a tae vile ~ a) sf yd eae on, i J _ MEMOIRS OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. ie 4 MEMOIRS OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. —_—— SECOND SERIES. FIFTEENTH VOLUME. LONDON : H. BAILLIERE, 219, REGENT STREET, AND 290, BROADWAY, NEw YORK. PARIS: J. BAILLIERE, Ruz HAUTEFEUILLE. 1860, fAlanchester : JBrinted by Charles Simms & Cn. 1S NPE The Authors of the several Papers contained in this Volume, are themselves accountable for all the statements and reasonings which they have offered. In these par- 4 ERRATA. Page 69, lines 4 and 6 (from bottom) ; for “5” read “ 4,” 70, — 9and11; for “7” read “6.” —— 71, — 6and8; for “9” read “8.” — 71, — 18 and 20; for “11” read “10.” — 73, — 2and 4; for “13” read “12.” —— 73, — 17 and 18; for “15” read “14.” —— 76, line 13; for “their” read “ these.” —— 84, — 14 (from bottom); for “then” sead “there.” —— 89, — 10; for “worked” read “ marked.” —— 132, — 1 of § 4; for “(1)” read “the general quintic.” —— 140, — 7 of § 27; omit “—-5” and “+K.” —— 140, — 9 of § 27; omit “and K is known.” CONTENTS. ARTICLE I, —On the Composition and Derivation of Rosolic Acid. By R. Anaus Smira, Ph. D., FoR. S. ........c0ceerereecesss II.— On the Consumption of Coal in Furnaces, and the Rate of Evaporation from Engine Boilers. By Mr. JOHN TIT. — On the Triedral Partitions of the X-ace, and the Tri- angular Partitions of the X-gon. By the Rey. THo. P. Kirxman, A.M., F.R.S., Rector of Croft with Southworth, and Honorary Member of the Literary and Philosophical Society of Manchester............... IV.— On Improvement in Meteorological Registration. By Mr. THOMAS HOPEINS......,....c0ccceecensee eee neeeeeneeens YV.— On the General Solution of the Problem of the Polyedra. By the Rev. Tuo. P. Krrxmay, M. Ae Bases Rector of Croft with Southworth, and Honorary Member of the Literary and Philosophicol Society GH MANCHESGER Ce ccconcccecs env -ccrerenescacocnsroncseesbanss VI.— Experiments to determine the Properties of some Mix- tures of Cast Iron and Nickel. By Wtttam WATRBATE NH, Bis S-tectencts -ddescocnsec«soecdesamecsnsantec. VII.— On the Hardness of Metals and Alloys. By F. Cracr Catvert, M.R.A. of Turin, F.C.8., and Ricnarp POHNAON, HOLS. 2.06. ioeisane.cecness oveceossnseecsorerscesee VIII. — On a Yellow Colouring Matter obtained from the Leaves of the Polygonum Fagopyrum or Common Buckwheat. By Epwarp Scuunox, Ph.D., F.R. 8. IX.— Researches in the Higher Algebra. By James Cockuz, M.A., F.R.A.S., F.C.P.S., Barrister-at-Law. Com- municated by the Rev. R. Hartry, F.R.AS. ......... X. —Note on Dalton’s Determination of the Expansion of Air by Heat. By J. P. Journ, uO De <)> Bs XI. — On the Utilization of the Sewage of London and other large towns. By J. P. Journ, LL.D., E.R.S., &e.... PAGE 43 75 93 104 113 122 131 143 146 viii CONTENTS. ARTICLE PAGE XII. — An Account of the Fall of Rain at Manchester, from the Year 1786 to 1857 inclusive. By Mr. Joun Curris.. 161 XIII. — Proposed Improvements in Pharology. By Ricnarp RoBEErs Vi Wnsts O.GN asc cccs-nscgtestvssesmaecteeeeenam 166 XIV. — On the Method of Symmetric Products, and its Applica- tion to the Finite Algebraic Solution of Equations. By the Rey. Ropert Hartey, F.R.AS. ...........0.0 172 XV.— On the Partitions and Reticulations of the R-gon. By the Rev. Tuo. P. Kirkman, M.A., F.R.S., Rector of Croft-with-Southworth, and Honorary Member of the Literary and Philosophical Society of Manchester...... 220 XVI.— A Contribution to Urology, embracing Observations on the Diurnal Variations in the Acidity of the Urine, chiefly in relation to Food. By Wixtiam RoBERtTs, B.A., M.D., Physician to the Manchester Royal EN PMNGT Ye in sso sok casaensevoslgrciti yes coe edrwomabder wate ee eae 238 APPENDIX. — PROCEEDINGS OF THE Microscopical SECTION OF THE LirzRary AND PurtosopHroat Society. Established 1859. Inaugural Address. By Professor C. Wini1AMson, F.R.S., President Ofte PECLION, « sieernssexcgescssaeaskeacee ene 301 On “Daphnia Pulex.” By Mr. Gro. Mostey............... 311 On the “ Structure of Starch Granules.” By Mr. H. A. ENC AG) Neve scene sso spaces eeaWeisacitsicibse eget eece ietae 313 Donations received. Sessions 1857-8 and 1858-9. Council and Members of the Society, 20th April, 1858, and 19th April, 1859. LIST OF ILLUSTRATIONS. Diagram showing the Fall of Rain in Manchester from 1794 to ESD. nos cc. etvna tennessee seevomees cour euteence to face page 164 Tables of the Fall of Rain in Manchester from 1786 to 1857...... 164 Plans of Mr. Roberts’s Floating Lights..................sseseceseveenes 169 i a MEMOIRS OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. I. — On the Composition and Derivation of Rosolic Acid. By R. Anecus Suiru, Pu. D., F.R.S. Read December Ist, 1857. Wuen engaged in making an examination of ammoniacal liquor from the Manchester Gas Works, I found that a solution, from which the sulphur united with the ammo- nium had been removed, became, after a week or two, of a very beautiful rose colour. The amount of substance necessary to produce this colour in a portion of liquid con- tained in a beaker glass, was extremely small, scarcely capable of being seen, much less of being examined care- fully. To have made use of portions of the gas water sufficiently large to obtain enough for examination would have required operations of very inconvenient magnitude. Some time after this, in 1855, Mr. Mc. Dougall found that the lime which was exposed to the currents of certain vapours in his works became red. These vapours arose VOL. XV. B 2 DR. ANGUS SMITH ON THE COMPOSITION from the operation of manufacturing the disinfecting pow- der called by his name, where carbolic acid or kreasote from coal is used. When he observed this red colour he renewed the surface of the lime and obtained a fresh portion. By continual renewals in this manner he ob- tained a crust of a very deep colour. This he brought to me. There were several pounds weight of it; but al- though the colour appeared so deep, a very small portion of colouring matter was found. When this lime compound was heated with alkalies, a portion of the colouring matter was removed. I used ammonia, but soda or potash answered the same purpose. It was, however, not possible, or at least it was exceedingly tedious, to remove the whole in this manner, and when the lime was dissolved by muriatic acid, a black resin was found first swimming in a melted state in the rather warm liquid, and afterwards sinking and cooling, a dark resinous substance. The power of giving colour was remarkable, as I did not obtain more of this resin than about an ounce to a hundredweight of the lime compound. Of course the colour in these cases led me to think of Runge’s rosolic acid, which he has described in 1834 in Poggen- dorff’s Annalen (vol. xxxi. p. 70) in the following manner : ** Rosolic acid, or rosadleic acid (Rosolsaure oder Rosadl- siure), is a product of the chemical decomposition of coal oil, and is so much the more remarkable on account of acting like a true pigment. With a suitable mordant it gives colours and lakes before which those of safflower, eochineal and madder must give way. “ Rosolic acid is a resinous mass, which may be reduced to a powder; it possesses a beautiful orange colour. I have not been able to find in the coal oils the substance from which it is produced; but to show its existence it is only necessary to shake the oil with milk of lime, to filter, and to boil for a few hours. The solution, at first colour- ——— AND DERIVATION OF ROSOLTC ACID. 3 less, becomes of a dark red colour. This colour arises from the rosolate of lime which, on standing, is precipi- tated as a highly coloured red powder.” From this it appears that Runge did not find its deriva- tion, although it has generally been put as an appendix to carbolic acid. In separating this resin from the lime by means of very dilute muriatic acid, there was at first a separation of fine light brown flakes; these at last gradually fell and united at the bottom of the vessel into a resinous mass. When this was dissolved in alcohol or ether, the solution had a deep brown colour, but no appearance of the rose colour; neither had it in any stage the beautiful orange colour mentioned by Runge. It is a question, there- fore, whether it is right to give it the same name; but in many particulars it is so like it that I have made no alteration. When treated with alkalies these shades of brown, or reddish-brown, gave way before the peculiar rose colour. The finest colour in solution appeared to me to be that with ammonia; in a solid state, the finest was certainly that of the lime compound. In order to find exactly its origin, I purified some car- bolic acid made from coal by heating it with caustic soda, decomposing with acid and distilling. OF COAL AND RATE OF EVAPORATION. p 4) | Fire-places each 3 feet by 6 feet, giving 18 square feet of fire surface. Internal flue 3 feet 9 inches in diameter. Fire-bars half inch thick with half inch spaces. The boiler plate 3**s of an inch in thickness. The experiments made with this boiler were continued under difierent circumstances and at various times of the year, and the average result obtained, working under the best conditions I could command, was 5.90 lbs. of water evaporated from the temperature of 84° by oue pound of coal consumed. Or if we add one-sixth to this result as equivalent to the heating of the feed water, the ultimate result will stand thus : In the breeches boiler 6.88 lbs. of water are evaporated from 212° by one pound of coal. In all the experiments made with this boiler, though my results were affected more or less by what was considered to be a slightly deficient draught, still each fire evaporated 24.429 Ibs. of water from 60° in the course of twelve suc- cessive working hours, and generally throughout the series it was found that this boiler was more affected by disturb- ing causes than the boilers of other shapes, possibly in consequence of the comparatively small size of the fire- places. The smallness of the yield from this boiler is a matter of surprise. I have, however, uo reasonable grounds for suspecting the accuracy of my experiments, since they were so often repeated and under such varied circum- stances. VI, Series or EXPERIMENTS. I may add that in a short series of experiments made on two other boilers of an analogous construction, namely, with two fire-places and two internal flues passing from 22 MR. JOHN GRAHAM ON CONSUMPTION end to end, (a construction of boiler which greatly recom- mends itself for strength, durability, compactness, and simplicity of form,) working with an excellent draught, and under a pressure of 57 lbs. on the inch, I received a con- firmation of the importance of disturbing causes. Both boilers were reported to me to be precisely the same, but that one yielded from the same weight of coals in the same time about one-third less steam than the other. However, it turned out that there were two slight differ- ences. The duration of the experiments, the quantity of coals consumed, and the draught, were the same for each boiler. No. 1. 7ft. diameter, 34ft. long, plates 3S of an inch thick. No. 2. Do. do. do. No. 1. Surface of fire-bars, 2ft. 8in. by 5ft. 6in. = 14.6 sq. ft. No. 2. Do. 3ft. by 6ft. = 18.0 sq. ft. No. 1. Evaporated 6.36 lbs. of water per pound of coal. No. 2. Do: 8.00 Ibs. do. do. No. 1 was then opened and found to be dirty and slightly scaled, No. 2 being clean. On No. 1 being thoroughly cleaned and again experimented upon a higher result was obtained, thus: No. 1 evaporated 7.39 lbs. of water per pound of coal. I could discover no other disturbing cause to account for the small yield from No. 1 beyond the comparative smallness of its fire-places, and accompaniment of dirt and slight scale. The evaporation is computed from water at the boil. VIL. Series of Experiments. The boiler next experimented with was of the form as shown in the drawing usually called James Watt’s “ wag- gon-shaped boiler.” It was 26 feet 6 inches long by 6 feet 6 inches in diameter. Fire-place 5 feet 6 inches by 6 feet, equal to 33 square feet of fire surface. OF COAL AND RATE OF EVAPORATION. 23 Fig. 9. GLELEILEL. Scale of feet. ee eee 12, 630 6 Feet Bars 16 inches from the top of the arch of the bottom of the boiler, half inch thick, with half inch spaces. Plates ~°;ths of an inch thick. Depth of water over fire 4 feet. A flue carried round the boiler, with capital draught. A corresponding series of experiments being made with this boiler gave a different result from that obtained with the last boiler; 8.80 lbs. of water from the temperature of 60° were evaporated by one pound of coal consumed; or, if we add one-sixth as equivalent to the heating of the feed water, the result will stand thus: Jn the waggon-shaped boiler 10.26 lbs. of water were evaporated from 212° by one pound of coal. In consequence of the facility with which dirt settles in the lower bends of this boiler it is more liable to be burned 24 MR, JOHN GRAHAM ON CONSUMPTION at these parts than any of the boilers of other shapes I have experimented with. Indeed this circumstance forms an objection of some weight to the boiler. The only mode I have tried to obviate this difficulty, was to place two blow- _ off pipes, one in each bend, extending at least as far as the length of the fire, with numerous holes in their under sides so that the dirt could be sucked in and discharged from the boiler. I have observed, apart from the injurious effects of dirt or scale, that any substance touching the boiler where it is exposed to great heat, such as cast iron, bricks, mortar &c., will cause, sooner or later, a serious deterioration of the strength of the plates at these parts. If these points of contact are excited by the presence of moisture, the deterioration proceeds with the greatest rapidity. VIII. Series or Experiments. Fig. 10. Se eae oS. fect 7260 1 2 3 : 5 ket The third boiler experimented with was a plain cylinder, 33 feet long by 5 feet 6 inches in diameter. a i eee OF COAL AND RATE OF EVAPORATION. 25 Fire-place 4 feet 8 inches by 6 feet, equal to 28 square feet. Fire-bars distant 12 inches at the front and 13 inches at the back from the bottom of the boiler. Bars half inch thick with half inch spaces. Boiler plate 38;*»s of an inch thick. Flame bed 5 inches from the boiler. There was no flue, but the flame-bed was carried up nearly to the water-mark. The draught was thereby re- duced to a thin and rapid current of 5 inches deep; under such circumstances the surface of the bottom of the boiler does not in any case coat with soot. Quantity of water evaporated from a temperature of 60°. lbs. When this boiler was worked at its ordinary speed, the average result from one pound of coal was, wita a fire kept’ anches; thick <2... 5.4ss0..-2-4s.: 5.60 Ditto 7 WHCHES ERICK. 6.c0022s020 jbres Oto Ditto Winches thick: .777....tha- tesd ess 5,93 When worked briskly, and apparently pushed with i MRC RIMEECS cin cc ss teacces] ¢o cats vemecnetne tat ay av bao 6.09 When in a dirty condition within, from a slight scale of sulphate of lime about the thickness of a sixpence or less, but in other respects under ordinary circumstances with a 7-inch fire......... 5.50 When made perfectly clean and keen internally by means of muriatic acid, with a 7-inch fire ... .. 6.45 This unusual activity produced by the muriatic acid con- tinued only for a day or two. The boiler by that time, with the water used, became glazed or smoothed, and the keen- ness was destroyed. This boiler at its ordinary speed evaporated, in 12 con- tinuous working hours, 27,267 lbs. of water from the tem- perature of 60°, and when pushed 38,678 lbs. The evaporating power of this boiler may therefore be VOL. XV. E 26 MR. JOHN GRAHAM ON CONSUMPTION considered as 6.09 Ibs. of water evaporated by one pound of coal. Or by adding one-sixth to that quantity for the heating of the feed water from 60° to 212°, the evaporating power will be 7.1 lbs of water evaporated by one pound of coal. I have already stated that as these experiments pro- ceeded, the deviations from uniformity became less per- plexing, and when such deviations occurred they were more readily accounted for. The following short table is given to illustrate the degree of accordance of such ob- servations. Table D. Water evaporated from 60° per pound of coal burned. By Thick Fires. By Thin Fires. Lbs. Observations, Lbs. Observations. 5.70 Monday. 5.63 6.20 5.65 5.97 5.74 6.04 5.57 Saturday. 5.79 5.85 Monday. 5.90 5.57 5.51 Monday. 5.36 Dirty coals. 5.93 5.64 5.20 New fireman. 6.00 Old fireman. &e. &e. IX. Series or EXPERIMENTS. Fig. 12. Scale of feet 20123. 5 20 25 let The next boiler experimented with was also plain and cylindrical, 42 feet long and 6 feet in diameter. OF COAL AND RATE OF EVAPORATION. 27. Fire-place 5 feet 1 inch by 6 feet 11 inch=35.1 square feet of fire surface. Fire-bars placed 14 inches from the bottom of the boiler at the front, half inch thick with half inch spaces. Boiler plate 35;t8s of an inch thick. Flame-bed, which was carried up to near the water mark, 6 inches from the bottom of boiler. No return flue. Water evaporated from 60° by one pound of coal. Ib When worked with the fire-bars 12 inches from the ss bottom of the boiler, the yield was ....... ee 6.14: With an arrangement to cause the draught to impinge against the boiler bottom................. 6.30 By lowering the bars 2 inches, and under ordinary Cee ats) 121 72): Ee er ea a 6.46 By the introduction of a series of half-bridges, whereby the draught was dispersed and caused fa impinge im all. directions,....6. ,.../2Psias ce vaie-as 6.69 By lowering the bars 114 inches more (whole dis- tance 254 inches), under ordinary circumstances 5.11 This boiler had been fitted up previously to these expe- riments, and worked for some time as seen in Fig. 14. 28 MR. JOHN GRAHAM ON CONSUMPTION It had been thus fitted to meet theoretical considera- tions in connection with smoke burning. When altered to the condition as seen in figures 12 and 13, a saving ac- crued of at least 30 per cent. on the consumption of coal, as practically estimated. The evaporating power of the present boiler, which is of the same class as the last, but greater in diameter and longer, may be considered as 6.46 lbs. of water at 60° evaporated by one pound of coal. Or adding one-sixth for the heating of the feed water from 60° to 212°, the power will be 7.54 lbs. of water eva- porated by one pound of coal. X. Serres oF ExpeRIMEnNts. Fig. 15. Scale of feet 42.5074 INS FOS : Fiyet 15 Feet Fig. 16. Fig. 17. The next boiler was, as shown in the drawing, of the shape usually called the “Butterly,” or, “Fishmouthed boiler.” Its dimensions were as follows : Length 25 feet by 7 feet in diameter. OF COAL AND RATE OF EVAPORATION. 29 Fire-place 4 feet 3 inches by 6 feet, equal to 25.4 square feet of fire surface. Fire-bars 21 inches below the crown or arch forming the bottom of the boiler. Bars half inch thick with half inch spaces. Internal flue 29 inches in diameter. Boiler plates 5;*8s of an inch thick. Flues were carried round the boiler. When this boiler was worked at its usual speed and in ordinary circumstances the evaporating power was 8.33 lbs. of water evaporated from a temperature of 60°. At its ordinary speed it evaporated 26,440 lbs. of water from the temperature of 60° in 12 successive working hours. By adding to the above result one-sixth for the heating of the feed water from 60° to 212°, the evaporating power of the boiler will be 9.72 lbs. of water evaporated by one pound of coal. This boiler became a general favourite, and was used with the smallest of the cylindrical boilers (fig. 10) in most of my subsequent experiments. In making these experiments I found it the more neces- sary as I proceeded to guard myself against all bias from reasoning of a purely speculative character, and to avoid the smallest approach to anticipating results. The theo- retical part of the inquiry will be best treated separately, and I hope to take it up on some future occasion and to have the pleasure of laying the results before the Society. The simple object I have hitherto had in view was to learn how to consume as little coal and evaporate as much water as possible, and to acquire that knowledge by ex- periment alone. My method was to alter the flues, damp- ers, draughts, bridges, fire-bars, fires, coals, and stokers, and everything else connected with a particular boiler outside and inside, and then, if necessary, to alter back again, 30 MR. JOHN GRAHAM ON CONSUMPTION until it appeared that I had obtained the most favourable condition for work, of the boiler under trial. To do all this required time and a considerable expenditure, as may be readily supposed. But the economical results which followed in greatly reduced consumption of coals, less fre- quent repair of boilers, and increased safety to the people employed about them, were always a sufficient motive to urge on the experiments. I may be allowed to say that the useful result in my case from these and other collateral experiments was a reduction of about 67 per cent. on the cost for coals per piece printed. XI. Series or ExPpeRiMENts. Attempts have been made to improve the boiler by en- larging that portion of its surface which is exposed to the direct radiation of the fire. A boiler possessing a contri- vance for that purpose came into notice a few years ago, and was strongly recommended on theoretical grounds, as it presented a very large absorbing surface as compared Fig. 18. Scale of fect € Heer: 12 6 0 Tatastestst OF COAL AND RATE OF EVAPORATION, 31 with the radiating fire surface. I was induced to have the cylindrical boiler, described under the VIII. head, altered to the new shape which is seen in Fig. 18. Under the cylinder boiler of 5 feet 6 inches in diameter, three smal] cylinders, each of 1 foot 6 inches in dia- meter, extending the length of the main boiler, were placed, and filled with water from the upper boiler. A line carried across the fire-place measured 4 feet 8 inches, while a line carried in the same direction, but over the surface of the boiler bottom and its appendages, measured 17 feet 7 inches. When again used, the altered boiler proved unequal to the work of the original plain cylinder. It was found, indeed, when properly tested, to possess a _ remarkably low evaporating power, namely, from 5.5 to 5.8 lbs. of water at 212° per pound of coal. The setting of the same boiler was then altered as seen in Fig. 19. Fig. 19. The lineal measurement of the absorbing surface was thus reduced to 10 feet 6 inches, while the line across the fire was the same as before. The new evaporating 32 MR. JOHN GRAHAM ON CONSUMPTION power was not numerically determined, but an improve- ment in the power of the boiler was most evident. It was now possible to go on with it at least, and the boiler was supposed at the time to have recovered about 20 per cent. in power. This result, however, being still unsatisfactory, I had the boiler altered as shewn in Fig. 20. fig. 20. The direct absorbing surface of the boiler was now re- duced to about 7 feet across, the fire remaining the same as before. Here again an improvement took place equal to at least an additional 8 or 10 per cent. as practically esti- mated. The boiler was employed in this last condition for some time, doing the full work previously allotted to it as a plain cylindrical boiler. The uniform impression of those who attended it was, that the boiler gave a slightly better result in this condition than as a plain cylindrical boiler. There were, however, practical objections to the boiler in this form, the brickwork over the tubes was constantly shaking loose, and the tubes themselves became scaled and OF COAL AND RATE OF EVAPORATION. 