. ALEXANDER GoopMAN Mors, M.R.LA. 2nd May 1887. H. A. WEBSTER. JOHN COCKBURN. Joun 8. Yo. A. S. Cummine, M.D. J. W. Capstiok. 6th June 1887. Cosmo Innes Burton, B.Sc., F,.C.S. Avotr P. Sosuuzt. APPENDIX—LIST OF MEMBERS DECEASED, ETC. 705 FELLOWS DECEASED OR RESIGNED DuRING SEssIon 1886-87. ORDINARY FELLOWS DECEASED. Witiiam Brown, F.R.C.S.E. D. Rutnserrorp Hatpans, M.D. Witiram Denny, C.E. JAMES PRINGLE, Provost of Leith. ALEXANDER Gipson, Advocate. ALEXANDER JAMES RUSSELL, C.S. Lorp Girrorp. Tuomas Stevenson, Hon. Vice-President. Rosert Gray. Rey. Francis Le Grix Waite. RESIGNED. James Tart Brack, Esq. Donatp Crawrorp, Esq., M.P. J. B. Brown Morrison, Esq. FOREIGN HONORARY FELLOWS DECEASED. Session 1886-87. Gustav Ropert KircHHOoFF. BERNARD STUDER. Hermann Ko.ese. Fe: eae OF THE ROYAL SOCIETY OF EDINBURGH. AS REVISED 20TH FEBRUARY 1882. ( 709 ) LAWS. [ By the Charter of the Society (printed in the Transactions, Vol. VI. p. 5), the Laws cannot be altered, except at a Meeting held one month after that at which the Motion for alteration shall have been proposed. | E THE ROYAL SOCIETY OF EDINBURGH shall consist of Ordinary and Honorary Fellows. 10m Every Ordinary Fellow, within three months after his election, shall pay Two Guineas as the fee of admission, and Three Guineas as his contribution for the Session in which he has been elected; and annually at the commencement of every Session, Three Guineas into the hands of the Treasurer. This annual contribution shall continue for ten years after his admission, and it shall be limited to Two Guineas for fifteen years thereafter.* III. Title. The fees of Ordi- nary Fellows resid- ing in Scotland. All Fellows who shall have paid Twenty-five years’ annual contribution shall Payment to cease be exempted from further payment. IV. The fees of admission of an Ordinary Non-Resident Fellow shall be £26, 5s., payable on his admission ; and in case of any Non-Resident Fellow coming to reside at any time in Scotland, he shall, during each year of his residence, pay the usual annual contribution of £3, 3s., payable by each Resident Fellow ; but after payment of such annual contribution for eight years, he shall be exempt * A modification of this rule, in certain cases, was agreed to at a Meeting of the Society held on the 3rd January 1831. At the Meeting of the Society, on the 5th January 1857, when the reduction of the Contribu- tions from £3, 3s. to £2, 2s, from the 11th to the 25th year of membership, was adopted, it was resolved that the existing Members shall share in this reduction, so far as regards their future annual Contributions. VOL. XXXIII. PART III. ok after 25 years. Fees of Non-Resi- dent Ordinary Fellows. Case of Fellows becoming Non- Resident. Defaulters. Privileges of Ordinary Fellows. Numbers Un- limited. Fellows entitled to Transactions. Mode of Recom- mending Ordinary Fellows. 710 LAWS OF THE SOCIETY. from any further payment. In the case of any Resident Fellow ceasing to reside in Scotland, and wishing to continue a Fellow of the Society, it shall be in the power of the Council to determine on what terms, in the circumstances of each case, the privilege of remaining a Fellow of the Society shall be continued to such Fellow while out of Scotland. V. Members failing to pay their contributions for three successive years (due application having been made to them by the Treasurer) shall be reported to the Council, and, if they see fit, shall be declared from that period to be no longer Fellows, and the legal means for recovering such arrears shall be employed. Al None but Ordinary Fellows shall bear any office in the Society, or vote in the choice of Fellows or Office-Bearers, or interfere in the patrimonial interests of the Society. WA, The number of Ordinary Fellows shall be unlimited. VIII. The Ordinary Fellows, upon producing an order from the TREASURER, shall be entitled to receive from the Publisher, gratis, the Parts of the Society’s Transactions which shall be published subsequent to their admission. IX. Candidates for admission as Ordinary Fellows shall make an application in writing, and shall produce along with it a certificate of recommendation to the purport below,* signed by at least four Ordinary Fellows, two of whom shall certify their recommendation from personal knowledge. This recommendation shall be delivered to the Secretary, and by him laid before the Council, and shall afterwards be printed in the circulars for three Ordinary Meetings of the Society, previous to the day of election, and shall lie upon the table during that time. * “A B., a gentleman well versed in Science (or Polite Literature, as the case may be), being “ to our knowledge desirous of becoming a Fellow of the Royal Society of Edinburgh, we hereby “ recommend him as deserving of that honour, and as likely to prove a useful and valuable Member.” LAWS OF THE SOCIETY. 711 Ke Honorary Fellows shall not be subject to any contribution. This class shall consist of persons eminently distinguished for science or literature. Its number shall not exceed Fifty-six, of whom Twenty may be British subjects, and Thirty- six may be subjects of foreign states. XI. Personages of Royal Blood may be elected Honorary Fellows, without regard to the limitation of numbers specified in Law X. XII. Honorary Fellows may be proposed by the Council, or by a recommenda- tion (in the form given below*) subscribed by three Ordinary Fellows ; and in case the Council shall decline to bring this recommendation before the Society, it shall be competent for the proposers to bring the same before a General Meeting. The election shall be by ballot, after the proposal has been commu- nicated viva voce from the Chair at one meeting, and printed in the circulars for two ordinary meetings of the Society, previous to the day of election. XIU. The election of Ordinary Fellows shall only take place at the first Ordinary Meeting of each month during the Session. The election shall be by ballot, and shall be determined by a majority of at least two-thirds of the votes, pro- vided Twenty-four Fellows be present and vote. XIV. The Ordinary Meetings shall be held on the first and third Mondays of every month from December to July inclusively ; excepting when there are five Mondays in January, in which case the Meetings for that month shall be held on its third and fifth Mondays. Regular Minutes shall be kept of the proceedings, and the Secretaries shall do the duty alternately, or according to such agreement as they may find it convenient to make. * We hereby recommend for the distinction of being made an Honorary Fellow of this Society, declaring that each of us from our own knowledge of his services to (Literature or Science, as the case may be) believe him to be worthy of that honour. (To be signed by three Ordinary Fellows.) To the President and Council of the Royal Society of Edinburgh. Honorary Fellows, British and Foreign. Royal Personages. Recommendation of Honorary Fel- lows. Mode of Election, Election of Ordi- nary Fellows, Ordinary Meet- ings. The Transactions. How Published. The Council. Retiring Council- lors. Election of Office- Bearers. Special Meetings ; how called. Treasurer’s Duties. 712 LAWS OF THE SOCIETY. XV. The Society shall from time to time publish its Transactions and Proceed- ings. For this purpose the Council shall select and arrange the papers which they shall deem it expedient to publish in the Transactions of the Society, and shall superintend the printing of the same. The Council shall have power to regulate the private business of the Society. At any Meeting of the Council the Chairman shall have a casting as well as a deliberative vote. XVI. The Transactions shall be published in parts or Fasciculi at the close of each Session, and the expense shall be defrayed by the Society. XVII. That there shall be formed a Council, consisting—First, of such gentlemen as may have filled the office of President ; and Secondly, of the following to be annually elected, viz.:—a President, Six Vice-Presidents (two at least of whom shall be resident), Twelve Ordinary Fellows as Councillors, a General Secretary, Two Secretaries to the Ordinary Meetings, a Treasurer, and a Curator of the Museum and Library. XVIII. Four Councillors shall go out annually, to be taken according to the order in which they stand on the list of the Council. XIX. An Extraordinary Meeting for the Election of Office-Bearers shall be held on the fourth Monday of November annually. 2G & Special Meetings of the Society may be called by the Secretary, by direction of the Council; or on a requisition signed by six or more Ordinary Fellows. Notice of not less than two days must be given of such Meetings. XXI. The Treasurer shall receive and disburse the money belonging to the Society, granting the necessary receipts, and collecting the money when due. He shall keep regular accounts of all the cash received and expended, which shall be made up and balanced annually ; and at the Extraordinary Meeting in November, he shall present the accounts for the preceding year, duly audited. LAWS OF THE SOCIETY. 713 At this Meeting, the Treasurer shall also lay before the Council a list of all arrears due above two years, and the Council shall thereupon give such direc- tions as they may deem necessary for recovery thereof. XX, At the Extraordinary Meeting in November, a professional accountant shall be chosen to audit the Treasurer’s accounts for that year, and to give the neces- sary discharge of his intromissions. ».O.G 508 The General Secretary shall keep Minutes of the Extraordinary Meetings of the Society, and of the Meetings of the Council, in two distinct books. He shall, under the direction of the Council, conduct the correspondence of the Society, and superintend its publications. For these purposes he shall, when necessary, employ a clerk, to be paid by the Society. XXIV. The Secretaries to the Ordinary Meetings shall keep a regular Minute-book, in which a full account of the proceedings of these Meetings shall be entered ; they shall specify all the Donations received, and furnish a list of them, and of the Donors’ names, to the Curator of the Library and Museum ; they shall like- wise furnish the Treasurer with notes of all admissions of Ordinary Fellows. They shall assist the General Secretary in superintending the publications, and in his absence shall take his duty. XXV. The Curator of the Museum and Library shall have the custody and charge of all the Books, Manuscripts, objects of Natural History, Scientific Produc- tions, and other articles of a similar description belonging to the Society ; he shall take an account of these when received, and keep a regular catalogue of the whole, which shall lie in the Hall, for the inspection of the Fellows. XXVI. All Articles of the above description shall be open to the inspection of the Fellows at the Hall of the Society, at such times and under such regulations, as the Council from time to time shall appoint. ».@.