33 dirty and could not easily be cleaned. When, therefore, the tubes were burned out, which took place in their third year, I decided to remove them, and replaced the boiler in its original simple form as seen below, which it will be remem- bered gave the result of 7.1 pounds of water evaporated. Fig. 21. CoNncLUSIONS. The more important conclusions to which I have been led from the experiments above described, may be stated as follows : 1. The boiler with the two internal fire-places, evaporated commonly called the “ breeches boiler,” worked as ¥ oat. herein described, will yield ...............2ceceeeeseee 6.88 lbs 2. The “waggon-shaped boiler,’ worked as herein described, will yield .........00.0 05.0: eccseeees 10.26 lbs 3. The smallest sized “plain cylindrical boiler,” worked as herein described, will yield ....... atk ee Fabs 4. The large sized “plain cylindrical boiler,” worked as herein described, will yield............... 7.54 lbs 5. The “Butterly boiler,’ worked as herein described, will yield, for one pound of coals con- UO ss as n) vcagadeaseer sehavengi cad bin peneee mgheu, AETEOs 9.72 lbs 34 MR. JOHN GRAHAM ON CONSUMPTION 6. A supplementary boiler, under the circum- stances above described, gives a saving of ......... 7. When numerous cast iron pipes are substi- tuted for the supplementary boiler, and mechani- cal means are adopted to keep their surface clear of soot, a saving may, under the circumstances deseribed, he*expected WO S05. o. es cc eves fea xcomeas 8. The flues round a boiler, when they come to be coated to the extent of one-eighth of an inch with soot, are of little or no use in raising steam. 9. If the sides and bottom of the boiler exposed in the flues (where such flues exist) be scraped once a week clean from soot, a saving will ensue Ol SUOOE -<.... dese tena ceca? rest oewske® pak atameen senses 10. It is advantageous to convert the side flues at once into a wide flame-bed (where it is practica- ble), the flame and draught being spread out thin and going up nearly to the water mark on each side, and led straight on to the chimney. By such means the bottom of the boiler is preserved for its whole length free from soot, and there is an effec- tive radiation from the hot bricks of the bed. 11. A very slight difference in the setting alone of the same boiler may readily produce a differ- ence in the result amounting tO ..........ceseeeeees In extreme cases of bad setting, as may be seen at Fig. 14, of course a much larger loss than this may be looked for. 12. The difference between a boiler of one shape properly set, and another boiler of a different shape improperly set, but both clean and in good order, may readily amount to ............:c:eeeeeeee In some cases even a greater difference is pos- sible. 15 pe 2pect 21 p ct 42 p ef OF COAL AND RATE OF EVAPORATION. 35 13. A difference in the mode of firing only may produce a difference in the result of ......... 13 p ct I refer here to a difference such as will be pro- duced by withdrawing the regular and prac- tised stoker, and substituting for him an equally intelligent and careful man, but who has not been accustomed to the boiler or the coal employed. A man who is careless or dishonest will occasion a loss, the extent of which can scarcely be calculated. All inju- rious influences upon firing are greatly ag- gravated in very large fire-places (say of six feet by seven feet) from the physical in- ability of the stoker to keep the bars equally covered. 14, The loss arising from the scale of sulphate of lime on the inner surface of the boiler, of not more than one-sixteenth of an inch, amounted to 14.7 p ct 15. Neither wet coals, nor coals which had been out of the pit for three years, nor wet weather, nor a variation of temperature of the atmosphere from 40° to 70°, produced any appreciable difference of result. 16. Windy weather invariably gave a high re- sult. 17. A moderately thick and hot fire, with a rapid draught, uniformly gave the best result. The coals used varying from the condition of dust to that of pieces four inches in diameter, it was found advantageous to keep the fire at a general thickness of from six to seven inches, in order to insure the bars being covered at all times ; and as it is found, that in all boiler fires the coals are consumed more quickly 36 MR. JOHN GRAHAM ON CONSUMPTION at some parts of the bed than at others, such parts were supplied with extra quan- tities. This irregularity in the burning of boiler fires is one of the troublesome “ dis- turbing causes” in boiler experiments, the more so that such quick burning localities shift slightly with variations of the wind and other changes. ‘The fire-bars were always kept open, so that the ashpit was always bright and clear. The fire itself was “clinkered” at least once, and often twice in a day. 18. This law, as regards rapid combustion, ‘was reversed on the small scale. In the usual evapo- rations commonly taking place in Print Works, where pans and open boilers are employed, vary- ing in capacity from ten to three hundred gallons, the rate of evaporation per pound of coal is very low, varying from two to four and a half pounds of water evaporated from 212° for one pound of coal; and the law of evaporation from such vessels is about as follows: If the time is reduced to one half, the evaporation per pound of coal will be reduced also, to the extent of from 10 to 20 p ct. 19. The difference in the results obtained -by a change in the coal used amounted to .... .......... 20. The same coal reputed to be from the same pit varied in its evaporating power to the extent of Il\pcé 6 pct 21. The higher the water stood in the boiler within cer- tain unknown limits, the better was the result. In some experiments this advantage was in the proportion of 1 per cent. for every six inches of depth of water. 22. As regards the effect of pressure on evaporation, the general result of these experiments is in favour of the sup- OF COAL AND RATE OF EVAPORATION. ou position that the rate of evaporation of water per pound of coals increases with and bears some ratio to the pressure under which the steam is generated. In connection with this idea I may mention a singular fact which may be supposed to give it support, or ‘may, perhaps, find some other practical explanation. Tf a boiler be used exclusively for the purpose of heating water and liquors in a dyehouse, and be capable of working that dyehouse just sufficiently when the steam is at 10 Ibs. pressure, it will be found unable to do so when the pressure falls to 7 lbs., and still less able to do its work when the pressure further falls to QL lbs. By repeated experiments on the large scale I find that the loss incurred by working steam at 21 lbs. as com- pared with steam at 10 Ibs., an equal quantity of coals always being burned in the same time, is as follows : — Available power of the boiler to heat dye cisterns : With the steam at 10 lbs. pressure—=100 = s 7 Ibs. pressure= 92 zy 5 2% lbs. pressure— 77 I have not experimented at higher or lower pressures than those stated. 23. While we may reasonably look for advantage from improvements in the construction of the fireplace, in the management of the fire itself, in the construction of the flame-bed and in adaptation of draught, in the form of the boiler, in the addition of separate supplementary heating surface, and in cleanliness, and by all these means effect a great saving in the consumption of coals, we cannot at the same time expect much saving from extension of flue space when allowed to coat with soot, nor from a greater length of boiler than four times the length of the fire-place. 94. With a view to the prevention of the “scaling” of boilers, or the coating with sulphate and carbonate of lime and mud, I have experimented with the following sub- stances: caustic soda, quicklime, muriatic acid, soap liquor, 38 MR. JOHN GRAHAM ON CONSUMPTION sawdust, spent madder, and logwood chips, with various degrees of advantage. Muriaticacid proved a most effec- tive agent, indeed somewhat too much so. Although I never added more than two-thirds of the quantity of that acid necessary to saturate the lime known to be in the water as carbonate, yet the action continued not only to the point of the disappearance of the remaining third of the carbonate of lime, but went further. It had an action, also, on the oxide of iron surrounding the rivets, and existing between the edges of the plates; so that after six weeks, the usual duration of an experiment, it was found that the iron plates were not ouly clean and bright like silver, but also that the joints had been acted on to a considerable extent. The rust did not appear to be dissolved, but only loosened, nor did the metal of the plates appear to be corroded, for water from the boiler tested daily showed no iron. The water used in these experiments was from the Lan- cashire “Red Rock,” hard in quality, containing both sulphate and carbonate of lime, with a little iron. It re- quired forty measures of Dr. Clarke’s soap test liquor to soften it. The scale, when allowed to form, was com- posed in 100 parts of 22.5 carbonate of lime, and 77.5 sulphate of lime. Two facts were noticed as regards the tendency of hard water to scale. First, that the sulphate of lime separates from the water only when in contact with solid objects, such as the bottom and sides of the boiler, or solid matters floating in the boiler, such as sawdust and logwood chips (hence their use); and that no precipitation takes place until the water has been concentrated by continued evaporation down to the condition of a saturated solution of sulphate of lime, or to that point which may be termed the “ salting point.” Secondly, carbonate of lime and mud are princi- pally deposited in the midst of the water, and have but little OF COAL AND RATE OF EVAPORATION, 39 disposition to adhere to the boiler, unless cemented by the sulphate of lime. It is found accordingly, that no scale of much conse- quence will be formed on engine boilers, with even such hard water as is used, if 100 gallons of water daily, and 200 additional gallons on Saturdays be run off through the usual mud machine, and if the boiler on every sixth Saturday be further entirely emptied and swept out. ‘There is little loss incurred by proceeding in this manner, as the chief discharge may take place at the close of each day, while a large profit accrues by the saving of coals and the increased durability of the boiler, with security from danger to all concerned with it. 25. In the five boilers most used in these experiments, the highest results were obtained when all the air required to consume the coals was introduced into the fire-place through the fire-bars. The introduction of additional air elsewhere uniformly occasioned loss, whether it was ad- mitted by means of the furnace door during two minutes after firing, or for a longer or shorter period, or through a regulating slit above the furnace door, or through apertures of larger or less size and more or less numerous, in the sides of the furnace, in the front of the bridge, or on the top or back of the bridge. 26. Perfect combustion of black smoke was never ar- rived at, without the introduction of additional air. The consequent loss was a little over 1 per cent. This loss, by the addition of air over the fire, will be uuderstood when we consider the smallness of the weight of black smoke. The following experiments may convey some information on this pomt. I had in view the possi- bility of working bleaching keirs with hot air from the flue of the furnaces instead of steam. The apparatus used is shown in the drawing : — 40 MR. JOHN GRAHAM ON CONSUMPTION Fig. 22. I drew the hot air and smoke out of the main flue by means of a pump at a temperature of perhaps 6002, forced it through the intermediate washing vessel, in which was kept a small supply of water, and where mechanical means were adopted to subdivide the stream of air, and passed it on to the keir. The result of these experiments was a failure as regards the heating of the keir. I passed the hot air at a high velocity, namely, about 130 cubic feet per minute, and found that the heat communicated by the mass of air was barely sufficient to sustain the coutents of the keir at a temperature not above 190° F., and that it took from six to eight hours to bring it up to that point. Incidentally, however, I had the means of estimat- ing the weight of black smoke. The hot air and smoke used were from furnaces that produced dense black smoke after each fresh supply of coals. The washing apparatus intercepted the soot in such a perfect manner, that the water in the keir was not more than discoloured at the termination of the experiment. The black residuum in the washer, composed of dust and carbon, was carefully ex- amined, but not analysed. I formed an opinion to the best of my judgment as to the amount of carbon, and as to its weight relative to the coal consumed in heating the air and smoke drawn by the pump, which was that the weight of black smoke does not exceed the one thousandth part of the coals consumed. The power of black smoke as OF COAL AND RATE OF EVAPORATION, 41 it issues from the chimney-top to discolour the atmosphere is prodigious, but its weight is inconsiderable. This discolouring power is rapidly abated and soon de- stroyed by the aggregation of the particles through electri- cal or other causes, and quick precipitation ensues, each particle being assisted in its descent by an associated burden of dust and other matters. The dense heavy cloud we see floating off to leeward of such large cities as Man- chester, is not composed of black smoke; indeed, it may be questioned if, at the distance of half-a-mile from its source, and unless drifted like sand before the gale, there is then a single particle of black smoke in the atmosphere. Those who have “ crofted pieces” under the current of their chimney can estimate the correctness of this remark as to the rapid precipitation of black smoke. The dust which descends, associated with the black smoke, has been invari- ably found to contain iron. This may be shown on the most minute scale by placing a particle of the soot be- tween the wet folds of a pink piece dyed with madder at that point of the process when the piece has been washed out of the “tin.” The chemical law of displacement in this experiment is beautifully shown by the iron taking the place of the alumina that formed the basis of the pink colour, and itself producing a black with the liberated alizarine. What we see floating in the atmosphere is dust associated with the volatile constituents of the coal which have been distilled off through imperfect combustion. It is scarcely in place in this paper to go further into this important subject; but I may remark that this rapid dis- appearance of black smoke by precipitation, more particu- larly in dry weather, is one of the marked tests of the proper action of the fires below, while if the smoke hangs lazily, keeps together, and is visibly black for a greater distance, we may be certain that something is going up VOL. XV. G A2 MR. JOHN GRAHAM ON CONSUMPTION, ETC. the chimney as well as black smoke, and in serious quan- tities, which ought to have been burned in the fire itself, The presence of such volatilized oils appears to have the effect of sustaining the black smoke by hindering its aggregation and precipitation. 43 III. — On the triedral partitions of the X-ace, and the tri- angular partitions of the X-gon. By the Rnv. Tuo. P. Kirxman, A.M., F.R.S., Rector of Croft with Southworth, and Honorary Member of the Literary and Philosophical Society of Manchester. Read November 17th, 1857. 1. Perhaps the simplest proof that can be given of the relation, F+S=E+2, among the F faces, S summits, and E edges of a polye- dron, is the following. Let e be any edge joining the summits a 6 and the faces A B, and let e vanish by the approach of 6 toa. If A and B are neither of them trian- gles, they both remain, though reduced in rank and no longer collateral, and the figure has lost one edge e and one summit 6. If B is a triangle and A no triangle, B vanishes with e into an edge through a, but A remains. The figure has lost two edges of B, one face B, and one summit 6. If B and A are both triangles, B and A both vanish with e, five edges forming those triangles are re- duced to two through a, and the figure has lost three edges, two faces, and the summit J. In any of these cases, whether one edge and one summit vanish, or two edges disappear with a face and a summit, or three edges with a summit and two faces, the truth or falsehood of the equation F+S=E+2 4.4 REV. J. P. KIRKMAN ON THE remains unaltered. By causing all the edges which do not meet any face to vanish, we reduce the figure to a pyramid upon that face. Now the relation is true of the pyramid ; therefore it is true of the undiminished polyedron. TuroreM A. In every (#@+1)-edron P having an x-gona, face X, there are at least two triangular faces. For there is no face of the (w+1)-edron not collateral with X: wherefore the edges of P which do not meet X can enclose no space, but must form a broken line whose terminations are p, P,..., two or more. From each point p, p,... two or more edges must pass to meet the base, since all the summits are at least 3-edral. That is Py; Pz... ave vertices each of one or more triangles collate- ral with X. Q. H. D. 2. The partitions of an w-ace, (w rays diverging from a point,) correspond to those of the #-gon. The diagonal joining the m™ and 2 summits of the latter answers to the intersection of the m‘ and n™ faces about the former. The number of triedral partitions of the #-ace is that of the triangular partitions of the z-gon, in which of course no two diagonals cross each other. A triangular partition of a non-planar or skew 2-gon, crowned by a simple 2-ace whose rays pass through its # summits, is an (v+1)-acron with only triangular faces. A triedral partition of an g-ace, standing on an unpartitioned x-gon, is an (#+1)- edron whose summits are all triaces. If we can enumerate the (v+1)-edra on an #-gonal base which have all their summits triaces, we shall have the solution of the problem of the triangular partitions of the x-gon, or of the triedral partitions of the x-ace. The ob- ject of this memoir is to show that this problem is very conveniently attacked by the consideration of these (v¥+1)- edra, and that this mode of solution is, where all modes seem difficult, as easy as any. And the view here taken of the question is entirely new. PARTITIONS OF THE X-ACE AND THE X-GON. 45 3. Let P be an (#+1)-edron in an #-gonal base, having only triedral summits. P has f triangular faces, f<¢2. If f=, no edge meets the base which is not the side of a triangle. If f<4v, there are x—2f edges (j) meeting the base which are not sides of triangles. And f cannot be > 3a, for if it were, there would be two triangles about the base having a common vertex out of it, which would be no triace, unless the figure were a tetraedron. Each of these v— 2f edges (j) connects a summit of the base with one (2) out of it. If we suppose that edge 7 to evanesce by the coincidence of the two faces about it, a summit disappears of the base, and a summit (2) disappears among those out of the base. Let all the x-—2f edges then disappear ; the base is now 2/-gonal, and has the same f triangles collateral with it, the greatest number possible, as has just been shown. And all the 2 -2/f summits 2 out of the base that have vanished have disappeared from edges & that do not meet the base. 4. It follows that all (#+1)-edra having f (< 42) tri- angles, can be constructed from the (2/+1)-edra having f triangles without touching those triangles, by joining points () in edges not meeting the base and not in tri- angles, to point in base edges, the face in which any such joining line (7) is drawn being supposed fractured into two faces thereby, and every such operation adding a face, two summits, and three edges, but altering and adding no triangles. Every (2f+1)-edron having / triangles is evi- dently wade by cutting away every base summit of a certain (f+1)-edron on an f-gonal base, which may have any number of triangles from 2 to $f, or 3(f-1), as f is even or odd. If the number of these (f+1)-edra is known, the num- ber of (2f+1)-edra having f triangles is also known; and if the whole number of ways be known in which m join- ing lines (7) above described can be drawn upon these 46 REV. J. P. KIRKMAN ON THE (2f+1)-edra, the number of (2/+m-+1)-edra on (2f+m)- gonal base having f triangles is determined. And thus the entire number of (#+1)-edra on an «-gonal base can be found, and classified according to the number f of their triangular faces. It is to be understood that in this enumeration no (#+1)-edron shall be either the repetition in a different position about the base, or the reflected image of any other. The avoiding of these repetitions is the great difficulty of our problem. 5. Polyedra, whatever be the rank of their faces and summits, as well as partitions either of the polygon or the polyace, fall into two leading classes, the reversible and. the reversible. An irreversible is in no position identical with its re- flected image. A reversible is in certain positions identi- cal with its reflection. Again, an irreversible regarded at one point of its base may be identical with itself regarded at another or at two other points of the base. If at one other only, it is called a doubly irreversible ; if at two others, it is called a triply irreversible. In the former, the sequence of faces about the base repeats itself once in the circuit; in the latter, it repeats itself twice: that is, it is three times read in the circuit. A singly irreversible is in no two positions iden- tical with itself. In like manner, a reversible may be singly, doubly, or triply reversible, exhibiting a configuration about the base that occupies the whole circuit, half the circuit, or one third of it. A singly reversible can be placed in two positions before a mirror so as to be identical with its image, in which position a bisecting axial plane is perpendicular to the mirror. A doubly reversible has four such positions in two axial planes at right angles to each other. A triply PARTITIONS OF THE X-ACE AND THE X-GON. 47 reversible has six such positions in three axial planes making equal angles with each other. All this requires the supposition, which we are at liberty to make, that the base of the polyedron is a polygon regularly inscribed in a circle, on which the whole figure is constructed as sym- metrically as possible. The three following are irreversibles, single, double and triple, standing on 7-gonal, 6-gonal and 9-gonal bases : | 54363538, 435435, 643643643, of which the third e.g. shows a hexagon, a 4-lateral and a triangle thrice read in that order about the base. The three following are reversibles, single, double and triple, on 5-gonal, 8-gonal and 6-gonal bases : 53443, 53635363, 535353. 6. A sequence four times read can never appear if the summits are al] triedral. For in such a sequence the tri- angles may be supposed to become infinitely small, and in the result of their evanescence the triangles again might be supposed to disappear, and so on till a figure appeared incapable of further reduction by the evanescence of tri- angles. But this figure would still have a four-fold se- quence, the original periods of configuration having been treated alike: that is, the figure would be a pyramid in a quadrilateral base, having a 4-ace, contrary to hypothesis. The effect of the evanescence of a triangle is to reduce by one side the rank of the face on either hand; thus, by the vanishing of the triangles, 643643643 becomes 535353, and 53635363 becomes 3434. 7. PROBLEM a. Let P be singly irreversible (2a + 1)-edron, having a 2a- gonal base and ~# triangular faces. Required: the number of (27+ m+ 1)-edra that can be constructed from it by the process of Art. 4, to have also w triangles. As all the S$ summits are triedral, or have 48 REV. J. P. KIRKMAN ON THE 2E=88, also F4+S=E+2, | whence 3F+2E=3E+6, E=38F -6=6z2 -3, of which 62-38 edges 2 are in the base, and 2 are sides of triangles, leaving 27-3 edges not meeting the base and none of them in a triangle. 8. If we draw in any non-triangular face h collateral with the base a line 7, from a point of the base to a point @ of any edge of h which does not meet the base, and con- sider h fractured along that line into two faces, the result- ing (2@+2)-edron (Art. 4) is still singly irreversible ; for the new summit of the base is the only one not in a tri- angle. If we draw another such line / either in 4 or some other non-triangular face, the result is still singly irrever- sible; for, by definition (Art 5), the new summit of the base is part of a configuration which nowhere else is read about the base: and in like manner, any number of such operations upon a singly irreversible subject will give a result singly irreversible, and no two of these results can be alike, because in our singly irreversible subject of operation the configuration read round the base from any face is different from that read round from any other. Our problem is to determine in how many different ways such operations can be performed on our subject PR, There will be added m summits 7 out of the base, which will be planted on the 2v-3 edges & not in triangles. The number of ways in which these m points can be dis- tributed on those edges, w on any edge, (wo) is, putting a 'i=a-a+1-a+2...(6 factors), (m+1)*4!1 [eit » And as the joining line 7 through any of these points can be drawn to the base through either of the faces meeting PARTITIONS OF THE X-ACE AND THE X-GON. 49 in the edge k containing the point 7, the whole number of arrangements of the m new lines 7 is (m a5 1D ie : 9m jer [1 If then I1(2x+1,x7) be the number of singly irrever- sible (27+ 1)-edra having 2 triangles, and Il(2e2+m-+1,2) denote the number thus constructible of (27+m-+1)- edra singly irreversible that have x triangles, we have proved that 1 Il(2¢-+m+1,2)=1(22+1,2)-2" —— For any one of the 1(2z%+1,z) being taken for the subject of operation, will give the same number of results as our subject P. The number II(2z+m-+1,z) is only part of I(24+m +1,#), the whole of the singly irreversible (27+m-+1)- edra that have 2 triangles, namely that part which reduces, by the process of Art. 3, to the irreversibles I(2¢+ 1,2). The remainder of the number I(22+m-+1,z) we shall find in the proper place. 9. Prosiem b. To determine the number of doubly ir- reversible (4¢-+m-+1)-edra having 2z triangles reducible (Art. 3) to doubly irreversible (4z+1)-edra having 2z tri- angles, It is to be understood here and everywhere in this memoir, that we are considering only (#+1)-edra which . have an x-gonal base, and all their summits triedral. A doubly irreversible must have an even number 2w of triangles, because any sequence of faces about the base is twice read in the circuit; that is, the triangles and the configurations about them are corresponding pairs. Let P be the doubly irreversible (4a +1)-gon, the sub- ject of our operations. That our results be also doubly VOL, XV. H 50 REY. J. P. KIRKMAN ON THE irreversible, it is necessary that m be even, for the lines 7 must form corresponding pairs. We have (Art. 6), 4v-—3 edges k not meeting the base, on which to distribute our m points 7, an odd number of edges. Now, our subject is such that if any plane per- pendicular to the base is drawn to bisect it, and to contain no edge k, the configuration on one side of that plane is a repetition of that on the other. There is therefore one edge k’ bisected by that plane, and for every point 7 planted on one half of k we must plant another 7 on the other half; and every operation on one side that plane must be repeated in the proper place on the other side, if our result is to be doubly irreversible. We have, then, to consider in how many ways 4m points z can be laid on 4(4a-3+4+1)=2e-1 edges k. The number of operations by the preceding article is m 2a—2|1 (5 +1) of Pen a which are to be repeated in direct order on both sides the bisecting plane. These results are all different, because no sequence can be read round the base to begin at two different points on one side of our bisecting plane. And no more results could be obtained by varying the position of that plane, since every possible disposition of $m lines 7 has already been made on half the figure. If the I°(4v+1,2x) be the entire number of doubly irreversible (4v+1)-edra having 2z triangles, and I’I?(4x +m-+1,22) be the number of doubly irreversibles thus constructible, we have a nm 5 +1) PP (4x +-m+1,27)=P(S2+1,2e)-2. sap Qx—2|1 for the portion thus obtainable of the whole number T’(4e-+m+1,2x2) of doubly irreversible (4¢+m-+1)-edra PARTITIONS OF THE X-ACE AND THE X-GON. 51 having 2v triangles. The remaining portion we shall find presently. 10. Observe that the number II’ above written is always to be considered zero, if 4m is not integer; and in all future : Ra; m — m-e expressions of this kind, the occurrence of = Oe not an integer in a factorial is to be considered equivalent to that of the factor zero before the factorial. We may thus dispense with the use of circulating constants. Proptem c. To determine the singly irreversible (4x +m-+1)-edra having 2 triangles reducible (Art. 3) to doubly irreversible (4v + 1)-edra having 2v triangles. If we distribute m points 7 in every possible way on the 402-3 edges j of the subject P operated on in the pre- ceding article, and draw m lines 7 in every possible way, we shall obtain (m+1)"2 1*#4+ {1 results. Among these will evidently be found all those of the preceding article, once, and once only. The re- maining results will be simply irreversible, and each will be twice constructed; for the result P’ presenting a single sequence of configuration read round the base from a face h, will be identical with another result P”, having the same reading round the base beginning at the face of P opposite to h. If then from our total number of results we subtract those of the preceding article, the remainder divided by two will be the correct number of single irreversibles required. That is, if II’(4v-+m-+1,2z) be this correct number, 44/1 TP(4e +m 41,20) =3{ fan 2a—2 | 1 Gave ie pert 2} -T?(4e +1,22), 52 REY. J. P. KIRKMAN ON THE a portion of the whole number I(4v +m-+1,2h), of (4a+ m-+1)-edra singly irreversible having 2 triangles. 11. Prosiem d. To determine the number of triply irreversible (67-+-m-+1)-edra having 3 triangles, which are reducible (Art. 3) to triply irreversible (67+ 1)-edra having 3w triangles. We must operate on the third part only of the 62-3 edges k (Art. 6), and draw 4m lines j. This gives us 2v—2 | 1 ae 122 {1 results. If we repeat these operations in direct order on the other two thirds of the subject, we shall, as is evident from the reasoning of Art. 9, have all possible configura- tions and none twice repeated among them. The number of triangles must of course be 32. If then [°I°(6z+m1,3z) be the triply irreversibles, thus constructible from the whole number [°(62+1,3z) of tri- ply irreversibles with 3 triangles, then is m 22—2|1 a ) m PR (62 +m+1,37)=1(6r 4+ 1,32) - r= rer 2°. a portion of I°(6v+m-+1,3x), the triply irreversible (62-+m-+1)-edra having 3z triangles. Others will here- after be found. No doubly irreversible can be constructed on a triply irreversible subject, for a plane bisecting the base does not here show the configuration on one side of it repeated exactly on the other. We proceed then to 12, Proptem e. To determine the number of singly irreversible (62+ m+ 1,)-edra having 3 triangles reducible to) 6v+1)-edra triply irreversible having the same num. ber. If we plant m points 7 on our 67-3 edges k in every possible way, and draw the lines 7, we obtain PARTITIONS OF THE X-ACE AND THE X-GON. 538 (m+ 1)*+! mae . jes results. Among these every result in the preceding problem will occur once, and once only, and every singly irreversible will occur three times, because the same se- quence will have been begun and carried round from three similar starting points, say three A-gonal faces, of our sub- ject. We have then (m+ 7 gah Sian 11 ID(6x+m+1,3x) =1°(6x +1,8x) +4 Qm _G? Ps 4 : a 3 12 [hed -2 as a portion of the number I(6z+m+1,8z). 13. Prosrum f. To determine the number of singly reversible (2v-+m-+1)-edra reducible to singly reversible (2+ 1)-edra with 2 triangles (Art. 3). In a singly reversible one vertical plane and one only can be drawn bisecting the base, such that the configura- tion on one side of it is a repetition in reversed order of that on the other. The number of edges k is 27-3, an odd number. This bisecting plane, which we may call the axial plane, or plane of reversion, will therefore either contain one edge 4, or bisect one. First let the axial plane contain an edge #’: this edge can receive no point 7; for a line 7 being drawn from it on oue side the plane would require, in the reversal of opera- tions on this other side, another line 7 to be drawn from the same point on the other side, which would make? a 4-ace. We have then 4m points 7 to plant on 4(227-3-1)=x%-2 edges & on one side of the plane. This gives us a—3 [1 (S41 ) 3 17311 results, so that mis of necessity even. 54, REV. J. P. KIRKMAN ON THE Let R(Qv+1, os ds dene the whole number of singly reversible (2041). “edra having # triangles, in which one edge x’ is axial or in the axial plane, (the subindex kaw=1 signifying one is axial,) among which uumber is the sub- ject of our operations. Then we have (3 +1) RR(2e2+m-+1,z) =R(Qwr +1,2)- sae aes kax=1 kax=1 for the number of our results, which form a portion of R(Qv+m+ aly pa (2e-+m-+1)-edra singly reversible having wv enh in which the axial plane meets none, i.e. contains one, of the edges not meeting the base. Next let the axial plane bisect an edge /’. We can now operate upon X’ on both sides the bisecting plane, without producing 4-aces. It is easily seen that our results are, if m is even, m (5+1) of Pen sae) and we obtain the equation He 2)1 st RR(2Qve+m-+1, a EEL 2) +2 kax=0 as a portion of R(Qn m+ 1,v), the whole number of (27-+m-+1)-edra singly roveruible eivihe x triangles, in which no edge & is axial, as denoted by the subindex kar =0. The base in both these sub-classes is even angled (2a +m)-gonal. If however m is odd, we can produce as just pointed out, with the even number m-—1 operations, m— ] aE |" jae m—1 results, and then plant one point more in the axial plane at the centre of the edge /’, from which of course two lines j can be drawn, doubling this number. PARTITIONS OF THE X-ACE AND THE X-GON. 55 The (22-+4+m-+1)-edra thus produced have no axial edge kK’, but they have all an axial point 7 and line 7, which neither of the previous sub-classes of this article have. We may write therefore, m1 RRQe+m+1,7)=R(2Qr+1,2)- Pan jax=1 kax=0 the subindex yar =1 denoting that these have all a summit z in the axial plane. This then is a portion of the number R(2Qve+m-+1,z2) of reversible (2e+m-+1)-edra having » triangles, and a the base odd-angled, (2#+)-gonal, as must necessarily be the case when the axis of reversion passes through one summit of the base. 14. Although no reversible can be obtained from an irreversible subject of operation, the converse is not true ; and we proceed next to Prosiem g. To determine the number of irreversible (2a@+m-+1)-edra reducible to singly reversible (2v+1)- edra having 2 triangles. These will all be singly irreversible, because the subject of operation stands alike in no two positions, read in the same direction. We have 27-3 edges k, on which to place m points, giving us (+1)! 9 ]24 }1 results, among which all those of the preceding problem will occur once and once only, while each of the remaining irreversibles will occur twice, in one result as the reflected image of another result, because of the reversible character of the subject. We obtain, therefore, dividing this re- mainder by 2, 2vu—4 | 1 IRQe+m+1,2)=IRQ0 41,2). tI. 2m m «a3 |1 m (z+1) — tR(2¢+1,x)-2?-———- — 4 R(274+1,2) kax=1 r—3 | 1 ae kaw=-0 56 REV. J. P. KIRKMAN ON THE m e—2|1 m= et n(o+1) om a } Siro pe Gt ees ae ehenpte which is a portion of the number I(2v7+m+1,z) of singly irreversible 27-+m-+1)-edra having w triangles. R(2e+1,7) here stands for the whole number of singly reversible (27+ 1)-edra having ~ triangles, 7.e. R(2e+1,7)=R(Qe+ bk a ae + vii 15. Prosiem fh. To Phos the ee a doubly reversible (4v-+m-+1)-edra having 22 triangles. No mention is made here of the subject to which these are reducible, because it must always be a doubly reversible, one of the number R?(4vw +1,22) ; for the lines 7, which dis- appear (Art. 3) in such reduction, are placed symmetrically about two axial planes: wherefore that symmetry must remain in the subject P to which the figures reduce them- selves. One of the axial planes (Art. 5) of P must contain £’, one of the 47-38 edges k, which x’ (Art. 13) can receive no point 7, if the result is to be doubly reversible and all the summits triaces. In any one of the quadrants between these planes will be seen 2-1 edges k, on which 47 points 7 are to be planted, and the configurations are to be completed so that every quadrant shall read as the reverse of that on either side of it. The number of results will be, EONS ™ eee ee Been “eyo Pen 5 ar wherefore eG » G+) R?(den +m + 1,20) = 24. Faq —— RX(4e + 1,22), which is the whole number R*(4v + m+ 1,22.) This vanishes if e=1, unless m=O, showing that the wedge is the only doubly reversible with two triangles. ee PARTITIONS OF THE X-ACE AND THE X-GON. 57 16. Prostem #. To determine the number of doubly irreversible (4v+m-+1)-edra reducible to doubly rever- sible (4s -+1)-edra having 22 triangles. Nothing prevents us here from depositing points 7 on the axial edge # of the subject, inasmuch as the opera- tions in one half of that edge can be repeated in the other half without introducing tessaraces. On one side of the axial plane which bisects k’ we see 2-1 edges k, including the half of #’, wherefore the number of ways of drawing 4m lines 7 is ene [== lanai & 2 ? wf which operations are to be repeated in direct order on the other side of the plane. This gives us only doubly irre- versible results, but every one is twice obtained because of the reversible configuration on which we have operated. To every face h on one side of our bisecting plane corre- sponds another face h’ on the same side, such that the sequence read from / in one direction is identical with that read from /’ in the opposite; so that every sequence is twice constructed. We have therefore to divide the above by two; hence 2¢—2 41 (sith) m=2 PR(4¢+m+1,22)= eC) Oe »R?(4e 41,22) ; which is a portion of the number ?’(4w+m-+ 1,22). No new results can be obtained by operating on the semi- solid cut off by the other bisecting plane; for if in P’ a result so gained we move the bisecting plane to its old position in our subject, we sce on one side of it a configu- ration before enumerated, since all possible ones have been before enumerated. 17. Prostem 7. To determine the singly reversible (4a VOL. XV. I 58 REV. T. P. KIRKMAN ON THE +m-+1)-edra reducible to a doubly reversible (4¢+1)- edron having 22 triangles. We can produce singly reversible configurations bisected by either of the axial planes of the doubly reversible sub- ject. Taking first that plane which contains the axial edge k’, we see on one side of our plane 27-2 edges & (Art. 13) on which we can deposit 4m points 7, thus ob- taining a {23 | ie ery 2” results, which are to be repeated in order reversed on the other side of the bisecting plane. Among these will occur once every doubly reversible (4e¢-+m-+1)-edron with 2v triangles enumerated under problem 4 (Art. 15); for if the bisecting plane be drawn through the axial edge # of any of these, we see a configu- ration that we have been producing here; and evidently this configuration, and therefore this (4a+m-+1)-edron has been only once here obtained. And every singly reversible (4v-+m-+1)-edron among our present results will occur twice, because. of the reversible character of the semi-solid on which we have been operating, as pointed out in the preceding article. Moreover all these singly reversibles have an edge # in the axial plane, and all the singly reversible (47+ m-+1)-edra having an axial edge K’ which are reducible to a doubly reversible have been here constructed. Therefore, subtracting the results of Art. 15 and dividing the remainder by two, we obtain 2a—3 | 1 zt) pk RR (424+ m+1,22)= jes 2 : kan=i a2|1 zt) m= | oe es saan 2% (R(4v4+1,22); ee PARTITIONS OF THE X-ACE AND THE X-GON. 59 which is a portion of BGe pm 1,22). = m is of ne- ran= cessity even, and the expression vanishes for x=1, if m> QO. 18. We consider next the bisecting axial plane of our subject which cuts the axial edge #’. On either side of it we see 27-1 edges f, including half of #’, on which, m be- ing still supposed even, we have to deposit $m points 7. The number of results is 2Qa—2 | 1 m (y +1) 2 yet 2. By reversing these operations on the other side of the axial plane, we shall again produce, and once only, every result of Art. 15; for if in any of these the bisecting plane be drawn to cut the axial edge #, we shall see one of the configurations here constructed. And evidently this has been constructed only once by the process of this article. Also by the reasoning of the preceding article it -appears that every singly reversible (4¢+m-+1)-edron here ob- tained has been twice obtained, and has no axial edge. Subtracting then and dividing by two as before, we get 2v—2 | 1 m (5 +1) m2 BR (Got m+ 1,20)— [= eS 2 m a—2|1 G +1) m—t — ea -2 7 [R(4e+1,22), which is a part of the number R(4v+m-+1,2z2). kax=0. Suppose next that m is odd, the axial plane still cutting the edge #’. We can with the even number m-1 produce as before Qv—2 | 1 a mes ee Pan « results. We can then deposit one more point 2 at the 60 REV. T. P. KIRKMAN ON THE centre of the edge i’, in the axial plane, and from this draw either of the lines j, in every figure constructed. But none of these constructions will be doubly rever- sible, being not the same on both sides the axial plane containing #. If then we multiply the above by two, we shall have, as shown in the preceding article, double the number of singly irreversibles when m is odd, and no sub- traction of doubly reversible results has to be made. That is, m—-1 Qw—2} 1 [eae 54 l) ml RR?*(4e+m+ Wee ee ies +1,22)- ea ae g2, a portion of R(4”+m-+1,27). In these the base is odd- jaz=1. angled, being (4a-+m)-gonal, and in these alone, having m odd, is there a line 7 in the axial plane, as denoted by the sub-index jaw=1. 19. Prosrem &. To determine the number of singly irreversible (4¢-+-m-+1)-edra reducible to a doubly rever- sible (4¢+1)-edron having 22 triangles. We can distribute m points on the 4¢-3 edges & (Art. 7). This gives us (m aE hee Jl o oe arrangements of lines 7. Among these results will occur every one of R@R? (Art. — 15) once, and once only; for the lines 7 in any of these can be laid on the subject in one way only: every one of I°R2 (Art. 16) will occur twice, in one result appearing as the reflected image of the other: every singly irreversible (RR?2)’ will be constructed twice, viz. the same configura- tion being laid on both sides of that axial plane of the subject which is not the axial plane of (RR?)’. Every singly irreversible now to be enumerated will present itself four times; for any sequence read about the base of the subject of operation is read four times, beginning In every PARTITIONS OF THE X-ACE AND THE X-GON. 61 quadrant, and being read in one direction in two quadrants and in another in the other two; wherefore every con- structed configuration will be begun at four different points of the circuit, and appear in two results as the reflected image of itself seen in the other two. If then from the above number of arrangements of lines we subtract the doubly reversibles of Art. 15, twice the doubly irreversibles of Art. 16, and twice the reversibles of 17 and 18, the remainder divided by four is the number of irreversibles here required. That is, 4y—4 | 1 TR2(4e +m +1,22) =1R?(4er + 1,22) {re MEL a—2|1 2r—2|1 at o B30) hi m oz Gt) os ioe “2 ae: pen “2 my Qr-2| 1 a m 221 ‘ wi 8) pp MARE) gs (eat 2Qa—2|1 a. no — eee ee | Bird +1a5| St eet ea QF m a2 1 m zt) sy he LY) m4 a 2 . Ae ee ee 2 2 = ile z = yan piace 2 ae 0 eT i |? {1 which is part of the number I(4a-+m-+ 1,22). 90, Prosiem J. To determine the number of triply re- yersible (6a -+m+ 1)-edra having 3z triangles. All these by the definition of a triply reversible, Art. 5, 62 REV. T. P. KIRKMAN ON THE have 3z triangles. If these triangles all vanish, the result will have 32’ (2 o and that a factorial containing us not integer is always zero. (4). P(2ve4+1,v) =P(w+1,2) + P(w + 1,3) + P(w+1,4) + =P(x+1). 2e—4 1 (8). II(2v7-+m+1x)=1(2e+1,2)-2”- eae » (5 +1) 2a—2|1 (9). PE(4e +m 41,22) =12(4e@ + 1,22) -2 j2217 (10). IP?(47+m+1,22)=P(4e + 1,22) 2 l 2a—2 | I x Fe 3 (m+ a teers ad ; (y+ ) : #4 [1 je }1 In (9) and (10) m is always even. m 2a—2| 1 » (¢ +1) (11). PR(6e24+m+1,32)=P(67+1,382)-2°- [=3n1 ; (m=8n). (12). nay 3x) =41°(67+1,32) m 2x—2 | 1 m, ss jaar we Gry) . x oe eee oF os cle «2—3 | nm (+1) (13). Bier Le R(2Qe+1,2)- ) 20 Ce oe kax=1. kax=0 x2}1 (5 +1) RR2e+m+1 Aim 2G og yt) 2 ae (22) eee m+ RR(Qa+m-+1,2) =RQe+1,2).2? 2 jax=1. kaa=0. ES (14). IR(Qe +m 41,0) = FRQx+1,2)-gm. (mtd) {ea—pir > m w—3|1 n (gt) — ?R(2x+ 1,2) x2?- yea kaxv=1. PARTITIONS OF THE X-ACE AND THE X-GON. 67 a—1|1 aE a—2|1 | = ce) aoe “) , ye 11 ¥ yet x tR(Qe4+1,2). ka st m x2] 1 » (¢ +1) (15). R?(4e-4+m+1,22) =R*(3x+ 1,22) - 27. a0 (16). PR(42+m+1,22)= R?(4¢+ 1,22) 2-2 | 1 wal +1) x Qe ees a a (17). RR*(4¢+m-+1,27) =R*(4@ + 1,22) kax=1. ig 2x3 | 1 v2 /1 wy Gt!) ot G we . x 2 (18). RR?(4e¢ +m +1,22) = R*(4e + 1,22) kax=0. a 27-2 | 1 on z—2}1 tat ©) Enirigicy, ye ea oe ee ae RR*(4¢ + m+ Ie) ED +1,22) ee weer : pan (m odd). (19). IR*(4¢+m+1,22)=R*(4e + 1,22) v—2|1 +1) m+1)#4") ays fore ee +27 = Seeniri: 2021 oe 2x3 | 4 oe Ge ) mt (5 +1) = 2 Jae ae : Fes Ga (= 2v—2 | 1 eB} “27-21 m w—2|1 m ee. (20). B°(6x-+m +1,82) =R6x+ 1,32) -2°» Sa —- 68 REV. fT. P. KIRKMAN ON THE (21), RRGe+m-+1,82) =R' (6x +1,32) aaa a2] 1 | : (5 +1) n(g +1) X27. ett a aay eo (22). PR9(6x +m+1,3x) = R%(6e + 1,32) 2a—2'| 1 s pa a 1) 1-2 }1 (28). IR%(62-+m+1,3x) = 4R3(62 + 1,32) m w—2|1 0 bia ae ona imenet? (e+) - yori Gee! 32-3 [1 2v—2 | 1 m—2 (541 we (5 +1) - ee Tee oe eer 26. If now we desire to enumerate P(7+1), the (@+1)-edra on an 2-gonal base haying triedral summits or, in other words, to enumerate the triangular partitions of the w-gon; we write first, (24). P(#+1)=P(7+1,2) + P(#+1,8)4+ (25). P(jw+1,f)=R(27+1,f)+R(e+1f)+R(e+Lf) +P(@+lLf)+P(e+1Lf)+l(e+L/f). Ri(v+1,f)=R*(2f+4 (w-2f) + Lf) is given by (20); Rw + 1,f) =R(+ (w-2f) +1,f), by (15); R(w+1,f)=(RR+ RR*+ RR’) (2/4 (@-2f) +1,f), by (18, 17, 18, 21). T(e+1,f) = (114+ 1? + I+ IR + IR? + IR*)(2f+ (w-2f)+1,f), by (8, 10, 12, 14, 19, 23); P(w+1,f) = (EP + PR 2+ (v-2f) +1,/), by (9) and (16); Met CP+TENAt+ (2-9) +12), by OD and (22). And thus P(#+1) can be found in terms of 2 /P(f+ Lf)=2 (B+ R?+R4P4+P4+DH(Q+LS), where f has every value from 2 to the integer next below da. PARTITIONS OF THE X-ACE AND THE X-GON. 69 We may either consider P(#+1,f) to denote the tri- edral partitions of the z-ace which leave f pairs of con- tiguous edges of the z-ace undisturbed, or to denote the triangular partitions of the w-gon, which leave f angles of the 2-gon untouched by a diagonal. 27. To show the use of these formule, we can begin at the beginning, thus: P(4)=1; a. P(5)—R(5,2)=1 : 17(56,2)=—0=RG6,2)—=R*6,2). b. B*(5+m,2)=0, if m>0, by (15)=0”. m—2 ce. [(5+m,2=2 7, by 16 and a). d. Brite) = i (18.) d. R(+m, of QF _e (18 and a); m even. kaa=0. m—1 d’, R(5+m,2)=2 ? , by (18 and a m odd. jax=1. 