@'4 bE A Register shall be kept, in which the names of the Fellows shall be enrolled at their admission, with the date. Auditor, General Secretary’s Duties, Secretaries to Ordinary Meetings. Curator of Museum and Library. Use of Museum and Library. Register Book. THE KEITH, MAKDOUGALL-BRISBANE, NEILL, AND VICTORIA JUBILEE PRIZES. The above Prizes will be awarded by the Council in the following manner :— I. KEITH PRIZE. The Keira Prize, consisting of a Gold Medal and from £40 to £50 in Money, will be awarded in the Session 1887-88 for the “ best communication on a scientific subject, communicated, in the first instance, to the Royal Society during the Sessions 1885-86 and 1886-87.” Preference will be given to a paper containing a discovery. II. MAKDOUGALL-BRISBANE PRIZE. This Prize is to be awarded biennially by the Council of the Royal Society of Edinburgh to such person, for such purposes, for such objects, and in such manner as shall appear to them the most conducive to the promotion of the interests of science ; with the proviso that the Council shall not be compelled to award the Prize unless there shall be some individual engaged in scientific pursuit, or some paper written on a scientific subject, or some discovery in science made during the biennial period, of sufficient merit or importance in the opinion of the Council to be entitled to the Prize. 1. The Prize, consisting of a Gold Medal and a sum of Money, will be awarded at the commencement of the Session 1887-88, for an Essay or Paper having reference to any branch of scientific inquiry, whether Material or Mental. 2. Competing Essays to be addressed to the Secretary of the Society, and transmitted not later than Ist June 1888, 3. The Competition is open to all men of science. APPENDIX—KEITH, BRISBANE, NEILL, AND VICTORIA PRIZES. (Us) 4, The Essays may be either anonymous or otherwise. In the former case, they must be distinguished by mottoes, with corresponding sealed billets, super- scribed with the same motto, and containing the name of the Author. 5. The Council impose no restriction as to the length of the Essays, which may be, at the discretion of the Council, read at the Ordinary Meetings of the Society. They wish also to leave the property and free disposal of the manu- scripts to the Authors; a copy, however, being deposited in the Archives of the Society, unless the paper shall be published in the Transactions. 6. In awarding the Prize, the Council will also take into consideration any scientific papers presented to the Society during the Sessions 1886-87, 1887-88, whether they may have been given in with a view to the prize or not. Ill. NEILL PRIZE. The Council of the Royal Society of Edinburgh having received the bequest of the late Dr Patrick Neu of the sum of £500, for the purpose of “the interest thereof being applied in furnishing a Medal or other reward every second or third year to any distinguished Scottish Naturalist, according as such Medal or reward shall be voted by the Council of the said Society,” hereby intimate, 1. The Netti Prize, consisting of a Gold Medal and a sum of Money, will be awarded during the Session 1888-89. 2. The Prize will be given for a Paper of distinguished merit, on a subject of Natural History, by a Scottish Naturalist, which shall have been presented to the Society during the three years preceding the 1st May 1888,—or failing presentation of a paper sufficiently meritorious, it will be awarded for a work or publication by some distinguished Scottish Naturalist, on some branch of Natural History, bearing date within five years of the time of award. IV. VICTORIA JUBILEE PRIZE. This Prize, founded in the year 1887 by Dr R. H. Gunnin, is to be awarded triennially by the Council of the Royal Society of Edinburgh, in recognition of original work in Physics, Chemistry, or Pure or Applied Mathematics. 716 APPENDIX—KEITH, BRISBANE, NEILL, AND VICTORIA PRIZES. Evidence of such work may be afforded either by a Paper presented to the Society, or by a Paper on one of the above subjects, or some discovery in them elsewhere communicated or made, which the Bue may consider to be deserving of the Prize. The Prize is open to men of science resident in or connected with Scotland. The first award shall be in the year 1887, and shall consist of a sum of money. In accordance with the wish of the Donor, the Council of the Society may on fit occasions award the Prize for work of a definite kind to be under- taken during the three succeeding years by a scientific man of recognised ability. =a “I — ba | — AWARDS OF THE KEITH, MAKDOUGALL-BRISBANE, NEILE, AND VICTORIA JUBILEE PRIZES, FROM 1827 TO 1887. I. KEITH PRIZE. lst Brenniat Periop, 1827—29.—Dr Brews7er, for his papers “on his Discovery of Two New Immis- cible Fluids in the Cavities of certain Minerals,” published in the Transactions of the Society. 2np BienniaL Pertop, 1829-31.—Dr Brewster, for his paper “on a New Analysis of Solar Light,” published in the Transactions of the Society. 3RD BrenNIAL Pertiop, 1831-33—TuHomas GrauaM, Esq., for his paper “on the Law of the Diffusion of Gases,” published in the Transactions of the Society. 47H Bienniat Periop, 1833—35.—Professor J. D. Forsss, for his paper “ on the Refraction and Polari- zation of Heat,” published in the Transactions of the Society. 57H BrenniaL Periop, 1835-37.—Joun Scorr RussE x1, Esq.,for his Researches “on Hydrodynamics,” published in the Transactions of the Society. 67TH Bienniat Periop, 1837-39.—Mr Joun Suaw, for his experiments “on the Development and Growth of the Salmon,” published in the Transactions of the Society. 7ta BienniaL Pertop, 1839—41.—Not awarded. 8TH Brenniat Psriop, 1841-43.—Professor James Davin Fores, for his papers “on Glaciers,” published in the Proceedings of the Society. 9TH BrennraL Periop, 1843—-45.—Not awarded. 107TH Brenntat Periop, 1845—47.— General Sir THomas BrisBane, Bart., for the Makerstoun Observa- tions on Magnetic Phenomena, made at his expense, and published in the Transactions of the Society. llrH Brenniau Pertop, 1847—49.—Not awarded. 127H Brenniat Periop, 1849-51.