25 m—2 é. 1(5-+4+m,2)=2 DF rei atta: ee a (19 and a); in which, of course, m>1 and sre the second or third term must=O, as m is odd or even. As P'(5+4+m,2)=R*(5+m,2) evidently=0, we have, by addition of c, d, d’, d", e, A. P(54+m,2)= an +m,2) + R(5+m,2) +1(5 +m,2) m— m—2 ns =2 pol. 2 vag + 2n-1° oe where m>0O, and 2,,=1-—2,, , is a ceeniinee of the usual , mM. : ; form ;»?.¢. 3,1 or=0,, as | 18 or is not integer. This P(5+m,2) is the number of triangular parti- tions of the (5+m)-gon in which two angles, and two only, are untouched by diagonals, or the number of tri- edral partitions of the (5+m)-ace in which two pairs of rays, and two only, of the #-ace remain each a united pair. J. P(7,8)=P(4) =R*(7,3)=1, by (4,25). 70 R37 -+m,3)=0", by (20). RR*(7 + m,8) =2?, es [8 3 REV. T. P. KIRKMAN ON THE by (21,f), (m>0). i. I5R9(7+m,8)=2°, by (22,/). j. in(7+ms)=1{ Gt 2” 8.28 28}, by (23,f). (h+i+J) gives, B. P(7+m,3)= ele) rein ae 42%, (m>0); the number of triedral partitions of the (7+m)-ace in which three pairs of rays remain each a united pair, or of triangular partitions of the (7 +)-gon in which three tri- angles are marginal. k. P(9,4)=P(5) =R%(5,2) =R2(9,4)=1, by (4,a,25). 1. R(9-+m,4)=24, (m>0), by (15,4). 2|1 Abie +1 ee ara : m-2 m= mn RR(O+m+4) = sige .27 —2*, by (17,h). 2\1 (F+1 ) mee m—4 n. RR*(9+m,4)=—7—— + 2 = _2F by (18,4) kax=0. m+1\?!} 3a! ee n. RR*(9 +-m,4) = >a - 2? (moda), by (18,4). jax=1. AN tate eo m—8 0. sae a a Seah ig 2\1 asaya x rete I ye m+1,?!3 (- 9 ) m—3 # * by: (194). PARTITIONS OF THE X-ACE AND THE X-GON. 71 The addition of (+m+n+n'+n"+0), remembering the observation (Art. 25), gives, 141) me m—8 C. P9+m4y= tN 9 — ity? m—8 +2, > (m+2)(m+6) - 2 ? m—9 42, 1° (m+1)(m+8) -2 2, m>0); the number of triedral partitions of the (9 +m)-ace having four pairs of rays united pairs, or of the triangular par- titions of the (9-+m)-gon having four marginal triangles. p. P(11,5) + P(6)=R(6,2 ) = i= R(11, au (4, d’); for when every base summit ‘of “R(6,2) 3 is nes as in Art. 4, the line j in the axis becomes a line & in the axis. 2|1 a) 2 eb g RR(OI1 +70) aba #6 22, by (18 and p). m i aE EP ae. a =? . TROL +m5)= GEV. 2 ‘apie ie? by (14,p). The addition of these gives us, D. PQ ae ee gn m—8 +2,,- (m+2)(m+4) + 2 Re (SO; the number of triedral partitions of the (11+ )-ace having five of the angles about the vertex undisturbed, or of the triangular partitions of the (11+mm)-gon which have five marginal triangles. gs. P(13,6)=P(7,2) + P(7,3) == (752) + R72) + R*(7,3) ar=C. Ly i Pe a a (4,¢,d',9), =1%13,6) + R(13,6) + B°(13,6), (Art. 4). kax=0, t. TI?(13+m,6)=2?->r1— » by (9,8). REY. T. P. KIRKMAN ON THE (m+1)8" Ae, (Gat 7 U. 11°(13+.m,6)=2”"" = elie x . kaa by (10,3). m 4|1 v. RR(13 +m,6)=2" - aE by (13,8) m+1)*'" mr (== Ww. RR(13 +m,6)=2 A mata , by (13,8). saan Oat x. IR(13+m,6)=2"" - Frees al ° 2 1a m+1)1!3 wa CO) _o a by (14,8). y. R13 + m,6) =6n- 25, by soni eh . 2 __6,,-2%, by (21,8). z. RR(18+m,6)= =2" ylt 2\1 : eu ada. TPR3(13 +m,6) =2° . SL els by (22,8). 8|1 m bb. IR3(13-+m,6) = t{an. (nt) 4.28 1 2\1 m—2 7 +1) m 3+) | ie Sgt Bonga E : by (23,8)- The addition of these nine quantities gives us, 7 gn, = ss iy a E. P(13+m,6)=3 + or) wo o|3 + oe Cole wo oo PARTITIONS OF THE X-ACE AND THE X-GON. 73 a oS ay a2 29 the number of triedral partitions of the (13+ m)-ace having six angles of the polyace undisturbed, or that of the triangular partitions of the (13+ 2)-gon in which are always six marginal triangles. Ge.4 P(15,7)=P(8,2) + P(8,3) =R(8, 2) Boks 2) +1(8,3) en a lage oe Ue by (2567) = R(15,7) +1(15,7) kax=1. = 2 + 2. (Vide p, and Art. 4.) dd. RR(15+m,7)=2:- 2, by (13,cc). kax=1. aL bot m+1 10{1 G ee, IR(5+m,7)=1- FP —.2m—1 : page yas ea . oF by (14,cc). 101 if. Ts oy a ee by (8,cc). Adding these three quantities, we obtain, (5 +1) 10|1 m F. P(154+m,7)=3-2"- —— : 42,72? —_—— 3 the number of these partitions of the (15+ 7)-ace, or of the (15 + m)-gon. And thus we can, without difficulty, proceed to find for any value of x and m, P(2Q2+m-+1,#), the triedral par- titions of the (2a -+m-+1)-ace having # pairs of rays united pairs, or the triagular partitions of the (2a -+m-+1)-gon having 2 marginal triangles. In the Philosophical Transactions for 1856, in a memoir “On the enumeration of the #-edra having triedral summits and an (#—1)-gonal base,” I have given a VOL. XV. L 74 REV. T. P. KIRKMAN ON X-ACE AND X-GON, ETC. solution of the problem of this paper in a form less calculated for obtaining easily the algebraic expression of P(2Qex+m-+l,2); and in the same Transactions for 1857, in a memoir “On the partitions of the polygon and the polyace,” I have given formulz for all these partitions, whether triangular or not. The method of this paper is unquestionably the best adapted for the deduction of the general expression of these partitions, and is capable of being extended to them all. IV. — On Improvement in Meteorological Registration. By Mr. Tuomas Hopkins. Read January 12th, 1858. Great numbers of persons are engaged in collecting in- formation respecting the changes which, it is well known, frequently take place in our atmosphere. Many indivi- duals register the alterations indicated by meteorological instruments with great care and industry, and govern- ments have established and maintained observatories for this purpose, at considerable cost; yet it has been said by eminent men, of whose competency to form opinions on the subject no doubt can exist, that the results obtained are of little value. This suggests the idea, that the course pursued in investigating the subject may be defective, and warrauts a particular examination of the methods adopted. The weight of the atmosphere, in each locality, is almost constantly varying, and the amount of that weight as well as the alterations which it undergoes are indicated by the barometer. But the most vigilant watching of the height of the mercury, combined with observations of the state of the sky, has failed to exhibit to our view the causes of the fluctuations of atmospheric pressure. There is a general impression that, in some way, heat lightens the atmosphere, and that cold makes it heavy, but in what way the changes, from the one to the other state, are effected, is not known. 76 MR. T. HOPKINS ON IMPROVEMENT The principal instruments used in meteorological inves- tigations are the barometer, which may be considered to measure correctly the various alterations that occur in the amount of atmospheric pressure, the dry thermometer, and the wet bulb thermometer. In addition to these the dew-point, or point of condensation of vapour is ascer- tained, from which is deduced the tension, or pressure, of vapour. The barometer shows the total atmospheric pres- sure, but that pressure is known to be made up of two parts,—one the pressure of the gases, and the other that of the vapour which is diffused through them ; and in inves- tigating the subject it is necessary that we should know how the separate pressures of their constituents are ascertained, in order that we may judge of the degree of reliance that may be placed on the results that are obtained. It has been found by experiment that aqueous vapour loses its aeriform state, and is converted into water, when exposed to a sufficient degree of cold, or, in other words, when heat is sufficiently abstracted from it by reduction of temperature. And the more dense the vapour, the more readily does a reduction of temperature convert a part of the vapour into liquid. Hence it is seen that the varying quantities of aqueous vapour found in the atmosphere may be condensed by different degrees of cold; and the particular temperature at which the vapour is converted into water indicates the degree of abun- dance, or the density, in which it exists in any part of it. The temperature of liquefaction, or the dew-point, there- fore shows the quantity of vapour present in the atmos- phere. And each quantity has a particular weight and elastic force, with which it presses on the earth, and of course on the mercury of any barometer placed on its surface. This force is generally called the tension of vapour, and it indicates the quantity of vapour in the atmosphere, at any and every period of time. IN METEOROLOGICAL REGISTRATION, 77 The tension of vapour is found to be different in various climates and seasons. In warm and moist tropical countries where vapour exists in great abun- dance, its tension is sometimes equal in force to the weight of about an inch of mercury. In colder countries there is less vapour in the air, and therefore its tension is less, until in very cold and dry countries the quantity is so small as to make it a difficult task to ascertain its amount. But whatever may be the tension of the vapour that exists in the atmosphere, it is considered to constitute a part of the general atmospheric pressure which is measured by the barometer; and, the vapour pressure being deducted from the total atmospheric pressure, the remainder is deemed to belong to the gases. Tables of the tension of vapour have been constructed to show what belongs to each dew-point, by which means the separate pressure of vapour has been given in meteorological registrations. This being subtracted from general atmospheric pressure, as shown by the baro- meter, gives the separate amount of gaseous pressure. The barometer shows that atmospheric pressure alters, from time to time, to the extent of three inches of mercury, making one-tenth of the total average weight of the atmosphere; and as tension of vapour seldom equals one inch, the cause of the greater part of the alterations im the weight of the atmosphere must exist in changes of the gases; hence the importance of ascertaining what those changes are. After finding what was due to tension of vapour, in order to account for the alterations which take place in the heights of the barometer, the temperatures of the gases, as ascertained by a thermometer, have been care- fully given. It is well known that heat expands gases in the open atmospheric space; and in the proportion in which they expand they diffuse themselves around to a distance, becoming lighter in the part from which they 78 MR. T. HOPKINS ON IMPROVEMENT have expanded, and consequently pressing there with less weight on the earth and on any barometer near it. Change in the temperature of the gases is therefore recognised as a cause of alteration of atmospheric pres- sure, as it is measured by the barometer, and, taken in conjunction with changes of vapour tension, is presumed by meteorologists in general, to cause all the fluctuations of the barometer, no other cause being acknowledged to influence atmospheric pressure. Yet the barometer occasionally sinks without either temperature or vapour tension showing alterations to account for it. Indeed sometimes, tension of vapour varies but little, and temperature declines; but the baro- meter, so far from rising, as might be expected from the recognised influences of these two forces, actually falls, often to a considerable extent. From these circumstances we may infer that neither reduction of vapour tension, as indicated by the dew-point, nor fall of the temperature of the gases, as shown by the thermometer near the surface, nor both together, are the real causes of the sinkings of the barometer that sometimes take place. The sun is undoubtedly the great scurce of the heat that disturbs the atmosphere, but its heating effect in different latitudes is unequal, and its influence on dif- ferent parts varies also with its passage across the tropics ; yet any partial effect thus produced on the weight of the atmosphere is, in no long time, counteracted and corrected by the operation of gravity, which imme- diately begins to act on the aerial ocean to restore the equilibrium of atmospheric pressure. Therefore no change of temperature as shown by the thermometer has furnished reasonable ground for believing that the great falls of the barometer that often occur can have been caused by such changes as are indicated by the thermometer. The sun has a daily, as well as an annual influence, but IN METEOROLOGICAL REGISTRATION. 79 the effects of daily changes may be traced better than those of annual ones. The atmosphere is disturbed by solar influ- ence every hour of the day, and these disturbances take place so regularly as to admit of their being followed and minutely examined, with a view to discover the precise way in which the solar influence is exerted. The daily changes therefore afford better means than the annual ones of inquiring into the causes of atmospheric disturbance. In most parts of the world, the atmosphere becomes heavier from four until ten o’clock in the morning, during which time the baro- meter rises; and the causes that must then have been in in action to produce these effects, being so regular, may be followed, and the forces may possibly be exhibited. This has been attempted for a considerable time, but the results obtained are by many persons deemed unsatisfactory ; it therefore becomes desirable that we should examine the means that have been employed to account for barometric movements, and if they should be found to have been defective, to point out their defects and suggest improve- ments. With the approach of the sun every morning, evapora- tion of water from the part of the earth’s surface heated by it becomes more active, and additional aqueous vapour is sent into the air, showing itself in a higher dew- point and an increase of atmospheric pressure, which of course raises the mercury of the barometer. The hours during which these phenomena occur vary to a certain extent with latitude and season, but the average time is from four to ten o’clock in the morning, and within that period the barometer rises in consequence of an increase of aqueous vapour. At a certain time, how- ever, say ten o’clock, the barometer begins to fall, although vapour continues to pass, even in greater abundance, into the atmosphere. In the present state of our knowledge of the subject we naturally ask, What can be the cause 80 MR. T. HOPKINS ON IMPROVEMENT of this fall? Meteorologists point to the movements of the thermometer, and to the changes in tension of vapour, and endeavour to find in them an adequate cause for the reduction of atmospheric pressure which then occurs. But the registrations of their instruments have been of such a kind as to make this reference unsatisfactory, and have consequently permitted vague and insufficient expla- nations to be given by some, and to be received by others. These explanations have proceeded on the assumption that the fall of the barometer was to be accounted for by the alterations which take place in temperature and in vapour pressure, though this assumption may from many of these registers be shown to be erroneous. Of these, the observations made at Toronto in the year 1846, and pub- lished by the government, may be deemed the most suit- able for our present purpose. The whole Toronto observations extend over the years 1846-7-8; but in order to have a case sufficiently simple to admit of being dealt with in the present paper, while it is a fair specimen of what occurs in many parts of the world, I will extract the hourly registrations of the baro- meter from four o’clock in the morning until ten at night of each day during the month of July in the year 1846, and then give in the same tabular form, for the same time, the recorded amount of tension of vapour, as deduced from the dew-point. The tension of vapour being then sub- tracted from the total atmospheric pressure as shown by the barometer, the remainder corresponds, according to the present views of meteorologists, to the separate pres- sure of the atmospheric gases, which is also given in another line. Having appended the corresponding tem- peratures as recorded from a standard thermometer, we may proceed to examine how far the results harmonise with the views of meteorologists. — (See the Table.) 81 IN METEOROLOGICAL REGISTRATION. $9.08 69-08 89-07 GZ-.9 96-02 CV.6 99..6 PE.06 18-08 69-08 T9-.8 10-08 PP-8 18-0G GhroV €9-08 LG0G 0-06 10-03 M “NOILVYOdV AT GE+069 8E009 86-019 E0-0F9 9G-089 P8-oS9 19-99 66-099 84-099 L609 FF-o99 08089 $4099 19-G9 BS-.F9 Zl-09 90-09 00-049 88-09 “AHLANOWYAAL €TOd TAM 06-069 L609 99-099 8004 Co0E4 66-094 L094 ChroGh G9-oGL CSL GOS T8c84 LloSL SS-ol4 FSo69 G4-099 €9-.69 G0-.69 68-089 “TA LANONWYAH “GEST. SS0I- TIZ0- TPO. OL80. 86E0- S8FO- I8G0- GIe0- Tsg0. ezc0. e890. 6160. FSS0- @F90- LSL0- G6IIT- TIST- TTPt- SOYOUT GF “HUOSSHUd SNOASVY ee ee ee ie ee OVOP. OOSP- O86). OGTS- O&GE- OGTS- OGTS- OFS. OSPS. OOS. OPPS. O9EG- O0GS. OSG. OSPS. OTEs- O88h O&Fh- OGhP ‘aN0dVA dO NOISNAS GL8&- SG8S- T69G- T€9S- O09G- S82. S49G. TI4G- G94. TSG. S969. FOO. 6109 F809. g609- 4909 666¢- TP6e. TEse. SOYOUT G& “TALAMOUV EL Dip SO 806 em 2 8s SVS 8 ee hi. i Ob 2G seme Meemegian 9G sm *sInoy] VOL. XV. 82 MR. T. HOPKINS ON IMPROVEMENT Here we may perceive that at four o’clock in the morning, the barometer stood at 29°5831 inches, and that by nine in the morning it had risen to 29°6084, being a rise of 0°0253 of an inch. At the first named hour, the tension of vapour was 0°4420, and by nine it had risen to 0°5550, or 0°1130 of an inch, leaving for the separate pressure of the gases at the first hour 29°1411 inches, from which it declined to 29-0534 at nine; so that the increase in the pressure of vapour must have counteracted the whole influence of increased temperature during the time, and also raised the baro- meter 0°0253 of an inch. The additional vapour pressure must have been a result of greater energy of evaporation, arising from the higher temperature sending more vapour into the air, and the increased evaporation is shown in the difference in the registrations of the wet and dry ther- mometers, as seen in the table above. This amounted to 2°01 at four o’clock, and increased to 5°81 by nine o'clock ; and we have seen that up to that hour enough vapour had been sent into the air to raise the barometer 0:0253 of an inch, as well as to counteract the influence of the increased temperature on the gases. A change now took place, and after nine o’clock the barometer fell until five in the afternoon, when it was as low as 29-5588 inches; and here we naturally ask: What was, at this time, the influence of the aqueous vapour which had just before shown itself so powerful in producing a rise in the barometer? It would seem to follow as a necessary consequence that as a rise of that instrument was caused by an increase of vapour, a fall would be accompanied by a decrease, or at least that the tension of vapour would show a reduction when the barometer fell equal to its increase when that instrument rose. At nine o’clock, the time when the barometer began to fall, we have seen that evaporation was at 5°81, but IN METEOROLOGICAL REGISTRATION. 83 instead of then becoming less, it became more energetic, and increased until five o’clock in the afternoon, when it was at 9°45, making an increase in the energy of evapo- ration from nine in the morning until five in the afternoon of 3°64, yet, notwithstanding this increase of evapora- tion, the barometer fell no less than 0°0496. It is, however, to be noticed, as may be seen from the table, that though evaporation had so greatly increased, the tension of yapour was somewhat reduced after nine o’clock; yet that reduction was small compared with the fall of the barometer. The tension was found at nine o’clock to be 0°5550, from which it became reduced until at five it was only 0°5190, but the ba- rometer fell during the same time from 29°6084 to 29°5588. Thus it required an increase of 0°1130 of vapour tension in the morning to raise the barometer 0°0253; but a fall of that instrument afterwards until five in the afternoon of 0:0569 was accompanied by a reduction of only 0:0360 of tension. So that tension of vapour is exhibited as being powerful compared with temperature, the other varying constituent of pressure, in the morning, and feeble in the evening, involving a mass of contradiction and confusion. From these statements, however, it beccmes sufficiently apparent that the weights of the constituents of atmo- spheric pressure cannot be inferred from the tension of vapour, seeing that the vapour which passes into the atmosphere between nine and five does not appear in the tension. I have shown elsewhere how the vapour that passes into the air at this time of the day reduces, instead of increasing, tension, but the aqueous matter in the atmosphere must evidently be increased by the evapora- tion; though there are no means of measuring the extent of that increase, unless we ascertain directly the quantities of water evaporated. Yet whatever the quantity may be, when it does not fall as rain, it remains in the air either as vapour or as cloud. And while in the air as cloud, it does 84 MR. T. HOPKINS ON IMPROVEMENT not lose its property of gravitation, but as ponderable matter must float in and be sustained by the gases. Indeed, it may generally be seen there as mist or cloud, and as both are sustained by the gases, the weight and pressure of the gases must be augmented. But from nine to five we have the fact that the barome- ter fell considerably, which proves that the gases must then have pressed on the surface with diminished force: and as they bear the weight of the clouds that float in them, they must themselves, from some cause, have been rendered so much lighter, as to be less heavy with the cloud in them than they were before the cloud was formed. That they are reduced in weight, whatever may be the cause of the reduction, is proved both by the falling of the barometer at the time, and by the kind of disturbance which takes place in the atmosphere. Solar heat makes the air light at the time, although it is evidently not the identical heat which at the same time affects the thermo- meter near the surface, It is, however, at the time that the barometer falls that the surface is heated, because solar heat is also then displayed in heating the surface ; but it does not then materially disturb atmospheric press- ure unless it finds aqueous matter to unite with. When, however, the heat forms an union with water, it passes into the upper regions of the atmosphere, where it acts in expanding the gases, diminishing atmospheric pressure, and producing winds. In meteorological registers the tension of vapour is sometimes deducted from total atmospheric pressure, and the remainder being treated as dry gaseous matter, is tabulated and occasionally exhibited in curved lines. These lines present an appearance to the eye the reverse of the curve of temperature as shown by the thermometer; and it seems not improbable that this form of exhibiting changes in the atmosphere has contributed IN METEOROLOGICAL REGISTRATION. 85 to the retention of the present unsatisfactory mode of treating the subject of meteorology. But it should be borne in mind that the middle of the day is the time at which the sun exerts its power to send vapour into the higher regions, and that at the same time the surface is heated. Yet the heat which is expended in raising the tem- perature of the surface is evidently not the same that has ascended in vapour to the upper atmosphere; and as the latter portion of heat produces the effects we are tracing, we should endeavour to follow zt, and discover its particu- lar mode of acting. Tc do this it may be necessary to point out, in the best way we can, in the present state of meteorology, the separate action of the solar heat on dry ground, and on the water which is spread so extensively over the earth. In places where there is little or no water to be con- verted into vapour, such as dry sandy deserts, it is well known that the summer sun greatly raises the tempera- ture of the land every day. Even on the sea coast of Lancashire in the summer I have observed that in the middle of the day, while the air over the moist sand of the shore was at 60° the sand itself was at 75°, and the dry sand among the contiguous hills was at 90°. In this case an equal direct supply of solar heat raised the dry sand 15° above that which was moist; but this difference was a consequence of much of the solar heat that reached the ground passing away in vapour from the moist portion. A hygrometer, held in the air at the time, showed that vapour was ascending from the moist shore in great abundance. In dry countries, far from the sea, as less water exists to receive the heat, less vapour is daily raised by the sun, and therefore the temperature of the surface is raised to a higher degree. Burns, when near Bokhara, says: ‘The heat of the sand rose to 150° and of the atmo- sphere to 100°;” but then no water was present in the 86 MR. T. HOPKINS ON IMPROVEMENT district around to be converted into vapour. Yet at the same time at more than 15° farther south, in the moist countries of Bengal and Birmah, the temperature of the wet grouud was seldom so high as 80° even when the sky was unclouded. Here we see that the solar heat which raised the temperature of dry ground in one part to 150°, raised that of wet ground in warmer latitudes not much more than one-half; but this difference evidently arose from the circumstance that much heat was taken from the wet ground by vapour which passed into the air. The solar rays make the surfaces of dry deserts, in warm latitudes, very hot, partly because dry air is a bad con- ductor of heat, and partly because the air does not ascend, as newly-formed vapour does, through another and a colder medium. Ascending currents of warm air are never found over hot deserts. Such parts receive and accumulate heat in the day and part with it during the night without creating material disturbance in the atmosphere. The great cold of the night, as compared with the heat of the day, has been long noticed in the deserts of Arabia, but no such atmospheric disturbances arise from the heat of those parts as are often experienced over certain tropical oceans that have a temperature below 80°, or over the compara- tively cool but wet countries of tropical parts. In some deserts the cold of the surface at night is sufficient to condense a part of the little vapour that is in the air into water or dew, but this water is generally reconverted into vapour by the morning sun, without enough being sent into the atmosphere to produce dense cloud and heat the air ; therefore, although the barometer might be raised by the morning evaporation, very little mid-day vapour can rise from the already dried ground. In parts of the great African desert where little or no rain falls, much dew is formed on the ground at night by cold, and where the land is level, it is probable that the evaporation of this IN METEOROLOGICAL REGISTRATION. 87 dew in the morning produces the appearance of mirage so often spoken of, but the whole of the dew is generally sent into the atmosphere in the form of vapour without creating a dense cloud during the day. In such countries the vapour of the morning may raise the barometer toa certain extent, but if there should not be sufficient condensation of that vapour, in the middle of the day no fall of the barometer can then take place. In other and more moist countries enough of water might be evaporated and raised and sufficient vapour be condensed to cause a small decline of the barometer; and in still wetter countries a greater decline might occur, until in such parts as Bombay we have the maximum amount of the daily disturbance. The Toronto observations were made in a country of lakes, where a large surface of water exists, from which evaporation in the summer takes place freely, and vapour is sent into the air in considerable abundance until rather a late hour of the day; that place therefore is not well calculated to show what occurs over the deserts of Asia and Africa. The central parts of southern Russia offer good sites for examining dry atmospheres, but unfor- tunately, in the registrations made at the numerous stations of that country, the heights of the wet bulb ther- mometer are not given in connection with those of the dry one. This omission marks the defective state of meteoro- logy at present, the importance of ascertaining the amount of vapour that is hourly sent into the air by evaporation evidently not being perceived. In Russia, no doubt, as in other parts, it has been supposed that the tension of vapour at all times showed the weight and pressure of all the aqueous matter that was in the atmosphere. But it has been shown that the pressure of vapour, as exhibited by its tension, though itself interesting, is but a part of what is desired to be known, which is, the causes of the great disturbances in the weight and pressure of the atmosphere in all parts of the world. 88 MR. T. HOPKINS ON IMPROVEMENT In order to obtain this knowledge it is necessary that the metamorphoses of water should be traced, as has been here attempted in the case of Toronto. We have seen, in that place, that evaporation of water increases atmospheric pressure up to nine o’clock in the morning. But after this hour the vapour parts with its heat of elasticity, which heat then adds to the elasticity of the gases, and lightens them, and we wish to know to what extent this is done. The energy, or rate, of evaporation is the best evidence that we have of the quantity of vapour that is sent into the atmosphere, and therefore this energy should as far as possible be shown. In July at Toronto it was at 2°01 at four o’clock in the morning. At nine it was at 5°81, and at five in the afternoon it was at 9°45. Yet after nine in the morning the quantity of vapour found in the air as shown by its tension became less, and it follows that all the vapour which ascended after nine o'clock, toge- ther with the portion that was no longer found in the tension, disappeared ; and there is no doubt that it must have been condensed into liquid in the higher regions. And as we presume that this condensation liberated the heat which caused the fall of the barometer at Toronto, we might infer the extent to which that instrument would sink in other parts, if we could ascertain how much vapour is sent into the atmosphere and condeused under similar circum- stances in those parts. Or we may proceed in the reverse course, and from the fall of the barometer infer the amount of vapour that had been condensed in the upper regions. The aqueous portion exercises so important an influence on the movements of the whole mass of the atmosphere as to require that its metamorphoses should be traced with as much accuracy as possible, and particularly that portion of vapour which ascends from the time that the barometer begins to fall in the morning, until that instrument turns IN METEOROLOGICAL REGISTRATION. 89 and rises in the afternoon. To contribute to the attain- ment of this object I have shown in the table the difference between the temperatures of the wet and of the dry bulb thermometers during the day, which difference expresses the energy of evaporation. Jn all meteorological observatories this should be attempted. The extent to which the wet bulb instrument is kept below the dry one shows the proportional amount of vapour sent into the air. And the hourly heights of each instrument may, during the day, be easily worked on paper ruled to a scale of degrees of temperature. By drawing a straight base line on such paper to represent the wet bulb thermometer, then dotting the height of the dry above the wet one at each hour, and connecting the dots by lines, there would be presented to the eye a curve, showing the relative quantities of vapour that ascend into the atmosphere during each hour. It would be easy to compare these curves of evaporation with others, representing the mid-day falls of the baro- meter, with a view to tracing their connection. If the same kind of registration were to be made when storms occur in any part of the world, it would almost certainly throw new light on the cause of storms. The movements of the barometer, as an indicator of atmospheric pressure, are well known, but the influence of vapour, while undergoing condensation into water, on that pressure, is not known, and therefore to this point atten- tion should be strongly directed. As long as the fresh vapour of the morning shows itself only in an increase of tension it may be considered to produce no disturbance of the gaseous atmosphere, though it may raise the barometer. It will, therefore, then be sufficient to register the wet and dry bulb thermometers, the tension of vapour, and the barometer, as these will exhibit what is taking place during the time. But when the barometer begins to fall at nine or ten o’clock the difference between the two thermometers VOL, XV. N 90 MR. T. HOPKINS ON IMPROVEMENT should not only be carefully tabulated, but graphically exhibited; and if it be found that as more vapour rises the barometer falls, and fo an extent proportioned to the quan- tity of vapour that is ascending, it may be taken as evidence that the vapour supplies the means of reducing atmospheric pressure. And if, in addition to this evidence, the tension of the vapour in the lower regions is found to be reduced, that fact may be considered a proof that the quantity of vapour in the air has become less because some of the quan- tity which had previously been there has been condensed ; and the barometer alone, by showing how much the atmo- sphere has been lightened, would be evidence of the total quantity of vapour that had been condensed. In tracing these facts it could hardly fail to be perceived that ascend- ing vapour is the important agent in reducing atmospheric pressure. From ten o’clock in the morning until four in the after- noon is generally the time when the barometer falls, and during this period the different heights of the wet and dry bulb thermometers show the energy of evaporation —when the tension shows the quantity that remains as vapour, — whilst the fall of the barometer indicates the quantity of heat liberated by condensation of vapour to warm and lighten the atmosphere. Such a system of recording atmospheric changes would present a tolerably clear view of the alterations that take place in the higher regions in the middle of the day, and the facts being presented in a tabular form any meteorologist would be enabled to com- pare the different phenomena, and to connect them with the working of known laws of nature. Hourly changes have been traced for the reason already given, viz., that we are better able to collect evidence of their nature than we are respecting those depending on the annual revolutions of the earth. But if we can prove what disturbs the atmosphere at particular hours so much IN METEOROLOGICAL REGISTRATION. 91 as to produce a palpable daily change in its weight and pressure, we may apply the knowledge thus obtained to enquiries into the larger alterations that occur irregularly. Little doubt can be entertained that the same cause which makes the barometer decline daily, reduces atmospheric pressure during storms to a larger extent. The daily reduction of pressure, when no rain falls, is not often more than equal to half an inch of mercury ; but if condensa- tion has begun, at any time when there is a sufficient sup- ply of vapour, it may go on until the atmosphere becomes much heated in one part to a great height, and its weight in that part may be so much reduced, that the barometer may fall one or two inches — or even more, — the vapour to effect this large decline being brought from distant parts by winds. 92 REV. T, P. KIRKMAN ON THE V.— On the general solution of the problem of the poly- edra. By the Ruv. Tuo. P. Kirxman, M.A., F.R.S., Rector of Croft with Southworth, and Honorary Member of the Literary and Philosophical Society of Manchester. Read January 26th, 1858. Lone as the question -— How many #-edra are there? — has been before the mathematical world, no method has hitherto been discovered, except that of tentative con- struction, whereby the number required can be completely assigned even for v=8. ll that has been effected, so far as I can learn, towards the solution of the problem by general methods, may be seen in four memoirs of mine printed in the Philosophical Transactions for 1856 and 1857, and in a later memoir “On the partitions of the y-pyramid,” of which some account is given in the Pro- ceedings of the Royal Society for 1857,* and in another “‘On the triedral partitions of the 7-ace” in the Manchester Memoirs for 1857. In these papers are discussed certain great classes of the x-edra. In the pages following is made the first attempt at the analysis and solution of the whole problem. It is known that if the Q’ faces of p’-acral Q’-edron be an A-gon, a B-gon---a Q-gon, and its p’ summits an a- * This memoir is now printed in the Philosophical Transactions. SOLUTION OF THE PROBLEM OF THE POLYEDRA. 93 ace, a b-ace---a p-ace, e being the number of its edges, the equation following will be satisfied. (A)* A+B+4C+4+---+Q=a+b+e4-+-4+p =2e=2(p'-14+Q’-1); and there are in general many p’-acral Q’-edra, having a common description in this equation. But to determine whether this equation, when values of AB, &c. are chosen, really describes polyedra, and how many it includes, all differing in the mutual arrangement of the faces and sum- mits specified, has hitherto been found a problem of insu- perable difficulty. This difficulty arises from the fact that the problem is chiefly tactical; and we have had no ¢actic calculus, nor any means of expressing tactical conditions in algebraic language. It is easy enough to multiply theorems in combination, when these are symmetrical and exhaustive; but to assign the number of combinations or systems of combinations that can be selected, so as to fulfil given unsymmetrical conditions, and to be all essen- tially distinct, has thus far been a task beyond the powers of our algebra. The question, How many z-edra are there? is not with- out interest for its own sake; for every x-edron is one out of a precise number of permanent facts in nature: its principal charm, however, has lain in its difficulty, and in the consideration that the answer has been part of the sublime unknown. It is precisely on problems like this that the modern analysis delights to try the edge and temper of her tools; or rather to exercise herself in the handling of instruments which, though lent to human workmen, are not of human workmanship, but whose fabric and finish are divine. The obstacle to research in this direction has been * Vide my paper “On the representation and enumeration of the Poly- edra,”’ — Manchester Memoirs, vol. xii. 94. REV. T. P. KIRKMAN ON THE mainly the difficulty of stating the conditions of the pro- blem in algebraical propositions, 7.e. in equations. In this paper the reader will find that this statement is effected, and that nothing remains to be done for the complete solution except a certain amount of laborious elimination. The first step is to invent a representation of a polyedron on paper in symbols that algebra can handle. The con- sideration of the solids in space will help us little. I have tried to refer the figure to its amplest, 7.e. most angled, face as a base, and the figure has obstinately refused to acknowledge one face rather than another for a foundation. Its amplest face may be a g-gon, but it may have a hun- dred of these g-gons, about every one of which may stand a different configuration of summits and faces. I have endeavoured to generate the figures by defined processes from pyramids; but not only is the solid in general dedu- cible from different pyramids, but often in a great number of ways from the same pyramid, no one of which will allow another to be a better way than itself. And when I have attempted by authority to put the work into the hands of some oue of these methods, all equally clamorous and con- fident, I have invariably found that, in spite of all I could do to prevent it, the operator would persist in carving the same polyedron over and over again in different postures, so that it was impossible to know how many really distinct ones had been generated. The true method of looking at the question turns out, as is usually the case, to the mortification of wasted inge- nuity, to be a very simple one. All that we require is to have a clear statement of the summits that are about every face, and of the faces about every summit, and of the order about each in which they stand. This order is fixed by a correct account of the SOLUTION OF THE PROBLEM OF THE POLYEDRA. 95 edges, showing the two faces and the two summits bound- ing each edge; for if the edges _E and F make an angle on the solid, they will have then both a common face and a common summit. Thus to form a paradigm of the 9-acral 8-edron here figured, which is one of those described by the equation, 5, +5pt+4e4+4e+3p4 304 324+3u =5,+4,+3,+3,4+3,4+8,4+3,+3,+3,=2-15, we mark the faces in any way A BC.-.-.-, and the summits abc--- Wenext form a table of eight columns, headed ABC..-, making nine rows headed by abc:- ay ye ae se A oe oh Om eel. Many te Pao ta, eyes) Ee, eee ob tot: teen oe, ie aaa Begone! Co) aa ag A wee 3 D,, F, E, 3 D, F, G Bak AG Down uiiGe arate Ses a gt idee pale ara gata aes 5 3 Rife Cadet wr hans Site putting M,, for the point common to the column M and the row n, and entering M,=1, or M,=0, according as the face marked M has or has not the summit marked n for one of those about it, 7.e. according as the summit has or has not the face for one of those about it. Thus we have entered A,=1=F,=B,, because the pentace a is in the pentagon A, the tetragon F and the triangle B; and D,=0=G,, because that pentace is not 96 REV. T. P. KIRKMAN ON THE in the pentagon D, nor in the tetragon G. Hence it comes to pass that we read five units under the pentagon A, showing all its summits, and five units opposite the pentace a, showing all its faces, &c. The fifteen edges of the figure are represented thus: AjA,B,5;=% AjA.C,C,=1, A,A,D.D,=—1, A,A,G63=5 A,A,H,H,=1, D,D.E,E.=1, B,B,C;=1, B,BzE,E,=1, C.C,D.Da=1, D,D,G,G,=1, D.D,F.F-=1, G,G,F,F;=1, G,G,0,0,=1, FP n,n = 1, 2b Bee 2) Each factor in every quadruplet is unity. The paradigm shows these edges each one as four points of a rectangle; thus A, A, D, D, are four such points, and F, F; H, H; are such also. The charming peculiarity of the paradigm is, that no displacement of any face and its column, or of any summit and its row, can break these edges, 7.e. spoil these rectangles ; so that the same polyedron is accurately exhibited, whatever be the order of the letters A B C---or abc--- And no rectangle occurs in the paradigm which is not an edge of the polyedron. The property of these quadruplets is, that no duad of two capitals or of two small letters appears a second time, while every duad of a capital and a small letter occurs exactly twice over, and always in two different quadruplets. If the reader will take any other 8 numbers, ABC.-. and some 9 adc.--., satisfying such an equation as (A), and endeavour by trial to form with them a paradigm showing fifteen edges, he will, whether he succeeds or not, obtain some idea of the tactical difficulty involved in this problem of the polyedron. Or he may try to satisfy him- self as to the possibility of forming other 9-acral 8-edra with the numbers above handled. The next step is to express all the conditions of the question in the shape of equations, which are to be true independently of all arrangement of their symbols. SOLUTION OF THE PROBLEM OF THE POLYEDRA. 97 We have p’Q’ unknown quantities M,, discontinuous variables, which can have any of them the value zero or unity, and no other value. The columns headed by A BC - ++, the faces in (A), give the equations, (M,) A=A,+A,+A,+++++A,=AZ+A5+--++A3, B=B,+B,+B,+---+B,=Bi+ Bj+---+By, Q=Q,+0,+Q,4+++-+Q,=Qi+Qj+---+Q), Q’ equations for any whole value >0 of 7, because M7, = M,,, whether this be unity or zero, and the rows give the p equations for any value integer and positive of 7. (M,) a=A,+B,+C,+---+Q,=A7+Bi+---+Qi, 6=A,+B,+C,4+---+Q,=A}+B+---+Q}, p=A,+B,+C,+---+Q,=A%+Bi4+---+Q5. The (p'—1+Q'—1)-edges have next to be expressed. If one of these is the intersection of the faces I J, and connects the summits m and n, we shall have LlndnIn=1, and Pike K=O = 3 Sede, whatever be K and p; because the three faces IJ K, having no common line, cannot all contain the same two summits m and n, and the three summits mn p, not being in the same line, cannot all be in the same two planes I and J. As the I-gonal face I has I edges, we have the following equations, by what has just preceded, (E) (A, ABnBrn+AmAnOnC,+AmAnDnDrat ++: +A,A,Q,Q,)=A 3(B,,B,AnAn+ BrBrCmCn+BrB,DnDnt +++ +B,,B,Q,,Q,)=B (Cp CpAmAn+ CnOnBmBn+ CmCnDmD, + + >> +C,,C,,Q,,Q,,) =C 3 (QQ AmAn a Q,,Q,B,B, “fs Q,,Q,D,,D,, a VOL. XV, i) 98 REV. T. P. KIRKMAN ON THE +Q,,Q,PP,)=Q, or more briefly, Dy) LE Cap, Eh where X is every letter in turn but I out of ABC---Q, and m n is every pair in turn of a bc-+ +p. In like manner, since the m-edral summit m has m edges, we must have (E) ¥(V.XaVeXe+ VaXaVeXe+ VaXaVadat*** +V,XiV,X») =a 3(V,X,VaXat VeXzVeX%e+ VeXeVaXat °° +V,X,VpXp»)=5 3 (VpXpVaXat VpXpVeXo+ VpXpVeXet+**: +V,X,V.X.) =P, or more briefly, SV Ska \ Mem, where a is every one in turn but m of a dc---p, and V X is every pair in turn of ABC.---Q. We have thus completely expressed both the tactical and arithmetical conditions of the problem. I fear that there is no shorter mode of arriving at the solution than by the way of elimination of this table-full of p'Q’ variables. This is possible; but the enormity of the operations implied is such as to give a very lively idea of the difficulty and vastness of this problem of the poly- edra. The p'+Q’ equations (M,,)(r=1) give the means of eli- minating as many variables from the p’ + Q’ equations (EF). The values thus found for those p’+Q’ are then to be substituted in the equations (M,,),(r=2), A=A?2-+ A? &., B=B2+B} &c., a=A?2+B? &., b=A}+ Bj &c., which will introduce certain duad products; after which all exponents may be erased from the variables, and prQ SOLUTION OF THE PROBLEM OF THE POLYEDRA, 99 more can be eliminated as before from equations (E). The values substituted for the expelled variables are next to be introduced in A=A3+ A? &., B=B3+B}3 &c., &c., a=A°+B &., b=A}+B} &c.; by which operation certain products of three, four, five * and six variables will be exhibited in the last written p+Q equations. We can then erase exponents and pro- ceed linearly to expel p'+Q’ more variables from equa- tions (E), and can thus operate till all are eliminated ; when an equation U=O will be the result, expressing a condition or conditions among the quantities A B---Q, ab -+ +p, which will be different from (A). The work will simplify itself as we proceed, by the erasure of all those products of variables which contain the capital J with more than J subindices, or the subindex 2 under more than 2 capitals; for we know that not more than J of J, J, J,, &c., can have value, nor more than 7 of A; B; C, &c. Or, if we wish to obtain a result in terms perfectly general, without any choice of value for J or i, we can content ourselves with erasing all exponents. Of course no product of more than Q’ different capitals, or more than p’ different subindices, can possibly remain. If we wish to construct and enumerate the polyedra of which (A) is the common description, we may do it thus. First eliminate all capitals but A. The mere inspection of the result in A, A, A,, &c., will show what systems of A values are possible, for A,,=0 A,,=1 are the only values that can arise, that is, no system of summits can be about the A-gon, which is not readable in that result; and if there be only one A-gon, it will often happen that only one set of summits is readable and admissible. We can pronounce with certainty that all are impossible but those (m n, &c.) readable in our result (A,,=1 A,,=1, &c.) 100 REV. T. P. KIRKMAN ON THE We next eliminate all capitals but A and B, obtaining an equation containing only A, A,---B, B,---. Putting in this for A,, &c., in turn the systems of A values not proved before impossible, we can read a certain number of sets of B values (B,,=1 B,=1, &c.) of which we can say that none besides are possible; that is, we can construct a certain number of sets of two columns under A and B, of” which we can say that every possible system is written down among them. Next, forming the equation contain- ing only A, A,---B, B,---C, C,-++, we can construct a limited number of sets of three columns, of which we can say with certainty that every possible polyedron is thus far constructed. And thus we can proceed till we have before us a certain number of sets of Q’ columns, of which we can pronounce that every possible Q’-edron is once, and only once, constructed; that is, we can, by a strictly de- monstrative process, both construct and enumerate the p’- acral Q’-edra described in equation (A). And this is a complete and rigorous solution, from the theoretical point of view, of the problem of the polyedra; for they are all comprised in descriptions of the form of (A). The result U=0, above mentioned, will show at once whether any polyedron is described in equation (A); for, if none exists, U=O will not be satisfied by the values of AB---Q ab---p under consideration. It appears highly probable that the same equation U=0 will be sufficient for the enumeration of the different poly- edra under (A), for the reasons following. Let }X,X,Y,Y, be the number of pairs of lines, edges or diagonals, drawn or drawable from the a-ace a to any two other summits 7 and s, a7 being in the X-gon and as in the Y-gon, which have the common summit s. No two different polyedra can have this sum alike at every summit a; for if they have, they will have the same faces in the SOLUTION OF THE PROBLEM OF THE POLYEDRA. 