—Professor Krtuanp, for his papers “on General Differentiation, including his more recent communication on a process of the Differential Calculus, and its application to the solution of certain Differential Equations,” published in the Transactions of the Society. 1378 Brennrat Periop, 1851—-53.—W. J. Macquorn Ranxine, Esq., for his series of papers “on the Mechanical Action of Heat,” published in the Transactions of the Society. 1478 Brenniat Periop, 1853-55.—Dr Tuomas Anpsrson, for his papers “on the Crystalline Con- stituents of Opium, and on the Products of the Destructive Distillation of Animal Substances,” published in the Trans- actions of the Society. 157TH Brenntat Periop, 1855-57.—Professor Booz, for his Memoir “on the Application of the Theory of Probabilities to Questions of the Combination of Testimonies and Judgments,” published in the Transactions of the Society. 16TH Brenntat Periop, 1857—59.—Not awarded. 177H Brenniat Periop, 1859-61.—Jonn Atian Broun, Esq., F.R.S., Director of the Trevandrum Observatory, for his papers “on the Horizontal Force of the Earth’s Magnetism, on the Correction of the Bitilar Magnet- ometer, and on Terrestrial Magnetism generally,’ published in the Transactions of the Society. VOL. XXXIII. PART III. 5M 718 APPENDIX—KEITH, BRISBANE, NEILL, AND. VICTORIA PRIZES. 18ra Brewntat Pertop, 1861—63.—Professor Wiii1am THomson, of the University of Glasgow, for his Communication “on some Kinematical and Dynamical Theorems.” 19H Brenntat Pertop, 1863—65.—Principal Forsss, St Andrews, for his “Experimental Inquiry into the Laws of Conduction of Heat in Iron Bars,’ published in the Transactions of the Society, 20rH Brenntan Pertop, 1865—67.—Professor C. P1azzt1 Smyta, for his paper “on Recent Measures at the Great Pyramid,” published in the Transactions of the Society. 21st Brennraut Pertop, 1867—69.—Professor P. G. Tart, for his paper “ on the Rotation of a Rigid Body about a Fixed Point,” published in the Transactions of the Soeiety. 22npD Brenntat Pertop, 1869—71.—Professor CrerkK Maxwew., for his paper “on Figures, Frames, and Diagrams of Forces,’ published in the Transactions of the Society. 23rp Brennrat Periop, 1871—73.—Professor P. G. Tait, for his paper entitled “ First Approximation to a Thermo-electric Diagram,” published in the Transactions of the Society. 247TH BirnntaL Periop, 1873-75.—Professor Crum Brown, for his Researches “on the Sense of Rota- tion, and on the Anatomical Relations of the Semicircular Canals of the Internal Ear.” 25TH BIENNIAL PERIOD, 1875—77.—Professor M. Forster Heppxz, for his papers “on the Rhom- bohedral Carbonates,” and “on the Felspars of Scotland,” published in the Transactions of the Society. 267TH BrenniaL Periop, 1877—79.—Professor H. C. Fiuremine Jenxiy, for his paper “on the Appli- cation of Graphic Methods to the Determination of the Effi- ciency of Machinery,” published in the Transactions of the Society; Part IT. having appeared in the volume for 1877-78. 271TH BrenniaL Periop, 1879-81.—Professor Groree Curystat, for his paper “on the Differential Telephone,” published in the Transactions of the Society. 28rH brenniau Periop, 1881-83 —Tuomas Muir, Esq., LL.D., for his “ Researches into the Theory of Determinants and Continued Fractions,” published in the Proceedings of the Society. 297H Brexniat Perrop, 1883-85.—Joun ArrKen, Esq., for his paper “on the Formation of Small Clear Spaces in Dusty Air,” and for previous papers on Atmospheric Phenomena, published in the Transactions of the Society. Il. MAKDOUGALL-BRISBANE PRIZE. Ist BiewntaL Perriop, 1859.—Sir Ropprick Impry Murcuisoy, on account of his Contributions to the Geology of Scotland. 2np Brennrat Periop, 1860—62.—Witiiam Sevier, M.D., F.R.C.P.E., for his ‘‘ Memoir of the Life and Writings of Dr Robert Whytt,” published in the Trans- actions of the Society. 3rD Brenniat Periop, 1862--64.—Joun Denis Macponarp, FEsq., R.N., F.R.S., Surgeon of H.M.S. “Tearus,” for his paper “on the Representative Relationships of the Fixed and Free Tunicata, regarded as Two Sub-classes of equivalent value; with some General Remarks on their Morphology,” published in the Transactions of the Society. 4rH Brenntau Periop, 1864-—66.—Not awarded, APPENDIX—KEITH, BRISBANE, NEILL, AND VICTORIA PRIZES, 719 5TH Bienniat Periop, 1866-—68.—Dr ALExanpDER Crum Brown and Dr Tuomas RicHarp Fraser, for their conjoint paper “on the Connection between Chemical Constitution and Physiological Action,” published in the Transactions of the Society. 6TH Bienniau Periop, 1868—70.—Not awarded. 77H Brenntau Periop, 1870—72.—Grorcs James Attman,M.D., F.R.S., Emeritus Professor of Natural History, for his paper “on the Homological Relations of the Ceelenterata,” published in the Transactions, which forms a leading chapter of his Monograph of Gymnoblastic or Tubu- larian Hydroids—since published. 8TH Brenniat Periop, 187 2—74.— Professor Lister, for his paper ‘‘on the Germ Theory of Putre- faction and the Fermentive Changes,” communicated to the Society, 7th April 1873. 9TH BrenniAL Pertop, 1874—76—ALExanpeR Bucuan, A.M., for his paper “on the Diurnal Oscillation of the Barometer,” published in the Transactions of the Society. 107TH BrenntaL Psriop, 1876—78.—Professor ARCHIBALD GEIKIE, for his paper “on the Old Red Sandstone of Western Europe,” published in the Transactious of the Society. 1) 7H Bienniat Periop, 1878—80.—Professor Piazzi Smyru, Astronomer-Royal for Scotland, for his paper ‘“‘on the Solar Spectrum in 1877-78, with some Practical Idea of its probable Temperature of Origination,” published in the Transactions of the Society. 127TH Brenniat Periop, 1880—82.-— Professor James GEIKIE, for his “ Contributions to the Geology of the North-West of Europe,” including his paper “on the Geology of the Farées,” published in the Transactions of the Society. 137H BreyntaL Pertop, 1882—84.—Epwarp Sane, Esq., LL.D., for his paper “on the Need of Decimal Subdivisions in Astronomy and Navigation, and on Tables requisite therefor,” and generally for his Recalculation of Logarithms both of Numbers and Trigonometrical Ratios, —the former communication being published in the Pro- ceedings of the Society. III. THE NEILL PRIZE. ist TrienniaL Periop, 1856—59.—Dr W. Lauper Linpsay, for his paper “ on the Spermogones and Pyenides of Filamentous, Fruticulose, and Foliaceous Lichens,” published in the Transactions of the Society. 2np TripnntAL PEriop, 1859-—62.—RosBert Kays Grevitwz, LL.D., for his Contributions to Scottish Natural History, more especially in the department of Cryp- togamic Botany, including his recent papers on Diatomacez. 3RD TRIENNIAL Periop, 1862—65.—ANprEew CromBik Ramsay, F.R.S., Professor of Geology in the Government School of Mines, and Local Director of the Geological Survey of Great Britain, for his various works and Memoirs published during the last five years, in which he has applied the large experience acquired by him in the Direction of the arduous work of the Geographical Survey of Great Britain to the elucidation of important questions bear- ing on Gevlogical Science. 47H Trisnniau Periop, 1865-68.— Dr Wittiam Carmicuart M‘Intosu, for his paper “on the Struc- ture of the British Nemerteans, and on some New British Annelids,” published in the Transactions of the Society. 720 APPENDIX—KEITH, BRISBANE, NEILL, AND VICTORIA PRIZES. 5a TrieNNiAt Pertop, 1868—71.—Professor Wint1am TurNER, for his papers “on the great Finner Whale ; and on the Gravid Uterus, and the Arrangement of the Foetal Membranes in the Cetacea,’ published in the Transactions of the Society. 67H TRIENNIAL Periop, 1871—74.—Cuaries WiLx14M Praos, for his Contributions to Scottish Zoology and Geology, and for his recent contributions to Fossil Botany. 71H TRIENNIAL Pertop, 1874—77.—Dr Ramsay H. Traquair, for his paper ‘on the Structure and Affinities of Tristichopterus alatus (Egerton),” published in the Transactions of the Society, and also for his contributions to the Knowledge of the Structure of Recent and Fossil Fishes. 8TH Trrenniat Prriop, 1877—-80.—Joun Murray, for his paper “on the Structure and Origin of Coral Reefs and Islands,” published (in abstract) in the Proceedings of the Society. 97TH TRIENNIAL Pertop, 1880—83.—Professor Herpman, for his papers “on the Tunicata,” published in the Proceedings and Transactions of the Society. 107TH Trienn1AL Pertop, 1883-86.—B. N. Peacu, Esq., for his Contributions to the Geology and Palaeontology of Scotland, published in the Transactions of the Society. IV. VICTORIA JUBILEE PRIZE. lst TRIENNIAL Pertop, 1887—90.—Sir Witi1am Tuomson, Pres. R.S.E., F.R.S., for a remarkable series of papers “on Hydrokinetics,” especially on Waves and Vortices, which have been communicated to the Society. PROCEEDINGS OF THE STATUTORY GENERAL MEETING, 22ND NOVEMBER 1886. iad CHOTUTATE srg eras STATUTORY MEETING. HUNDRED AND FOURTH SESSION. Monday, 22nd November 1886. At a General Statutory Meeting, Mr Gray in the Chair. The Secretary read apologies for absence from the Earl of HappineTon, Sir DoucLas MACLAGAN, and Professor M‘INTOSH. The Minutes of last General Statutory Meeting of 23rd November 1885 were read, approved, and signed. The Secretary intimated that Voting Papers as to the Hour of Meeting had been issued to the Fellows of the Society in the following form, with the result now declared :— I. Hour of Meeting to remain as hitherto, . : ; : 110 II. Hour to be changed from 8 P.M. to 4 P.M., : 50 III. One of the Meetings in each Month to be held at A P.M., i other at 8 P.M., 2 ; ; : : : i ’ 83 It was stated on behalf of the Council that occasional extra Meetings, to be held at 4 P.m., would be introduced when a sufficient number of Communications happened to be in the Secretary's hands. The Chairman named Mr ANDREW YouNG and Professor DUNS as Scrutineers of the Balloting Lists. They reported that the following Council was unanimously elected : — Sir Wittr1am THomson, LL.D., F.R.S., President. A. Forzes Irving, Esq. of Drum, Joun Murray, Ph.D., Davin Mityr Homs, LL.D., Professor Sir Dougnas Macnagan, The Hon. Lorp Mactaren, The Rev. Professor Fiint, D.D., Professor Tart, M.A., General Secretary. Professor Sir Wa. Turner, F.R.S., Professor Crum Brown, F.R.S., Apam Gites SmirH, Esq., C.A., Treasurer. ALEXANDER Bucuan, Esq., M.A., Curator of Library and Museum. Vice-Presidents. \ Secretaries to Ordinary Meetings. 724 APPENDIX—-PROCEEDINGS OF STATUTORY MEETINGS. COUNCILLORS. Professor CHRYSTAL. Tuomas Muir, Esq., M.A., LL.D. Professor Dickson. Professor M‘Intosu. Professor SHIELD NicHOLSON. Rosert Gray, Esq. T. B. Spracus, Esq., M.A. ArtTHUR MITCHELL, Esq., C.B. Professor Butcurer, M.A. Srair A. Acnew, Esq., C.B., M.A. Professor M‘Krnpricx, F.R.S. Rosert M. Frreuson, Esq., Ph.D. The TREASURER’S Accounts for the past Session, with the Auditor’s Report thereon, were approved. On the motion of Professor CkuM Brown, the Auditor was reappointed. Mr Fores IRVINE proposed a vote of thanks to the Chairman, which was unanimously agreed to. WILLIAM THomson, Pres. The following Public Institutions and Individuals are entitled to receive Copies of the Transactions and Proceedings of the Royal Society of Edinburgh :— London, British Museum. Royal Society, House, London. Anthropological Institute of Great Bri- tain and Ireland, 3 Hanover Square, London. British Association for the Advancement of Science, 22 Albemarle Street, London, Society House. 