101 same order about every summit, i.e. they will not be dif- ferent polyedra. It is easy to form equations containing such sums. Thus, if we multiply the square of A by that of B, we have A?= J A?+235A,A,=3A,4+25A,A,=A+4254,A, B?= 3 B?+25B,B,=>B,+2>B,B,=B + 2>B,B,, 4(A?—A)(B?-B) =3A,A,>B,B,, where 7s is every pair in turn of a0---p, in each of the sums of this product. Hence, keeping A invariable, and giving to X 7's every possible value, 31 (A? A)(X?-X =F A,A,X,X,+ FA,AX,X,+ FA,A,X,X,, =A+3A,A,X,X,+ SA,A,X;X,, 5 which, by adding up Q’ such equations, one for every face ' in turn of A B---Q, becomes 33(X?-X)(Y?-Y) =2e + 25X,X,Y,Y,+22X,X,Y,Y,, where X and Y are every pair in turn of AB---Q, and rstu are every set in turn of four out of ad---p. In like manner it is proved that 43\(7?—1r)(#—s)=e+ 2X,Y,Y,Z,+ 5X,Y,V,Z, where 7's are every pair in turn of @b-+-p, and XYVZ are every four in turn out of A B---Q. So far as I have examined cases, I find no two different polyedra having the same value of the sum 3X,X,Y,Y, taken over all the summits s. Now there will be for every polyedron a certain equation S=0, containing only symmetric functions of the variables of different degrees in them, and true for no other poly- edron. This will be translateable into an equation V=O, containing only symmetric functions of the numbers A B -+-Q ab---p, This condition V=O will therefore, in all probability, be a factor of our result of elimination, U=O. If so, the number of factors of distinct forms (such that the evanescence of one does not for all values of A B, &c., follow from that of the other), into which U=O can be 102 REV. T. P. KIRKMAN ON THE broken, and which are satisfied by the values of A B, &c., under review, will be the number of p’-acral Q'’-edra de- scribed under equation (A). There is reason to believe that the result of this investi- gation may be made to take the shape H=Iv+Jy+Kz+--- where HI J K---are symmetric functions of A B---Q ab..-p, and x y z---are symmetric functions of the vari- ables M,,; and such a shape that the number of solutions of this equation in whole values of wy z---is the exact number of polyedra which it is our object to discover. This form, if it be attainable, would certainly be the most elegant issue that we could desire to our investigation. As the number of these symmetric functions is limited, and as they appear in the expression of symmetric func- tions of the numbers ABC---adc---, it is plain that all but certain ones can be eliminated; but I have not been able to obtain a final equation having no solutions in xv y z -- «foreign to the question. The equation above found, 13 (K-X)(V2-Y) -e=wty is true for more values of w and y than those of which we are in quest. But as I have not formed the equation ob- tained by eliminating from all the equations of this kind that can be written, I cannot say whether a result, show- ing symmetric functions of higher degrees, can be obtained of the form desired. I have a demonstration that the number of p’-acral Q’-edra can be easily found, whatever be the numbers of AB.---Q ab---p, if that of the p’-acral Q’-edra be known for another set of numbers A, B,---Q, a ,:-+p,, of which either A, =B,=C,=.-- -=Q;=8, or else a,=),;=¢j/=*- - =p,=8. And thus the problem is reduced to the deter- mination of the number of Q-edra having only triedral summits, or, which is analytically the same thing, of p- SOLUTION OF THE PROBLEM OF THE POLYEDRA. 103 acra haying only triangular faces. But I do not see any way to the solution of the problem, even in this shape, more direct than that given in the pages preceding. In fact, the question of the #-edra having only triedral sum- mits is identical with this problem in Tactics : To form triplets of three different letters with A A’s, B B’s, C C’s, &c., so that all these symbols shall be once, and once only, employed, and that every possible duad, AB, AC, &c., shall be either not or twice employed. The analysis above given supplies a complete solution hereof; and this is, I believe, the first instance in which a problem of this nature has found either an entire solu- tion, or even a translation into algebraical language. I have my hopes that the method here opened will suggest the proper expression and solution of a multitude of tactical problems, which have hitherto bidden defiance to our analysis. 104, MR. W. FAIRBAIRN ON THE PROPERTIES OF VI.— Experiments to Determine the Properties of some Mixtures of Cast Iron and Nickel. By Wiuiiam Farrsarrn, F.R.S. Read March 9th, 1858. Some of my chemical friends in London had got an impression from a careful analysis of meteoric iron, that it could be produced artificially by the combination of some of the same elements that were found to exist in the spe- cimen analysed, containing about 24 per cent of nickel. In order to determine whether it would be possible to obtain an artificial compound of this nature, and to ascer- tain the effect produced by mixing a certain proportion of nickel with cast iron, the following experiments were instituted. They consisted, in the first instance, in the extraction of the nickel from the ore which is found in the mines of the Duke of Argyle, near Inverary in Scot- land. The metal having then been purified by repeated meltings, was mixed with cast iron in such proportions as to form a compound, containing the same quantity of nickel as the specimen analysed. This was done by melting 24 per cent of nickel with carefully selected South Welsh cast iron from the works of Blaenarvon and Ponty- pool. The mixtures were fused in crucibles, and run into ingots or bars, which were then tested in regard to their mechanical powers of resistance to a transverse strain. SOME MIXTURES OF CAST IRON AND NICKEL. 105 This was done with great care, and the results which follow give unmistakeable evidence of the effects produced upon cast iron by an admixture of nickel, however small the quantity of the latter that may be introduced. Me- teoric iron is, above all others, the most ductile, and it is recorded by travellers that the Esquimaux have instru- ments made from this description of iron so ductile that they may be made to bend round the arm. ‘The ingots prepared on the occasion of these experiments were, how- ever, widely different, as their power to resist impact was nearly one half less than in those composed of pure iron. It is uncertain what might have been the results had the castings produced been treated as cast steel, and hammered out until they were rendered malleable and magnetic. This process was not, however, attempted, as, judging from the appearance of the fracture, they were more likely to crumble under the hammer than attain malleability. The nickel for these experiments was prepared from the ore, by fusing at a very high temperature in a crucible or steel pot, 30/ds. of roasted ore, 5/bs. of fine sand, 2lbs. of charcoal, 2lbs. of lime. This mixture was kept in the furnace six hours, and then taken out and allowed to cool. The metal was then sepa- rated from the slag, and again melted with half its weight of roasted ore and one quarter its weight of green bottle glass ground to powder. As before, the mixture was kept for six hours, at the temperature of a cast steel furnace. The metal had by the end of that time collected at the bottom of the cru- cible. It contained about 25 per cent of nickel, and was VOL, XV. P 106 MR. W. FAIRBAIRN ON THE PROPERTIES OF of sufficient purity to be fused with cast iron. 10/bs. of it melted with 112/bs. of cast iron gave a mixture containing about 24 per cent of nickel. The object of these fusings was to reduce the metallic oxide by means of the charcoal, while the lime and sand removed the oxide of iron, silica, sulphur and other impu- rities by forming a fusible slag. Mixtures of nickel with cast iron (No. 1) and Blaenarvon iron (No. 3) and Pontypool iron, each containing about 24 per cent of nickel, and also some pure Blaenarvon cast iron (No. 1) were cast into bars about one inch square and two feet six inches long, and subjected to the follow- ing experimental tests : — SOME MIXTURES OF CAST IRON AND NICKEL, 107 TABLE I. Breaking weights and deflections of bars of cast iron and of alloys of nickel and cast iron, when subjected to a transverse strain. Experiment IV. Experiment V. Blaenarvon Iron, | Pontypool Iron No. 3. with pure Nickel. Experiment III. Blaenarvon Iron No. 3, with Nickel. Experiment I. Experiment II. Cast Iron No.1, | Pure Cast Iron, with Nickel. No.1. Section of bar Section of bar Section of bar Section of bar Section of bar in inches, in inches, in inches, in inches, in inches, 1x1. 1x1. 1x1. 1.02 x 1.07. 1x1. Weight ; Deflec- | Weight Deflec- | Weight | Deflec- laid on | tion in | laid on | tion in | laid on | tion in in lbs. | inches. | in lbs. | inches. in lbs. } inches. 147 | 0:05 147 | 0:05 259 |O1 259 | 01 815 | 0-14 871 | 016 Weight | Detiec- Weight, Defiec- laid on | tion in | laid on | tionin inches. | inlbs. | inches. in lbs. 147 |0125 | 147 | 01 259 |0175 | 259 | 07155 315 | 0°19 371 | 0°25 371 | 0-2 371 |0115 | 371 | 0225 427 |0°275 427 |6:18 427 427 |0'237 483 | 0°34 483 | 0225 | 483 | 0-2 483 |0°2 483 | 0:25 539 |0°375 | 539 | 0:25 539 |0:237 | 539 |0'237 | 539 | 0262 567 | 0:25 567 | 0:25 595 | O41 595 | 0°31 595 |0:255 | 595 | 0:262 609 | 0325 623 |0'425 | 623 | 0:33 623 |0:277 | 623 | 0:275 637 637 | 0°35 Broke with 651 | 0°36 651 |0:285 | 651 |0°3 651 | 0°35 679 |0°325 | 679 | 0312 707 |0°337 | ‘707 | 0°325 679 | 0:29 this weight. | 679 | 0365 707 | 03 707 | 0:367 721 |0°375 735 | 0-4 749 | 0°325 749 | 0362 763 | 0°41 763 | 0°34 763 | 0°35 777 | 0-425 791 | 0-425 777 | 0375 791 791 | 0-44 819 | 0-46 833 | 0°45 819 | 0-4 798 847 | 0°46 847 |0'42 | Broke with 861 861 this weight. Broke with 875 875 | 0437 this weight. | Broke with | 917 0°45 this weight. | 959 |0°5 1043 | 0°525 1099 | 0°575 1155 | 0°675 With this weight it re- mained twenty minutes. 1267 | 0°75 Broke after some seconds. Ultimate de- | Ultimate de- flection 0°75” |flection 0°366” Ultimate de- flection 0°58” Ultimate de- | Ultimate de- flection 0°434’| flection 0°47” 108 MR. W. FAIRBAIRN ON THE PROPERTIES OF TABLE If. Results reduced to bars 1" square. | Distance between the supports 2’ 3”. Strength Break- Power of compared ing Ultimate jresisting __ with weight |deflection| impact Blenarvon (0) (d) (bxd) | =1000 Experiment I. bar D. pure Blaenarvon No. 3.......... 1131 | 0°75 | 848-2 | 1000 Do. II. ,, C.Blaenaryon No.8 andnickel,,.| 875 | 058 |507°5 | 773 Do. III. ,, B. pure cast iron No.1............ 861 | 0:47 |404°7 | 761 Do. IV. ,, A.castiron No. 1and nickel.......| 637 | 0434 | 2764 | 563 Do. YV.,, E.Pontypoolironand purenickel| 798 | 0°366 | 292°1 | 705 052 |465°7 | 760 Rabeatiemas dee 860 From the above it is evident that an admixture of nickel in the proportion of 24 per cent does not increase but diminish the tenacity of cast iron. To what extent it might be improved by augmenting or lessening the proportion of nickel, a more extended series of experi- ments alone can determine. Mixtures of the two metals in the proportion used in the above experiments are de- cidedly inferior to the pure metal in their power of resist- ance to a transverse strain and to impact. In the first and second experiments on Blaenarvon iron there is a loss of nearly one-fifth the strength; or the strength of the pure metal is to that of the mixture as 1000: 773. And in Experiments III. and IV. with a more fluid iron there is about the same loss, the relative strengths being as 761:563; or as 1000: 740. From these facts it is evident that nickel in the proportion of 24 per cent seri- ously injures the strength of cast iron, and moreover has injurious effects on its power of resisting impact, as the columns in the above table indicating those properties clearly show. It is difficult to account for the serious deterioration and loss of strength which the above experiments indicate. It may probably arise from the improper treatment of the nickel ore during its calcination and subsequent reduction SOME MIXTURES OF CAST IRON AND NICKEL. 109 in the crucible. When cast into ingots after the second melting the nickel had not the appearance of a pure metal, but exhibited a dull fracture, as if fine sand or particles of earth had been mixed with the crystals, showing the pre- sence of impurities which it would be almost impossible to get rid of. It remained, therefore, a question for consi- deration whether the results would be different if nickel, properly prepared and of greater purity, were employed. To clear up doubts on this point a mixture of pure nickel with No. 3 Pontypool iron was made, but the result was a bar of white silvery metal, which broke when the weight of 798/bs. was laid on, as will be seen by referring to the Tables above. I obtained from London a number of other bars, con- sisting of iron and nickel, which on being submitted to the same tests as before, gave better results than those obtained in the previous experiments when nickel pre- pared from the ore was employed. The results obtained from this second series of bars are given in the following table: 110 MR. W. FATRBAIRN ON THE PROPERTIES OF TABLE IIT. Breaking weights and deftections of bars of nickel and cast iron, when subjected to a transverse strain. The bars were about one inch square, and the distance between the supports was two feet three inches. Anes Expermgnt Experiment Experiment See eee Experiment Ix. Bar ¥ 1 not Bar F 2 not B Bar H 1 marked|Bar H 2 marked 5 ar G1 marked|Bar G 2 marked marked with | marked with ithiodenocnd waibnondenocehe with two with two notches. notches. oes notches, Section of bar | Section of bar | Section of bar | Section of bar | Section of bar | Section of bar in inches, in inches, in inches, in inches, in inches, in inches, 1°02 x 1°02. 1°02 x 1°02. 1°02 x 1°02. 1°2x1°2, 1°15 x 1°15. 1°15 x 1°15. Weht. |Defiec- | Weght. |Defiec-| Wght. Deflec-| Wght. |Deflec-| Wght. )Deflee-| Wght. |Deflec- laid on | tion in {laid on | tion in | laid on | tion in } laid on | tion in| laid on | tion in | laid on | tion in in Ibs. |inches. | in lbs. | inches.| in lbs. |inches,| in lbs. | inches.) in lbs. | inches,| in lbs, | inches. 147 | 0:02 | 147 | 0005} 147 | 0:02 259 | 0:06 | 259 |0°05 | 259 | 0:06 | 259 | 0:06 | 259 |0:06 | 259 | 0:08 871 {01 | 371 |O1 | 371 {012 | 371 | 0:08 | 371 {01 427 |0°12 | 427 |0:115) 427 483 |014 | 483 | 0°14 | 483 |0°17 | 483 |0-11 | 483 | 0:14 | 483 | 0°14 539 |0°16 | 539 | 0:18 | 539 539 | 07155) 539 | 0°165 595 | 0°18 | 595 |0°19 | 595 | 0:23 | 595 |0°17 | 595 |0-17 | 595 | 0-18 623 | 0°19 651 |0°2 | 651 |0:21 | 651 | 0:26 | 651 | 0-19 | 651 |0°18 | 651 | 0-2 679 | 0:21 | 679 | 0:22 707 | 0:22 | 707 {0:23 | 707 |0:29 | 707 |0'2 | 707 | 0-19 735 |0°24 | 735 | 0°24 763 | 0:25 | 763 | 0°255) 763 | 0:32 | 763 |0:215| 763 |0:2 | 763 | 0:23 791 |0:26 { 791 |0:27 | 791 805 | 0'265| 805 | 0:28 Broke. 819 | 0°23 | 819 | 0:23 | 819 | 0:25 819 | 0°27 | 819 | 0°285 875 | 0°25 | 875 |0:24 | 875 | 0:27 833 | 0°28 | 8838 | 0°29 931 | 0°27 | 981 |0°27 | 931 847 | 0:29 | 847 | 8295 959 | 0:28 | 959 | 0:28 Broke. 861 | 0°3 861 |0°3 987 | 0:285| 987 875 |0°31 | 875 | 0°31 1043 |0°3 Broke. 903 903 | 0°32 1057 | 0°31 Broke. 917 | 0°325 1071 | 0:317 931 | 0°33 1099 | 0'°321 959 | 0°345 1127 | 0°33 987 | 0355 1155 | 0°34 1015 | 0°375 1183 | 0°35 1029 | 0°38 1121 | 0°36 Broke. 1289 |0'375 1267 | 0°385 1295 Broke. Ultimate Ultimate Ultimate | Ultimate | Ultimate Ultimate deflection | deflection | deflection | deflection | deflection | deflection 0'315” 0°38” 0°331” 0°41” 0°286” 0°29” SOME MIXTURES OF CAST IRON AND NICKEL. 111 TABLE IV. Results reduced to bars \"x1". Distance between the supports 2’ 3”. Break- | Ulti- | Power o Rates of ing mate de- resisting strength weight | flection impact. Bar F 2 (b) id) | (6xd) =1000 xperiment VI. Bar F 1 without notches} 867 |0°315 | 273 | 1000: 876 D VII 0. ar); 989 |0°38 | 376 |1000: 1000 Dor VIE"... Gel with one notch. 760 |0°331 | 231 | 1000: 768 Ds.0 EXiteG2 899 |0-41 | 368 | 1000: 908 Do. >. ae H1 withtwonotches 746 |0°286 | 213 | 1000: 754 Wor CTO Fale | » | 703 |o29 | 203 | 1000: 810 Mearttiirr. costes. 829 |0'335 | 280 | 1000: 838 I have been unable to ascertain the precise composition of these bars, but assuming it to have been similar to that of the first series of bars, the greater powers of resistance shown by them would seem to indicate that the nickel employed in their preparation possessed a higher degree of purity than that used for the first series. Much, how- ever, depends on the quality of the cast iron with which the nickel is mixed. The results derived from the fore- going experiments are conclusive, both in regard to those made on the first and those made on the second series of bars. Further experiments may, however, lead to different results; but judging from what has already been done, I am inclined to believe that chemical combi- nations of a different nature are required, and probably a totally different process of manufacture will have to be adopted before a sufficiently strong and satisfactory com- pound can be obtained. In attempting to ascertain the effect of a mixture of nickel with cast iron, the principal object was to determine to what extent the compound gave positive or negative re- sults. It is well known that meteoric iron is peculiarly ductile, and it was assumed that nickel, added to cast iron in such proportion as to produce a compound similar 112 MR. W. FAIRBAIRN ON THE PROPERTIES, ETC. to meteoric iron, would impart to it increased ductility. The foregoing experiments lead, however, to the conclu- sion that an admixture of nickel produces an exactly op- posite effect, and it now remains to be determined by a more extended series of experiments, whether by mixing nickel with malleable iron in the same relative proportion more satisfactory results would be obtained. In prose- cuting these experiments it would be interesting to know the extent to which these metals are capable of combining chemically, and how closely such combinations would ap- proximate to meteoric iron. Besides endeavouring to obtain a metal of greater duc- tility, another object of equal importance was aimed at in these experiments, namely, to produce a metal of increased tenacity suitable for the casting of cannon and heavy ordnance. During the last two years innumerable expe- riments have been made for this purpose, with more or less success; but the ultimate result appears to be, that for the construction of heavy artillery there is no metal so well calculated to resist the explosion of gunpowder, as a perfectly homogeneous mass of the best and purest cast iron, when freed from sulphur and phosphorus. 113 VII. — On the Hardness of Metals and Alloys. By F. Cracu Catvert, M.R.A. of Turin, F.C.S. &c.; and Ricwarp Jounson, F.C.S., Se. Read April 6th, 1858. Tue process at present adopted for determining the com- parative degree of hardness of bodies consists in rubbing one body against another, and that which indents or scratches the other is admitted to be the harder of the two bodies experimented upon. Thus, for example, Diamond, Tron, Topaz, Copper, Quartz, Tin, Steel, Lead. This method is not only very unsatisfactory in its results, but it is also inapplicable for determining with precision the various degrees of hardness of the different metals and their alloys. We therefore thought that it would be useful and interesting if we were to adopt a process which would enable us to represent by numbers the comparative degrees of hardness of various metals and their alloys. To carry out these views we devised the following appa- ratus and method of operating. The machine used is on the principle of a lever, with this important modification, that the piece of metal experimented upon can be relieved from the pressure of the weight employed without re- moving the weight from the end of the longer arm of the VOL, XV. Q 114 MR. F. C. CALVERT AND MR. R. JOHNSON lever. The machine consists of a lever H with a counter- poise B and a plate C, on which the weights are gradually placed. The fulcrum L bears on a square bar of iron A, passing through supports E. The bar A is graduated at a, and has at its end a conical steel point F, 7mm. or 0°275 of an inch long, 5mm. or 0°197 of an inch wide at the base, and 1:25mm. or 0:049 of an inch wide at the point which ~ ON THE HARDNESS OF METALS AND ALLOYS. 