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All the Honorary and Ordinary Fellows of the Society are entitled to the Transactions and Proceedings. See Notice at foot of page 730. The following Institutions and Individuals receive the Proceedings only :— SCOTLAND, Edinburgh, Botanical Society, 5 St Andrew Square. Geological Society, 5 St Andrew Square. Scottish Geographical Society. Mathematical Society, 8 Queen Street. Edinburgh, Scottish Meteorological Society, 122 George Street. Pharmaceutical Society, 36 York Place. Geological Society of Glasgow, 207 Bath Street. The Glasgow University Observatory, Berwickshire Naturalists’ Club, Old Cambus, Cockburnspath. APPENDIX. - 729 ENGLAND. London, Geoloyists’ Association, University College. Mathematical Society, Street, London, W. Institution of Mechanical Engineers, 10 Victoria Chamhers, Victoria Street, Westminster. Meteorological Office, 116 Victoria Street. The Meteorological Society, 25 Great George Street, Westminster. Nautical Almanac Office, 3 Verulam Buildings, Gray’s Inn. 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The Lick Observatory, Mount Hamil- ton, va San José, San Francisco. Cincinnati, Observatory. Society of Natural History. Ohio Mechanics’ Institute. Colorado, Scientific Society. Connecticut, Academy of Arts and Sciences. Davenport, Academy of Natural Sciences. New Orleans, Academy of Sciences. New York, The American Museum of Natural History. The American Geographical and Sta- tistical Society, No. 11 West 29th Street, New York. Philadelphia, Wagner Free Institute of Science. Salem, The Essex Institute. Peabody Academy of Science. Trenton, Natural History Society. Washington, Philosophical Society. American Museum of Natural His- tory, Central Park. United States National Museum. Wisconsin, Academy of Sciences, Arts, and Letters. SOUTH AMERICA, Rio de Janeiro, Museu Nacional. Santiago, Deutscher Verwissenschaftlicher Verein. MEXICO. Mexico, Observatorio Meteorologico-Magnetico Central. Sociedad Cientifica, “Antonio Alzate.” Tacubaya, Observatorio Astronomico. NOTICE TO MEMBERS. All Fellows of the Society who are not in Arrear in their Annual Contributions, are entitled to receive Copies of the Transactions and Proceedings of the Society, provided they apply for them within Five Years of Publication. Fellows not resident in Edinburgh must apply for their Copies either personally, or by an authorised Agent, at the Hall of the Society, within Five Years after Publication. TN DES BO“ VOER.* XXX ET: A AITKEN (JouN), F.R.S.E. On Dew, 9. Dew on Grass, 13. Dew on Soil, 19. Dew on Roads, 23. Dew and Wind, 26. Dew and Vegetation, 27. Radiation in relation to Dew, 33, 45. Agonus cataphractus, Ova of, 124. Alcicornopteris convoluta, 152. Alethopteris, 384. ALEXANDER (P.), M.A. Expansion of Functions in terms of Linear, Cylindric, Spherical, and Allied Functions, 313. Alosa sapidissima, Ova and Larva of, 113. Alternating Functions. By Taomas Muir, LL.D., FE.R.S.E., 309. Amiurus albidus, Eggs of, 109. Ammonium, Chloroplatinate Method for the De- termination of. By Professor W. Ditrmar, F.R.SS. L. and E., and Joun M‘Arruour, 561. Ammotrypane aulogastra, 654. Ampharete gracilis, 659. Amphicora, 670. Amphitrite, 661. Anarrichas lupus, Ova of, 125. Anguilla, Reproduction of, 113. Annularia, 343, 344. Antennarius, Ova of, 123. Arenicola, 648. Argyropelecus henigymnus, Eggs of, 111. Aricie naidine, 635. Aricude, 642. Arius, Eggs of, 109. Arnoglossus megastoma, 118. Aspredo batrachus, Eggs of, 109. Atherinichthys, the Young of, 125. Axiothea, 678. B Batrachus tau, Young of, 122. Bdellostoma, the Reproductive Organs of. CunnineHam, F.R.S.E., 247. By J.T. Brepparp (Frank), M.A., F.R.S.E. On the Minute Structure of the Eye in certain Cymothoide, 443. Belone longirostris or truncata, Eggs and Larva of, 110. Blennius galerita, Ova of, 125. pholis, Ova of, 125. Brook (Grorce), F.R.S.E. The Formation of the Germinal Layers in Teleostei, 199. BurnsipE (Professor W.). On the Partition of Energy between the Translatory and Rotational Motions of a set of Non-Homogeneous Elastic Spheres, 501. C Calamocladus, 343. Callichthys, Eggs of, 109. Callionymus lyra, Ova of, 124. Calymmatotheca affinis, 145. asteroides, 148. bifida, 140. Capitella, 679. Capros, Ova of, 120. Carassius auratus, Eggs of, 109. Cardiocarpus, 403. Carpolithus, 404. Caulopteris, 392. Centronotus gunnellus, Ova of, 125. Cheetozone, 647. Chone, 669. Cirratulade, 643. Clupea harengus, Eggs and Larva of, 97, 112. sprattus, Eggs and Larve of, 112. Cocconeidece, 432. Colours of thin Plates. By Lorp Rayueicu, 157. Conger, Reproduction of, 112. Copper, Iron, and German Silver, Thermal Con- ductivity of. By A. Cricnton MirowE.t, 535. Cordaittes, 402. Coregonus oxyrhynchus, Eggs and Larva of, 112. 732 Corynepteris, 381. Cottus gobio, Eggs and Larvee of, 123. quadricornis, Eggs and Larve of, 123. scorptus, Eggs of, 103. Cryptoraphidiee, 440. Ctenolabrus adspersus, Ova and Larve of, 128, 129. Cunnineuam (J. 'T.), B.A., F.R.S.E. The Eggs and Larve of Teleosteans, 97, 247. ——— The Reproductive Organs of Bdellostoma, and a ‘T'eleostean Ovum from the West Coast of Africa, 247. CunnineHaM (J. T.), BA. F.R.S.E., and Ramace (G. A.) On the Polycheta Sedentaria of the Firth of Forth, 635, Curves. The Determination of the Curve, on one of the Coordinate Planes, which forms the Outer Limit of the Positions of the Point of Contact of an Ellipsoid, which always touches the three Planes of Reference. By G. Puarr, Docteur-és-Sciencés, 465, Cybium maculatum, Ova and Larva of, 122. Cyclopterus lumpus, Eggs and Larva of, 104. Cyclotella antiqua, 440. Cylindric Functions. By P. Atexanper, M.A., 313. Cymbellee, 421. Cymothoide, On the Minute Structure of the Eye in certain. By Frank E. Brpparp, M.A., F.R.S.E., 443. Cyprinodontide, Larva of, 110. Cyprinus idus, Eggs and Larva of, 109. lenciscus rutilus, Eggs and Larva of, 109. D Dacetylotheca, 381, 382. Deutocaulon, 461. Dew. By Joun Aitken, F.R.S.E., 9. Dew-drop, 56. Diatomaceous Deposit from North Tolsta, Lewis. By Joun Rarrray, M.A., B.Se., F.R.S.E, 419. Dicksonites, 383. Dictyopteris, 361. Ditrmar (Professor W.), F.R.SS. L. and E., and Fawsirr (Cuartes A.). A Contribution to our Knowledge of the Physical Properties of Methyl- Alcohol, 509. Dirrmar (Professor W.) F.R.SS. L. and E., and M‘Artuur (Jonny), F.R.S.E. Critical Experi- ments on the Chloroplatinate Method for the Determination of Potassium, Rubidium, and Ammonium; and a Redetermination of the Atomic Weight of Platinum, 561. Dodecaceria, 647, 648. INDEX. E Electrical Properties of Hydrogenised Palladium. By Careitt G. Kwort, Professor of Physics, Tokayo, Japan, 171. Lpithemia, 433. Ereutho Smitti, 666. Esox, Eggs and Larva of, 111. Eucalamites, 340, Eumenia crassa, 655, —— Jeffreysit, 655, Eunotia, 435. Excipulites callipteridis, 339. F Fawsirr (CHares A,), and Dirrmar (Professor W.) F.R.SS, L. and E. A Contribution to our Knowledge of the Physical Properties of Methyl- Alcohol, 509, Ferns from the Carboniferous Formation—their Fructyication. By Roserr Kinston, 137. Fierasfer, Ova and Larva of, 116, 119, 122. Filigrana, 672. Flabelligera, 676. Flora (Fossil) of the Radstock Series of the Somerset and Bristol Coal Field (Upper Coal Measures). By Rosert Kinston, F.R.S.E., 335. Fossil Flora of the Radstock Series of the Somerset and Bristol Coal Field, Parts I. and II. By Rozsert Kipston, F.R.S.E., 335. Fow.er (Dr G. H.), B.A., and Marspatn (Dr A. Mines), F.R.S. Report on the Pennatulida dredged by H.M.S. “ Porcupine,” 453. Fragilaria virescens, 437. Functions (Alternating). By Tuomas Murr, LL.D., F.R.S.E., 309. —— Expansion of Functions in terms of Linear, Cylindric, Spherical, and Allied Functions. By P. Avexanper, M.A., 313, Fundulus nigrofasciatus, Larva of, 110. Funiculina, 460. G Gadus eglefinus, Eggs and Larva of, 102, 114, 117. merlangus, Eggs and Larva of, 114. —- morrhua, Eggs and Larva of, 114. Gastrosteus, Ova of, 126. German Silver, Copper, and Iron, Thermal Con- ductivity of. By A. Cricuton Murcuet, 535. Gobius minutus, Ova of, 124, Ruthensparri, Ova of, 124. Gomphonemec, 427. INDEX. H Hawlea, 378. Heliasis chromis, Ova of, 127. Hemitripterus Americanus, Ova and Larve of, 119. Hippocampus, Embryo of, 129. Hydrogenised Palladium. See Palladium. Hydrogenium, Electrical Resistance of, 171. Electromotive Force of, 181. iL Instability in Open Structures. LL.D., F.R.S.E., 321. Iron, Electrical Resistance of, 195. Iron, Copper, and German Silver, Thermal Con- ductivity of. By A. Cricaron Mircuett, 535. By Epwarp Sane, J Julis vulgaris, Ova of, 128. K Kinston (Rosert), F.R.S.E. On the Fructification of Ferns from the Carboniferous Formation, 137. On the Fossil Flora of the Radstock Series of the Somerset and Bristol Coal Field (Upper Coal Measures), Parts I., II., 335. Kinetic Theory of Gases. By Professor Tart, 65. Part II., 251. Kwort (Carcitu G.), F.R.S.E., Professor of Physics, Imperial University, Tokayo. On the Electrical Properties of Hydrogenised Palladium, 171. —— The Electrical Resistance of Nickel at High Temperatures, 187. Kophobelemnon, 460. L Labrax lineatus, Larvee of,118. Lacuome, 669. Lanice, 663, 664. Lepadogaster Decandolit, Ova of, 126. Lepidodendron, 393. Lepidostrobus, 395, Leucadore, 641. Linear Functions. By P. Auexanpsr, M.A., 313. Liparis Montagui, Eggs and Larva of, 103. LIipobranchius, 655. | Lophius piscatorius, Eggs and Larve of, 122, 123. Lota vulgaris, Eggs and Larve of, 115. Lycopod macrospores, 401. M M‘Arrnur (Jonny), and Dirrmar (Professor W.) See Dirrmar (Professor W.) | VOL. XXXIII. PART III. 733 M‘Laren (The Hon. Lorn), F.R.S.E. Tables for Facilitating the Computation of Differential Refraction in Position, Angle, and Distance, 279. Macrosphenopteris, 352. Macrostachya, 344. Magelona papillicernis, 642. Mariopteris, 363. MarsHatt (Dr A. Miuysgs), F.R.S., and Fow.er (Dr G. H.), B.A. Report on the Pennatulida dredged by H.M.S. “ Porcupine,” 453. Megaphyton, 390. Melinna, 659, 660. Merlucius, Ovum of, 115. Methyl-Alcohol, the Physical Properties of. By Professor W. Dittmar, F.R.SS. L. and E., and Cares A. Fawsitt, 509. MircneLt (A. Cricuton). Thermal Conductivity of Iron, Copper, and German Silver, 535. Motella argentea, the Young of, 114. Eggs and Larva of, 115. cimbria, Ova of, 115. — mustela, Eggs and Larva of, 114, 115, Muir (Tomas), LL.D., F.R.S.E. On a Class of Alternating Functions, 309. Myzxicola, 671. N Naviculee, 422. Nerine, 636. Nerophis, Ova of, 129. Neuropteris, 354. Neuropteris heterophylla, 150. Nickel, Electrical Resistance of, 189-192, 195. Nicomache, 678. Notomastus, 681. Odontopteris, 363. Ophelia, 654. Osmerus eperlanus, Eggs of, 98, 112. mordaz, Ova of, 112, 113. Owenia, 656. le Palladium, On the Electrical Properties of Hydro- genised. By Carami G. Knorr, F.R.S.E., Professor of Physics, Tokayo, Japan, 171. — Electromotive Force of, 181. Electrical Resistance of, 190, 191, 195. Pecopteris, 365, 387. Pectinaria belgica, 656. | Perca fluviatilis, Ova and Larva of, 118. 