115 bears on the piece of metal Z to be experimented on, and this is supported on a solid piece of iron G. The support or point of resistance W is lowered or raised by the screw M, and when, therefore, this screw is turned the whole of the weight on the lever is borne by the support I and the screw M. When it is necessary, by turning the screw M, the weight on the lever is re-established on the bar, and experimented upon. When we wished to determine the degree of hardness of a substance we placed it on the plate G, and rested the point F upon it, noticing the exact mark on a on the bar A, and then gradually added weights on the end of the lever C until the steel point F entered 3-5mm. or 0°128 of an inch during half an hour, and then read off the weight. A result was never accepted without at least two experi- ments were made, which corresponded so far as to present a difference of only a few pounds. The following table gives the relative degree of hardness of some of the more common metals. We specially confined our researches to this class, wishing the results to be practically useful to engineers and others who have to employ metals, and often require to know the comparative hardness of metals and alloys. Weight Calculated Cast Names of Metals. employed. Tron = 1000 Staffordshire Cold Blast Cast Tron — Grey, No. 3 Steel Wrought Iron* Platinum Copper — pure Aluminium Silver — pure Zine GOs weaves teetett capes. Gold GOs ichecapaasovcestes Cadmium do Bismuth do. ...ccccccceccecsveeee Tin Or % i Ser deiieteduars's Lead OG. vx4 cas aathe cosaeaene * This wrought iron was made from the above-mentioned cast iron. 116 MR. F. C. CALVERT AND MR. R. JOHNSON This table exhibits a curious fact, viz., the high degree of hardness of cast iron as compared with that of all other metals, and although we found alloys which possessed an extraordinary degree of hardness, still none were equal to cast iron. The first series of alloys we shall give is that of copper and zine. Obtained | Calculated* Formule of Alloys and per centages. eet eet eos ores ae / lbs. hs ZiOagitiy ter be Ww kss..d 2050 | 427-08 | 28083 bad te aie. fd 2250 | 46875 | 276-82 Pant geen ker 2250 | 468-75 | 276-04 LAC pyc ae 2270 | 472-92 | 261-04 Zn Cu a sos} ae (eae 2900 | 60417 | 24333 Cu Zn, Va pat seseaeeeees| Broke with 1500 Ibs. without Cu 2464. the point entering. Cu Zn, vo 75.36 Marere ty 95 8 Broke with 1500 lbs. with an Pan ipees impression $ mm. deep. Cu Zn, ics 80°43 se dele ine Entered a little more than the Cu 16:30 ) above; broke with 2000 lbs. Cu Zn; \Zn asin SPR: Entered 2 mm. with 1500 lbs.; B ) broke with 1700 Ibs. These results show that all the alloys containing an excess of copper are much harder than the metals com- posing them, and what is not less interesting, that the increased degree of hardness is due to the zinc, the softer metal of the two which compose these alloys. ‘The quan- tity of this metal must, however, not exceed 50 per cent. of the alloy, or the alloy becomes so brittle that it breaks as the steel point penetrates. We believe that some of these alloys, with an excess of zinc, and which are not found in commerce owing to their white appearance, deserve the * To calculate the hardness of an alloy we multiplied the per centage quantity of each metal by the respective hardness of that metal, added the two results together, and divided by 100. The quotient is the theoretical hardness. ON THE HARDNESS OF METALS AND ALLOYS. 117 attention of engineers. There is in this series an alloy to which we wish to draw special attention, viz., the alloy Cu Zn composed in 100 parts of Wonper? yilis 228s .ds 49°32 PUB O ye «Bia. EGRET. Ses 50°68 100:00 Although this alloy contains about 20 per cent. more zinc than any of the brasses of commerce, still it is, when carefully prepared, far richer in colour than the ordinary alloys of commerce. The only reason that we can give why it has not been introduced into the market is, that when the amount of zinc employed exceeds 33 per cent. the brass produced becomes so white that the manufacturers have deemed it advisable not to exceed that proportion. If, however, they had increased the quantity to exactly 50°68 per cent. and mixed the metals well, they would have obtained an alloy as rich in colour as if it had con- tained 90 per cent. of copper, and of a hardness three times as great as that given by calculation. In order to enable engineers to form an opinion as to the value of this cheap alloy we give them the degrees of hardness of several commercial brasses : — Tl Commercial Brasses, Weight Cast Iron = 1000. employed, | Obtained. | Calculated. lbs. Copper 82°05 *Targe Bearing” ......... {sti 12°82 Zinc 513 | 2700 562 259 Copper 80 “GI Se js 10 Zine 10 3600 750 262 “y, ” Copper 64 Yellow Brass” .,........-. { Fae 36 2500 520 258 Copper 80:0 *Ti : “Pumps and Pipes” ...... Sa oe Lead 75 | 1650 343 257 eC TE EL LTE EE TD * These alloys all contain tin. 118 MR. F. C. CALVERT AND MR. R. JOHNSON The alloy Cu Zn possesses another remarkable property, viz., the facility with which it is capable of crystallising in prisms half an inch in length, of extreme flexibility. There is no doubt that this alloy is a definite chemical compound, and not a mixture of metals, as alloys are generally considered to be. Our researches on the con- ductibility of heat by alloys, which we have recently presented to the Royal Society, leave no doubt that many alloys are definite chemical compounds. On Bronze Alloys. Formule of Alloys and pe: tag Wiseht & Bg os ted r centages. employed. ast Iron ast Iron : = 1000 =1000. ] lbs. Gulliag 4 or er ped. wails 400 83°33 | 51-67 (Cu 11°86} Cu Sn, (Sn 88.14 f tte 460 95°81 59°56 Cu 15°21) ! r Cu Sng { Peasaca pester ag: 500 10417 68°75 Cu 21°21 ) 5 ; Cu Sn, { =e ae 650 135-42 84°79 Cu Sn ' ae ae j bc > eaecpen ate At 700 Ibs. the point entered one half Cu 48-17 and the alloy broke. Sn Cu, \Sn 51°83} Me tleds« ih. At 800 lbs. the alloy broke without Cu 61-79 the point entering. Sn Cu, {Sn 38-2] } 1A ee At 800 lbs. the alloy broke into Ou 68-27 small pieces (blue alloy). SLT peak oo ie ai 1300 bs. divided the alloy into2 pieces Cu 72:90 withoutthe point havingentered lmm. Sn Cu; \Sn 27°10 phlei sie saat The same as the preceding. (Cu 84°32 Sn Cujo} s, 15.68} 50h Cecens 4400 916-66 | 257-08 { Cu 88°97 f ; SL peace Se ee 3710 77292 | 27083 Bn. Ougg fen Big tannieehe 3070 | 63958 | 277-70 Cu 93:17 . . Bu Ong i, gga poten: 2890 | 60208 | 279-16 The results obtained from this series of alloys lead to several conclusions deserving our notice. First, the marked softness of all the alloys containing an excess of tin; secondly, the extraordinary fact that an increased ON THE HARDNESS OF METALS AND ALLOYS. Its quantity of so malleable a metal as copper should so suddenly render the alloy brittle, for the Alloy Cu Sn2 or Copper ee 21 hes is not SHEER, Tae eaens Yaro whilst the alloy Cu Sn or Copper wean 34°98 in brite, fT ae ae 65:02 Therefore the addition of 14 per cent of copper renders a bronze alloy brittle. This curious fact is observed in all the alloys with excess of copper, Sn Cuz, Sn Cuz, Sn Cus, Sn Cu;, until we arrive at one containing a great excess of copper, viz., the alloy Sn Cujo, consisting of copper 84°68 and tin 15°32, when the brittleness ceases ; but strange to say this alloy, which contains four-fifths of its weight of copper, is notwithstanding nearly as hard asiron. This remarkable influence of copper in the bronze alloys is also visible in those composed of Sn Cu,;, containing 88°97 of copper. Sn Cugo, - 91°49 a Sn Cuz;, P| 93°17 3 Copper acquires such an increased degree of hardness by being alloyed with tin or zinc that we thought it inter- esting to‘ascertain if alloys composed of these two metals would also have a greater degree of hardness than that indicated by theory; we accordingly had a series of alloys prepared in equivalent quantities, and these are the results arrived at: 120 MR. F. C. CALVERT AND MR, R. JOHNSON a el | | Calculated Formule of Alloys and per centages of each. | enlisted. Cast Hon Cast Iron phe ee | NOE eye "300 6450 | 6083 Zactie sey Oe eh ie 330 6875 | 82°70 SER acy Neamt 400 83:33 | 110-00 Se a eee 450 9370 | 12458 Be Zing | oe ease tee 505 | 10520 | 131-22 Zn ea tid ane 600 | 125-00 | 142-08 BZ ae ee 580 | 12083 | 15838 These results show that these metals exert no action on each other, as the numbers indicating the degrees of hard- ness of their alloys are rather less than those required by theory. Our researches on the conductibility of heat by the three above series of alloys throw, we believe, some light on the great difference which the alloys of bronze present as compared with those of tin and zinc; for we have stated above that the latter conduct heat as a mix- ture of metals would do, and not as the former series, which conduct heat as definite chemical compounds. We shall conclude by giving the degrees of hardness of two other series of alloys, viz., those composed of lead and antimony, and lead and tin. In the series of lead and tin we find that tin also increases the hardness of lead, but not in the same degree as it does that of copper. ON THE HARDNESS OF METALS AND ALLOYS. 121 Lead and Antimony. Formulz of Alloys and per centages. Weight employed. lbs. SL an eee Entered 2-5mm. with 800 Pb 28°64) lbs.; then broke. Bb, Shyla oe. Ce) Nae eat Cae ee Entered 2-7mm. with 800 Pb 34:86 lbs.; broke with 900 lbs. Pb Sb, { Sb e514} eS TERE 875 Pb Sb, { ae eek TH ada asedOse Be Aipsanechde Entered 2°5mm. with 500 . Pb 61-61 lbs.; broke with 600 lbs. BSB Fae or) bovdi Sh cote 500 Pb 76°32 ae Ea ee 385 Pb 82:80) oe ed 310 Pb 86°52 Sb Pb, {Sp ie. 48} pee, 300 Pb 88°92 Sb Pb, { au 08 Meas 295 Lead and Tin. Weight Obtained Calculated Formulz of Alloys and per centages, employed. Dest Cast 7 Pb 26° ‘oF lbs. 200 41°67 105 40°62 Pb 36:99 Sn 63°01 160 32°33 125 26°04 100 20°83 125 26°04 135 28°12 125 26°04 110 22°92 We have great pleasure in thanking here Mr. Siméon Stoikowitsch, F.C.S., for his valuable assistance during these long researches. VOL, XV. R 122 MR. E. SCHUNCK ON A COLOURING MATTER VIII.— On a Yellow Colouring Matter obtained from the Leaves of the Polygonum Fagopyrum, or Common Buckwheat. By Epwarp Scuunck, Pu. D., F.R.S. Read October 6th, 1857. Amone the many plants which have been supposed to contain or yield indigo-blue, the Polygonum Fagopyrum or common buckwheat, a plant extensively cultivated in some countries for the sake of its seed, which is used as an article of food, is mentioned by some authors. In the Mechanic’s Magazine for November 1830, for instance, the following directions are given for obtaining a blue colour from this plant : — “Take the buckwheat out of the ground before the seed has become quite hard and lay it on the ground in the sun, until it is dry. Then throw the plant into heaps, moisten it and allow it to ferment, until decomposition commences, when it will assume a blue colour. It must now be formed into cakes, which may be dried either in the sun or ina stove. It imparts to boiling water a blue colour, which is not changed either by acetic or sulphuric acid. A durable blue may be dyed with it.” Though it does not follow from this that the blue colouring matter thus obtained was really indigo- blue, still it seemed not improbable that it might be identical with the latter, since another species belonging FROM THE POLYGONUM FAGOPYRUM. 123 to the same genus, the Polygonum tinctorium, is remark- able for the large amount of indigo-blue which it affords. Nevertheless on submitting the leaves of the plant to the same process as that previously employed in the case of the Isatis tinctoria and other plants producing indigo, I was unable to obtain a trace of that or any other blue colouring matter. The examination led, however, to the discovery of a crystallised yellow colouring matter, the method of preparation and properties of which I shall now proceed to describe. The plant, which is one easily cultivated and will grow in the poorest soils, having been allowed to attain its full size, the leaves are separated from the stalks, and treated for some time with boiling water. The decoction, which is muddy and of a greenish colour, is strained through calico, and the mass remaining behind is well pressed in order to remove all the liquid. A little acetate of lead is now added to it. This produces a bulky yellowish-green precipitate, consisting of chlorophyll and other impurities in combination with oxide of lead. Care must be taken to add only so much acetate of lead as to render the super- natant liquid clear and transparent, as an excess would precipitate a portion of the colouring matter. The liquid, having been again raised to the boiling point, is filtered boiling. It has a fine golden yellow colour, but on being mixed with a quantity of acetic acid it becomes pale yellow, and on being now allowed to stand for some time it deposits a voluminous mass of small yellow crystalline needles. These are collected on a filter, slightly washed with cold water and then re-dissolved in boiling water, to which a little acetic acid is added. A small quantity of white matter is usually left undissolved, which is separated by filtration. The filtered liquid again deposits, on cooling and standing, a mass of crystals, which, after being col- 124 MR. E. SCHUNCK ON A COLOURING MATTER lected on a filter and washed with cold water, are dissolved in boiling alcohol. The alcoholic solution is filtered from a small quantity of insoluble matter, and then distilled until the greatest part of the spirit has passed over. The dark yellow liquid left in the retort is then poured into a dish, and allowed to stand for some time, when it deposits the pure colouring matter as a mass of crystalline needles of a pale primrose-yellow colour. These are collected on a filter, washed with a little cold alcohol, and allowed to dry spontaneously. Its properties are as follows : — It is perfectly tasteless, and its solutions are neutral to test paper. When heated on platinum foil it melts to a brown transparent liquid, and then burns with a yellow flame, leaving much charcoal, which, on being heated, burns slowly away without leaving any ash. When heated in a tube it melts, gives off fumes having a strong empyreu- matic smell, and yields a yeliow oily sublimate, in which nothing crystalline is formed even after several days. It is hardly soluble in cold water, and only sparingly soluble in boiling water. The boiling watery solution deposits it on cooling in yellow silky needles, which, when dry, form a compact silky mass. It is more easily soluble in boiling alcohol than in water. It does not separate from the alcoholic solution on cooling, but is left on evaporation in star-shaped masses of a darker yellow colour than the needles obtained from the watery solution. Strong muriatic acid changes its colour to a deep yellow without decompos- ing it. When concentrated sulphuric acid is brought into contact with it, the acid acquires at first a greenish colour, and then dissolves it entirely, forming a deep yellow solu- tion, from which the greatest part of the colouring matter is precipitated by water in yellow flocks. If, however, the solution in acid be heated it becomes black and gives off sulphurous acid in abundance, the colouring matter being FROM THE POLYGONUM FAGOPYRUM. 125 decomposed. When the substance is dissolved in boiling water to which a little sulphuric acid has been added, and the solution is boiled for some time, no decomposition apparently takes place, proving that this colouring matter is not, like some others, a copulated compound. Nitric acid of ordinary strength dissolves it even in the cold, forming a dark orange-coloured liquid, which, on being boiled, evolves nitrous acid, and becomes pale yellow; the liquid on evaporation yielding a large quantity of oxalic acid. When suspended in water, and exposed to the action of chlorine gas, it gradually dissolves, forming a brownish-yellow solution, which on evaporation leaves a brownish-yellow glutinous residue. This residue has an astringent taste, which is, however, partly disguised by the muriatic acid formed during the decomposition; its watery solution gives with gelatine a curdy precipitate, similar to that produced with the latter by tannin. The colouring matter dissolves easily in caustic potash and soda, liquid ammonia, baryta water and lime water, form- ing deep yellow solutions, which, on the addition of an excess of acid, become pale yellow, and on standing again deposit the colouring matter in pale yellow needles. If, however, these solutions be left to stand for some time exposed to the air, a decomposition of the colouring matter seems to take place, for on now adding an excess of acid, the solutions remain yellow, and deposit no crystals. If the ammoniacal solution be left exposed to the atmosphere for some time and then evaporated, it leaves a residue consisting of crystals of colouring matter mixed with an amorphous substance. On adding water to the residue the latter dissolves, and the solution, after being filtered from the crystals and evaporated, leaves a trans- parent, yellow, brittle mass resembling gum, which contains only a trace of ammonia, and is probably a 126 MR. E. SCHUNCK ON A COLOURING MATTER product of decomposition formed by the action of the atmospheric oxygen on the colouring matter. The watery solution produces, with different reagents, the following reactions. With acetate of alumina it gives a bright yellow flocculent precipitate ; the filtered liquid is still very yellow, but on the addition of ammonia it deposits some more yellow precipitate, and then appears colourless. With protosulphate of iron it turns of a greenish colour, but on standing exposed to the air the colour changes to a dark green, whilst a dark green powder is deposited; the addition of ammonia now produces a dark brown precipitate, the liquid becoming colourless, On the addition of perchloride of iron it turns of a brownish- olive colour. Both the watery and the alcoholic solution give, with acetate of lead, a precipitate of a bright yellow colour, inclining to orange, which very much resembles that of chromate of lead; the filtered liquid in either case still retains some of its yellow colour. With basic acetate of lead the watery solution gives a precipitate of a rather darker yellow colour than that with neutral acetate, the filtered liquid being colourless. The watery solution when cold becomes, on the addition of acetate of copper, greenish- yellow, and remains clear, but on being boiled it deposits an abundant greenish-yellow precipitate, which dissolves again almost entirely when the liquid cools. The watery solution gives no precipitate with nitrate of silver, but on standing it becomes muddy and black, and deposits a fine black powder. On the addition of chloride of gold it also soon becomes muddy, and deposits bright spangles of metallic gold; when caustic soda is added at the same time, the reduction takes place instantaneously, the gold being deposited partly as a black powder, partly as a metallic mirror covering the sides of the glass. Proto- chloride of tin produces in the watery solution a bright FROM THE POLYGONUM FAGOPYRUM. 127 yellow precipitate, the supernatant liquid remaining muddy for a long time. Printed calico, when immersed in the warm watery solution of this colouring matter, becomes fully dyed, the alumina mordant acquiring a dark yellow colour, the tin mordant a light yellow, and the iron mordant various shades of yellowish-brown according to the strength of the mordant employed. Silk and wool do not acquire any colour in the boiling watery solution, unless they have been previously prepared with some mordant. The substance contains no nitrogen, for on being treated with boiling caustic soda lye, or heated in a tube with soda-lime, it evolves no ammonia. Its composition was determined by the following analyses : — I. 0:4145 grm. of the substance, crystallised from alco- hol, dried at 100°C. and burnt with chromate of lead, gave 0.7550 grm. carbonic acid and 0:2200 water. II. 0:4085 grm. crystallised from water, gave 0°7450 grm. carbonic acid and 0°2130 water. III. 0:4205 grm. of another preparation gave 0°7715 grm. carbonic acid and 0°2255 water. IV. 0:4170 grm. gave 0.7640 grm. carbonic acid and 0°2225 water. These numbers correspond in 100 parts to E II. STE. IV. Carbon........ 49°67 49°73 50:03 49°96 Hydrogen..... 5°89 5:79 5°95 5°92 4 ee 44044. 44:48 44-02 44°12 100:00 = 100:00 100:00 100-00 The lead compound was prepared by adding to the alcoholic solution an alcoholic solution of acetate of lead. The dark yellow precipitate was collected on a filter, 128 MR. E. SCHUNCK ON A COLOURING MATTER washed with cold alcohol, and dried over sulphuric acid. It yielded on being analysed the following results : — 0:9555 grm. gave 1:1315 grm. carbonic acid and 0°3090 water. 0:7630 grm. gave 0°4025 grm. sulphate of lead. From these numbers may be deduced the following composition : — Equivs. Calculated. Found. Carbongys. is: %3 2 ss) a0 er 180 31°83 32°29 Hydrogen .......... 18 18 3°18 3°59 Oxygen .. sees 18 144. 25-48 25°31 Oxide of Lead...... 2 223°4 39°51 38°81 565°4 100:00 100:00 If the formula of the lead compound is Cy) Hyg Oy + 2 Pb O, as the above analysis seems to indicate, then that of the substance in its uncombined state must be Cy) Hy Ox and its theoretical composition will be as follows : — Equivs. Carbon’... :..wexbies 30 180 50-00 Hydrogen......... 20 20 5°55 WOXVSER 2. .ctscetes 20 160 44°45 860 100:00 A comparison of the properties and composition of this substance with those of Rutine or Rutic Acid, the colouring matter discovered by Weiss* in the Ruta graveolens or common rue, and by Rochleder and Hlasiwetz+ in capers, leads to the conclusion that they are identical. A colouring matter, having the same properties and composition, and being apparently the same substance, has also lately been * Pharmaceut. Centralblatt, 1842, s. 903. + Annalen d. Chemie u. Pharm., Bd. LIII. s. 385. FROM THE POLYGONUM FAGOPYRUM. 129 discovered by Moldenhauer,* in the leaves cf the Ilex aquifolium, or common holly, and called by him Llixanthine. The quantity to be obtained from the buckwheat leaves seems, however, to be much more considerable than that procurable from any other source. In one experiment I found that 30lbs. of fresh leaves yielded 240 grains of crystallised rutine, or a little more than the thousandth part of their weight. In countries where the plant is cultivated it might be worth while to collect the leaves as a dyeing material, since the seeds are the only part at present employed for any useful purpose. * Annalen d. Chemie u. Pharm., Bd. CII. s. 346. VOL. XV 8 131 1X. — Researches in the Higher Algebra. By James Cocxte, M.A., F.R.A.S., F.C.P.S., Barrister-at-Law. Communicated by the Rry. R. Hartzy, F.R.A.S. Read October 5th, 1858. Sf. Tue higher algebra conducts us to results, the complex- ity of which is unfavourable, not only to the progress of research, but sometimes perhaps to intelligibility. The object of this paper is to exhibit some of them in a form which, without detracting from their real generality, will render their import comparatively easy of recognition. § 2. Let i be an unreal fifth root of unity, and let i+=4(i), ?4+8=4(°), 84+ 4@)=4, 14246) =4, 142¢(i)=c, 3+ 4()=d, then $() +o) =$@) -G@)= -1, 6+e=0, and ad=BP?=c?= — be=5. § 3. Again, let v, w, 2, y and 2 be the roots of a given equa- tion in uw of the fifth degree, and let v+iw+?e+iy+iz=f(i), fd -f@ SE) -f ) =9 and (we + vy) +w (ve + yz) + 2(vz 4+ wy) + yew + #2) +2°(vy +we)=U=W(u), VOL. XV. te 132 MR. JAMES COCKLE; RESEARCHES then, if the given equation be of the form + —5Qu'4-H=0 . 0... sheen eee (1), there subsists the relation 7S Vea | i a eel (2). § 4. When one of the six values of @ vanishes, (1) admits of finite algebraic solution. It is unnecessary to repeat the investigations in the course of which I arrived at this criterion of solvibility, or to exhibit the complicated formula by which, in the general case, it is expressed.* But the solution, if any exist, of the general equation of the fifth degree, will con- tain a term of the form 7”@, where © vanishes with @. § 5. When (1) holds, the conditions of homogeneity indicate 65+ aQh& + BQ‘ + yQVE’# + (SQ°E + eH’) 0 22 GA iQ * ERO) 0 i A 8 38. A (3) as the form of the equation in @. And in determining a, 8, &e., we may make 7=0, for E* does not enter into the criterion alluded to. § 6. Let y(w—h) or V be the value of U when each of the roots of (1) is diminished by er Then V=U-DU.- h+DU +5 57 DU: Tt = gt he. where D is the differential ae = 4 tate But DU =Se'w4+ 25vwe =- 5Q, D?U=450?4 103vw=0, DU =4830=0, D*U = 240, consequently WW SORA 10h Wk > femee e (4). * See Phil. Mag., May 1858, p. 389. IN THE HIGHER ALGEBRA, 133 § 7. Next, let (1) become which is equivalent to 3 (u— 1)°( 2? + 2u? + 8u+5 ) ==4 or, if u«—1l=p, to 15 2 3 2 — (p+ 5p +10p+-")=0 pO Wibe. 6). Then, if g, 7 and s be the roots of 5 p+bpe+ lop +7 =0 j Piet tol LIA. (7), we have Lye tes — gees hate ee (PD (8) § 8. For, since h is arbitrary, we may assume it equal to unity, and replace (4) by U=y(u-1) -5Q-10, a relation which, when (5) is satisfied, becomes 25 U=¥(p)-- ry But, by (6), two of the values of p are zero, and, therefore, v(p)=9"rs, and (8) is verified. § 9. Permuting g, 7 and s in (8), and determining the sym- metrical functions by means of (7), we find that the six values of U are the roots of 3 53 54 2 {U+—- U-s} aif Consequently, by (2), the corresponding equation in @ is { gs 3-5" y 5”)? _ 6 ( ) ot “oz a" 915 et Ee Oy 6m St ake a) es (9). 