50 734 Pennatulida dredged by H.M.S. “ Porcupine.” By Dr A. Mitnes MarsHatt, F.R.S., and Dr G. H. Fowter, B.A., 453. Prarr (G.), Docteur és-Sciencés. On the Deter- mination of a Curve, on one of the Coordinate Planes, which forms the Outer Limit of the Positions of the Point of Contact of an Ellipsoid which always touches the three Planes of Reference, 465. Plates, Colours of Thin. By Lorp Rayueicu, 157. Platinum, Electrical Resistance of, 190, 195. a Redetermination of the Atomic Weight of. By Professor W. Dirrmar, F.R.SS. L. and E., and Jonn M‘Artuour, 561. Platinum-Hydrogenium, Electromotive Force of, 181. Pleuronectes Americanus, Eggs and Larve of, 117. cynoglossus, Eggs and Larva of, 101. —— flesus, Eggs and Larva of, 99, 117. limanda, Eggs and Larva of, 100, 117. microcephalus, Eggs of, 102. —— platessa, Eggs of, 99, 117. Polyacanthus viridiauratus, Ova of, 127. Pomatocerus, 673, Porcupine (H.M.S.). See Pennatulida. Potassium, Chloroplatinate Method for the De- termination of. By Professor W. Dittmar, F.R.SS. L. and E., and Jonn M‘Arruovr, 561. Proptoptilum Carpenteri, 463. Pseudoraphidier, 433. Pseudorhombus, Eggs and Larve of, 117, 128. Ptychocarpus, 350. R Radiation in Relation to the Disposition of Dew. By Joun Arrken, F.R.S.E., 33. Radstock Series of the Somerset and Bristol Coal Field, Fossil Flora of. By Rogerr Kinston, F.R.S.E., 335. Ramace (G. A.), and Cunninenam (J. T.), BA, F.R.S.E. On the Polycheta Sedentaria of the Firth of Forth, 635. Rattray (Joun), M.A., B.Sc, F.R.S.E. On a Diatomaceous Deposit from North Tolsta, Lewis, 419. RayLEicH (Lorp). 157. Refraction. Tables for Facilitating the Computation of Differential Refraction in Position, Angle, and Distance. By the Hon, Lorp M‘Largn, 279, Rhabdocarpus, 404, Rhacophyllum, 387. Rotation of a Sphere about each of the Principal Axes, 505, On the Colours of Thin Plates, INDEX. Rubidium, Chloroplatinate Method for the Deter- mination of. By Professor W. Duirrmar, F.R.SS. L. and E., and Jonn M‘Arruour, 561. Ss Sabellaria, 652. Sabellide, 668. Salmo, Eggs and Larva of, 111. levenensis, Larva of, 98. Sane (Epwarp), LL.D., F.R.S.E. On Cases of Instability in Open Structures, 321. Schizostachys, 351. Scione maculata, 665. Scolecolepis, 639. Scoleples armiger, 642. Scomber, Ova of, 121. Scorpenda, Ova of, 118. Serolide, Structure of the Eye in. By Frank E. Bepparp, M.A., F.R.S.E., 448, 451. Serpula, 672. Serranus cabrilla, Ova of, 118. Sigillaria, 396. Siluride, Eggs of, 109. Siphonostoma typhle, Ova of, 129. Solenostomide, Ova of, 129. Sorocladus antecedens, 143. Sphenophyllum, 344. Sphenopteris, 345. Spheres, The Partition of Energy between the Trans- latory and Rotational Motions of a Set of Non- Homogeneous Elastic Spheres. By Professor W. Burnsipe, 501. The Averages of Rotation of a Sphere about each of the Principal Axes are equal, 505. The whole Average Energy of Rotation of a Sphere is twice the Average Energy of Translation, 505. —— One Set of Equal Spheres. 67, —— Mean Free Path among Equal Spheres. Professor Tart, 71. Spherical Functions. 313. Spinachia vulgaris, Ova of, 126. Spio, 640. Spionide, 635, 642. Spiropteris, 387, Spirorbis, 674. Spongelia, Note on Peculiar Capsules found on the Surface of. By J. AntHuR THomson, F.R.S.E., 241. Sporangia, 400. Stauroneis, 427. Stigmaria, 401. By Professor Tair, By By P.° Avexanper, M.A., INDEX. Stromateus triacanthus, Young of, 121. Structures, Cases of Instability in Open. By Epwarp Sane, LL.D., F.R.S.E., 321. Stylocalamites, 342. Suberites domuncula. By J. AntHuR THomson, 241. The Ciliated Chambers, 242. The Connective Tissue, 242. Reproductive Elements, 243. Surtrellece, 438. Svava glacialis, 459. Synedra ulna, 437. Syngnathus, 129. T Tabellaria, 438. Tait (Professor), Sec. R.S.E. The Kinetic Theory of Gases, 65. I. One Set of Equal Spheres, 67. II. Mean Free Path among equal Spheres, 71. IlI. Number of Collisions per Particle per Second, 75. IV. Clerk-Maxwell’s Theorem, 77. V. Rate of Equalisation of Average Energy per Particle in two mixed Systems, 82. . On some Definite Integrals, 84. . Mean Path in a Mixture of Two Systems, 86. Pressure in a System of Colliding Particles, 65. IX. Effect of External Potential, 91. Part II.—The Foundations of the Kinetic Theory of Gases, 251. VIL. X. On the Definite Integrals, fF and 0 fy r vw ay LOO €, + 2, 0 XI. Pressure in a Mixture of two Sets of Spheres, 258, XII. Viscosity, 259. XIII. Thermal Conductivity, 261. XIV. Diffusion, 266. Appendix. Table of Quadratures, 277. 735 Tarr (Professor), Sec. R.S.E. Introduction to Mr A. Cricaton MitcHett’s Paper on the Thermal Conductivity of Iron, Copper, and German Silver, 535. Tautoga onitis, Ova and Larva of, 128. Teleosteans, Eggs and Larva of, 97, Maturation and Fertilisation of the Teleostean Ovum, 130. By J. T. Cunnineuay, F.R.S.E. Teleostean Ovum from Africa. By J. T. Cunnine- HAM, F.R.S.E., 247. Teleostei, The Formation of the Germinal Layers in. By Grorce Brook, F.R.S.E., 199. The Ripe Unfertilised Ovum, 199. Viscous Layer of Ovum, 200. Formation of the Germinal Mound, 206. Segmentation, 213. The Part played by the Parablast, 222. Temnodon saltator, Ova and Larva of, 120. Terebella, 663, 665. Terebellides, 667. Thelepus, 665. Thelethuside, 648. Theodisca mamelata, 642. THomson (J. ARTHUR). On the Structure of Sube- rites domuncula, 241. —— Note on the Peculiar Capsules found on the Surface of Spongelia, 241. Thymallus, Ova of, 112. Tolsta Diatomaceous Deposit. M.A., B.Sc., F.R.S.E., 417. Trachinus vipera, Ova and Larva of, 122. Trigla Gurnardus, 122. Trigonocarpus, 403. Trophonia, 674. By Joun Ratrray, Tungsten, the Atomic Weight of. By Joun Waovpe.t, B.A., D.Sc., F.R.S.E., 1. Ww WavDELL (Joun), B.A., D.Sc., F.R.S.E. The Atomic Weight of Tungsten, 1. Z Zeilleria Avoldensis, 148. Zoarces viviparus, Ova of, 125. PRINTED BY NEILL AND COMPANY, EDINBURGH. = 7 . aot P i ‘5 se oR pill ye ganna. Seg. syepingtlia- we mine b+ ; io > The Transactions of the Royat Society or EpinsurGH will in future be Sold at the following reduced Prices :— Pri 1 i Price to th Price to > Vol. Pabhie. 4 Felice Vol. Public. : Fellows. I. II. III. | Out of Print. XIV. Part 1.) 21° 5b 0 le 0) IV. Sol eS 0) cere Te 0 > wearti2: Poe A) Ie SiG iV: 011 0 Oresen0 35. bart 3; 110 0 sis 0 VE 0-16 0 946 SROVee bart le 018 O 0138 6 WIL. | 87180 015 0 , Part2.| 2 2 0 Ltr’ © VIII. Ce 014 0 XXVI. Part 1. 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