134 MR. JAMES COCKLE; RESEARCHES § 10. When a=; and E=3 (5) is identical with (1), and (9) with (3). Therefore, de- veloping (9) and comparing coefficients, we have 3a 8-5* F 57 oy oO sp” Sp ime toes ot. ae =o Or VSP. oe toe BF ; a= op? OF a § 11. In (1) let Q=O and E=-1 and replace v, w, 2, y, 2, by 1, i, ?, @, #, respectively. Then, for all integral values of m, f(v”)=0 and @ vanishes. But if we make v=7, w=1, the other roots remaining undisturbed, we find Ad =ai, f@)=bi', {@) =c?, fU)=d, and 6=abedi?=-5*. Consequently the equation in 6 will, on the present supposition, be ETE | hae amaile air (10), whence, comparing (10) and (3), e= Blo and, by the last section, §= 2.5% — 23%8e= 2.55? — 2. 3°) =— 58-5". § 12. The equation in @ is, therefore, 65 + 2QE5°4 + 2Q'578 + QH*5"F — (58Q°— BP Bb pores 0 | ok cine (11) § 13. We may obtain (9) from (5) thus; let SGjH=14itPo+ Py +Hz, then z, y and ¢ are the roots of 4 2024 Bu +5=0 ROR otis. (12), IN THE HIGHER ALGEBRA. 135 and, by means of symmetric functions, we are conducted to, Ni) fl) =2c+ (wz +y), f@) -f@)=2b + (wz +y). It follows from this, that @=-5(a@z+y — 2)? or, eliminating vz and reducing by the aid of (12), @-5*(3y+1)=0, whence, permuting and determining the symmetric func- tions by means of (12), RD 7 g425'9, 5 which verifies (9). § 14. Again, let 5Q=1 and E=O and replace 2, w, 2, y, z by 1, 7, 7”, 0, O respectively, 7 being an unreal cube root of unity. Then SO JMO=HF, [QO FOYH=9, and, consequently, 0=bcj, or 64+57=0, whence (7? + 5°)?=0, Developing this equation, and comparing its coefficients with those of (3), we have 2, or, B=2:5' 4 ee ra) val eV Jae as before. § 15. The equation (11) may be put under the form (6° + 5°QH@ + 5’Q*)?=5"(108Q°E — E*)0, and this may be readily transformed into (9°+AS?-+ B)?= Cs, or 136 MR, JAMES COCKLE; RESEARCHES ’ Sah AOS QB Oi pie, eaeguy x (13), where 108AB_ A C= ae ag ee ee eee (14). § 16. Determining A by the equation A=B)’, and assuming AM=B)M=5H, we obtain from (14) c2— 108(Ax') (BA) par Andt 52,0 5°(BAS)A” or Cr®=(Cr y= 108 H*— li*. Consequently, when (14) is satisfied, (13) may, by means of the assumption f=A%, be transformed into ®45Hf+,/108—H-Hi+5H=0..... (15), or F {@4-6H(P+1)}"=(08-H) AH? 2S (16), or, making ?=s, di? POEs 4 1)g*= (108) H's *.' 5 (17). § 17. Tf, therefore, we determine T and F from the equations ig! by Qe § ard is which in fact lead to PS pts, the solution of (17) is made to cat upon that of Spe om =O" aes swank (18), a form into which the assumptions FPO wt=E, pé=u suffice to convert (1). § 18. The following remarks arise out of this discussion : — le. The functions a, 6, c and d seem to be entitled to rank as canonical expressions in the theory of equations of the fifth degree. IN THE HIGHER ALGEBRA. 137 2°. Since Sir W. Rowan Hamilton’s “ Inquiry into the Validity of Mr. Jerrard’s Method” has established the fact that the general equation may be reduced to the form (1), that form imposes only a seeming, not a real, restric- tion on the generality of our investigations. 3°. The same remark applies, when C is unrestricted in value, to (13); for to such a form the general equa- tion of the sixth degree may be reduced by Mr. Jerrard’s method. 4°, The condition (14) appears to restrict the general- ity of the sextic. But the solution of the restricted sextic is reduced to that of (1), or, which amounts to the same thing, the solution of (15), (16) or (17) is reduced to that of (18.) 5°. Were it possible to solve (15), (16) or (17) by means of equations of degrees lower than the fifth, we should probably be advancing a step towards clearing up the difficulties which encircle the question of the finite algebraic solution of (1), and, therefore, of the general quintic. 6°. If the roots of (15), (16) or (17) are connected, symmetrically or otherwise, with those of the final sextic in a proposed method of solution of a quintic, the solvi- bility or insolvibility of those equations may afford a test, perhaps the simplest which the case admits, of the suf- ficiency of the proposed method. § 19. Mr. Harley, in whose hands I was so fortunate as to place these investigations, has since verified all the co- efficients of the equation in 0, with the exception of the last two. He has effected this by a direct, original and general process, of independent interest and of intrinsic value and importance. His results, perhaps carried still further, will, I hope, be soon laid before the Society. 138 MR, JAMES COCKLE; RESEARCHES § 20. It is easy to evolve all the roots of vu’ — 5 Mut -5Pu}-5Qv’-5Ru+E=0..... (19) from a single expression, if unsymmetric functions are employed — thus {imfli) PME) + OPM) +i%/) +E) may, by assigning appropriate values to m, be made to yield all the roots in succession; but the possibility of evolving them from an expression of the form M +2” O'+ 2" Oi + Pit + jer Ov where 6', 6%, G1, ©, are functions of the coefficients of (19) is by no means an obvious consequence. § 21. But, when any one of the six values of @ vanishes, all the roots of (19) are included in an expression of the form M +776! +70" 418+ m@l” where m, as in the last section, is an integer, 7 is 0 or 1 according to circumstances, and 0’, 0”, 0”, are all known or ascertainable functions of the coefficients of (19). § 22. If, then, by any practicable transformation of (19), @ could be made to vanish, all the roots of the transformed, and consequently of the original, equation would be determined. § 23. There are, however, other modes of proceeding. Super- adding relations obtained from other sources — the second form of Euler, for example — we may examine the condi- tions which the evanescence of @ discloses, and endeavour to express w as a function of 6. Thus, in the elementary case in which (19) takes a binomial form, assuming IN THE HIGHER ALGEBRA. 139 u=x(4,, O,, O35, O4, 95; 4); we find —— ft ———————————— ay ewe and, hence, conclude that is a constituent of y. If we were conducted to unsym- metric or, by the assumption w=F(@), to general functions of 6, it would become necessary to inquire into the solvi- bility of the equation in @. § 24. A third mode, also involving that inquiry, would be to seek the values of uv in terms of those of @ by elimination. And here, as it would not be difficult to show, Mr. Harley’s investigations, as yet unpublished, substantially change the data into four quadratics, involving four unknowns, but with certain relations existing among the coefficients. § 25. Writing for a moment 6(u) in place of 9, the transition from 6(u) to 6{R(u)}, where R(w) is a rational and general or unique function of w—i.e. a rational function of any and only one value of ~—though perhaps interesting, does not seem to demand our attention at present. § 26. A finite algebraic solution of the general equation of the fifth degree may be impossible or unattainable. If the hypothesis of its existence lead to contradictory or inconsistent conditions, or to a self-contradictory result, such a solution is impossible or unconditionally unattain- able. If the hypothesis conduct to the conclusion that every known process for solution leads to an illusory re- sult, e.g. to vanishing fractions unassignable in value, or VOL. XV. u 140 MR. JAMES COCKLE; RESEARCHES to equations which present as much or more difficulty than the given one, then the solution is, not absolutely, but conditionally or relatively, unattainable. The latter con- clusion may afford probable evidence that the solution is impossible ; but the impossibility cannot be said to be demonstrated. § 27. Let vo” + iw” + Pa” + By” +2" =f"(i), f° @®F/"@H“*)=7,(u”), and PB) FP) =r(w"). Then 7 (Ww) + 72(u”) = 230 — So™w", — 5ry(u") x 7,(w") = 0(w") + K where 6(w”) is the same function of w” that @ or O(u) is of u, and K is known, and, consequently, 7 is known when @ is known. The function 6(w”) is known whenever 6(w) or @ is known. § 28. Let Le) yi (2*) +f"(#) t'() = L, POLO FLO LE) =S, then I aL AL =4> 7" ie Sv"w", and =3{(+I~-@4+9)}. But PPS LPOPOP+RPOLOPHPOCE LOY +1 fr@ Pe oy os Mena "|\POPLPOAPOLOLOLES Sut ) 7y(w) + T9(w”) To(w”) a re = is a function of tr. Consequently I and J are known when 7, which depends upon 0, is known. IN THE HIGHER ALGEBRA. 141 § 29. The relations POL M+ OLO=L LOL?) XPOL?O=1E) 1); enable us to determine f”(7) f"(#) and f”(#*) /”(2) mm terms of Iand7,. Similar relations hold for J and 7. ‘ ow tela = I vip wiplatp'y+p ztp This, on reduction, becomes 3 f@®@ a, J') ect Ors Pay Py P+ 8 FHS = and the relation fi) _ P@Or@) PO ru’) will enable us to express p in terms of I and 7. § 31. When, therefore, 0 is known, p can be determined ; and, if in (1) we substitute : —p for u and reduce the result to the usual form, the equation in ¢ will be solvible, for a factor of the symmetric product vanishes. The solution of (19) may be made to depend upon that of (1), and the latter, in its turn, upon that of (11) and of the equation in ¢. § 82. Mr. Harley has discovered that in certain cases U be- comes a square, and equals 7. This circumstance may throw a light upon the question of the solvibility of the equation in 0. Norg. — Mr. Cockle’s original communication, consisting of the 142 MR, JAMES COCKLE; RESEARCHES, ETC. first 18 §§, is dated May 17th, 1858. The 19-26 §§ are added by a postscript under date of September 10th, 1858; and $§ 27-32 are added by another posteript dated September 22nd, 1858. 143 X, — Note on Dalton’s Determination of the Expansion of Air by Heat. By J. P. Joute, LL.D., F.R.S., &c. Read November 2nd, 1858. In the twenty-first volume of the Memoirs of the Academy of Sciences, p. 28, Mr. Regnault, in the course of a discussion of the various co-efficients given for the expansion of air, by different experimenters, remarks that Rudberg had] brought to recollection an observation made by Gilbert, in his Annals, to the effect that the experiments of Dalton and Gay Lussac, which had been considered as giving almost identical results, differed on the contrary very considerably from one another. Then, referring to Dalton’s experiment, related in the Man- chester Memoirs, vol. v. part ii. p. 599, Regnault shows that if 1,000 measures of air at 55° Fahrenheit expand to 1,325 at 212°, 1,000 measures taken at 32° will become 1,391 at 212°. Upon this he goes on to remark that Dalton did not appear to have been aware of the error which had crept into his calculations, for he says in his New System of Chemical Philosophy that the volume of air, according to Gay Lussac’s and his own experiments, being taken 1,000 at 32° becomes 1,376 at 212°. On reading the remarks of the eminent French phy- sicist, the extreme improbability that a man so notoriously exact and careful in his mathematical and arithmetical computations as Dalton should have made the gross error 144 MR. J. P, JOULE ON DALTON’S DETERMINATION imputed to him, at once occurred to me. I therefore, on consulting Dalton’s works, was not surprised to find that his commentators had entirely misunderstood the facts of the case. These are as follow: — Dalton, in his Experi- mental Essays, read before this Society in the month of October 1801, describes experiments on the expansion of air by heat, the results of which, referred to the freezing point, are accurately stated by Regnault. But in the New System of Chemical Philosophy published in 1808, under the article Temperature, Dalton, while explaining his New Table of Temperature, writes —“The volume at 32° is taken 1,000, and at 212°, 1,376 according to Gay Lussac’s and my own experiments. As for the expansion at inter- mediate degrees, General Roi makes the temperature at mid-way of total expansion, 1164° old scale; from the results of my former experiments (Manch. Mem., vol. v. part ii, p. 599) the temperature may be estimated at 1194°; but I had not then an opportunity of having air at 32°. By more recent experiments I am convinced that dry air of 82° will expand the same quantity from that to 117° or 118° of common scale, as from the last term to 212°.” The first part of the above extract contains the passage quoted by Regnault, but its meaning is obviously not that which he infers. The experiments which Dalton states to agree with Gay Lussac’s are clearly some unpub- lished ones made subsequently to those described in the Manchester Memoirs. Ue nowhere that I can discover advances the assertion, attributed to him by Gilbert and adopted by Rudberg and Regnault, that his former “ ex- periments” agree exactly with those of Gay Lussac. They were, however, highly important at the time when they were made, and justified the approximately correct conclu- sion he drew, that all elastic fluids under the same pressure expand equally by heat. Dalton was at once aware of the immense importance of OF THE EXPANSION OF AIR BY HEAT. 145 this law, and in a sentence prophetic of the advancement of the theory of heat in recent times, pointed to the force of heat as the sole and immediate source of expansion in elastic fluids, and predicted that a study of their pheno- mena would ultimately lead to general laws respecting the absolute quantity and the nature of heat. 146 MR. J. P. JOULE ON THE UTILIZATION OF THE XI.— On the Utilization of the Sewage of London and other large towns. By J. P. Jouve, LL.D., F.R.S., &e. Read November 30th, 1858. I nave learned with regret that a system of metropolitan drainage has been adopted, and is about to be attempted by the Metropolitan Board of Works, which I consider to be a stride in the wrong direction, and which, if persevered in and copied by other towns, must be fraught with disas- trous consequences to the national prosperity. I cannot however say tnat I felt much surprised at the intelligence, as I knew that the advice and assistance of scientific men had not been sought, except in one or two solitary in- stances, and that the professional engineers consulted had been limited to only a few, however eminent, individuals, who differ among themselves as to the contemplated works. On a question of such vital importance as that which has been raised, I think all ought to contribute what experi- euce, information, or common seuse they possess, and so form a concentrated expression of opinion which cannot be disregarded. I therefore, though the subject is somewhat new to me, hesitate not to introduce it to the Society, in the hope of eliciting the opinions of those who may have studied it better than myself. From the Report* of Messrs. Hawksley, Bidder and * Report presented to the Metropolitan Board of Works, ordered by the House of Commons to be printed 13th July, 1858. SEWAGE OF LONDON AND OTHER LARGE TOWNS. 147 Bazalgette, I find that the history of the present question of London sewage dates from the year 1847, when, instead of the eight separate commissions which had previously existed, a consolidated one was appointed. This was shortly followed by a second, which advertised for and obtained 116 plans. A third and fourth commission re- ported against those plans, and appointed Mr. Forster as their engineer, who, after preparing a plan for the drain- age of the north district of the Thames, died in conse- quence of the anxieties of his position. A fifth commis- sion was embarrassed by the plans of the ‘“ Great London Drainage Company,” which, after occupying a great part of the session of 1853, were ultimately rejected by parlia- ment. In 1854 Messrs. Bazalgette and Haywood prepared a scheme, but another proposal by Mr. Ward having re- ceived the sanction of the Secretary of State, a sixth commission was appointed, which invited plans but arrived at no conclusions. In 1856 the Metropolitan Board in- structed their engineer to report and prepare plans. Sir B. Hall proposed modifications which were adopted. This final plan was submitted to three referees, viz., Captain Galton R.E., James Simpson C.E., and Thomas E. Black- well C.E., who reported thereon in July, 1857,* and, on their report being objected to by the Board, suggested material modifications of the plans proposed in the report submitted to parliament by Her Majesty’s first Commis- sioner of Works. A further communication from Messrs. Galton and Simpson, involving a third plan with further modifications, was made in January and February, 1858. Finally, at the request of the Metropolitan Board, Messrs. Bidder, Hawksley and Bazalgette reported on the 6th of April, 1858, on the plans of the government referees as from time to time modified. This last report, which in * Report on Metropolitan Drainage, ordered by the House of Commons to be printed 3rd August, 1857. VOL, XV. xX 148 MR. J. P. JOULE ON THE UTILIZATION OF THE general opposes the plans of the referees, appears to be the one finally adopted by the Board of Works on June the 29th, 1858. In justice to the eminent engineers I have named it is needful to premise that the duty they were called on to perform was rather to carry out a system predetermined by the hasty voice of public opinion, than to devise a plan entirely agreeable to their own views. It would be beyond my province as well as my ability to describe the vast works which are now being attempted in conformity with the last resolution of the Metropolitan Board. It will however be sufficient to describe the gene- ral principle, in order to enable us to decide how far the two great objects, which any reasonable person must place before him, will be met. These are first, and T say especially, as it to a great extent includes the second, the economical use of sewage; second, the beauty and healthfulness of the metropolis. And here it is most deeply to be regretted that the projectors generally, instead of applying themselves to the fair consideration of both the above objects, have hastily abandoned the first one, so that, even if the plans answer the intention of the designers, the first great object will be further than ever from its realization. In fact, to illustrate how steadily, and I may say determinedly, the opposition to economy has been carried on, I have only to quote the following language of the government referees in page 38 of their Report: — “ We consider it very inex- pedient for the Metropolitan Board of Works to adopt any plan which is based upon the deodorization or the utiliza- tion of sewage; that if an attempt is to be made to utilize London sewage, it should be made by private enterprize ;” and in page 43 —‘“ That the value of the fertilizing matter contained in London sewage is undoubtedly great; but that the large quantity of water with which it is diluted precludes the possibility of separating more than about SEWAGE OF LONDON AND OTHER LARGE TOWNS. 149 one seventh part of this fertilizing matter by any known economical process; that a copious dilution of the sewage is necessary to the health of the inhabitants of the metro- polis; and that therefore the sacrifice entailed by the dilu- tion must be endured.” — The plain meaning of all this I take to be: We will take care to dilute and remove the sew- age, and then when, as we have shown, private enterprise will be unremunerative, we will invite it. Sketch of the scheme of the Metropolitan Board.— On the north side the scheme consists of a main high-level sewer, to intercept the fall from the higher parts, extending from Hampstead; a main middle-level sewer from Kensal Green; and a main low-level sewer from Vauxhall Bridge Road. All these terminate near Bow, whence the united streams pass in a channel formed of a triple culvert of brickwork to Barking Creek. On the south side a similar system of high-level and low-level sewers is to extend from Clapham and Putney to Greenwich, and thence to be carried forward in one main through Woolwich to Cross- ness Point, a place midway between Woolwich and Erith. Pumps are to be employed to raise the sewage at certain points, and storm-overflows are to enable the mains to dis- charge themselves through the previous system of sewers into the river within the limits of the metropolis, when- ever in consequence of a sudden fall of rain the former are overcharged. Two large reservoirs are to be placed at the outfalls of the two great mains, with the object of retaining the sewage until after full tide, when it is to be discharged into the Thames. The utilization of sewage is virtually ignored in the scheme of which I have just given an outline. Will it answer the object for which it is solely designed — that of purifying the Thames, and increasing the healthfulness of the district? To reply to this question, we must consider lst the operation of the principle of intercepting and di- 150 MR, J. P. JOULE ON THE UTILIZATION OF THE verting the sewage from its original course. The present sewers in their usual functions will have to be considered as taking their rise at the points where they are crossed by the mains. Hence their size will be larger than it ought to be for the diminished current, and accumulations will result, which latter will be carried in time of storm-over- flow into the Thames. A striking proof that such accu- mulations are, even under the present system, liable to take place and be carried off during storms is adduced by Dr. Hofmann and Mr. Witt. These chemists state that when after a sudden and heavy fall of rain the flow of the Savoy Street sewer had increased sixfold, they found that instead of the sewage being thereby diluted, a given volume actually contained more than twice the quantity of solid constituents which it contained under normal circum- stances. 2nd. I doubt whether mains built of brick, however well cemented, can be depended upon to convey sewage. Brick is usually porous, and in that state it cannot be doubted that sewage-water will filtrate through it and thus gradually contaminate the adjacent ground. The injurious effects of such infiltration ought not to be overlooked in a system of mains extending to a total length of sixty miles. 3rd. That portion of sewage which arrives at the out- falls will not be entirely prevented from returning to the metropolis. I arrive at this conclusion from the fact that the sea-water penetrates occasionally as far as London Bridge. The river is frequently brackish at Barking Creek and Woolwich. Experiments with floats may induce falla- cious conclusions in this respect, since it is probable that the scour of the flood-tide at the bottom of the estuary is greater than that of the ebb-tide. 4th. The Thames will be rendered particularly nox- ious at the point where so vast a quantity of offensive matter is to be concentrated. By what justice a nuisance ee SEWAGE OF LONDON AND OTHER LARGE TOWNS. I51 can be removed from ourselves to be placed under the noses of our neighbours I know not. Nor can I appreciate the wisdom of sacrificing the purity of the air inhaled by the inhabitants of Greenwich, Woolwich, Gravesend, &c., and the immense floating population,* in the doubtful at- tempt to make the air of the metropolis more wholesome. Sth. The air confined in the new drains will be a serious increase to the already enormous volume of putrid gases in the sewers. The government Referees make just and forcible remarks upon this evil. They state that “the effect of trapping the street gully-drains, without providing other ventilation of the sewers, is, that the noxious gases generated in the sewers are forced into the houses when the flow of sewage increases, the syphon-traps of water- - closets and sinks being the points at which the least resis- tance is presented to their escape from the sewers. To ebviate these evils, the plan has been partially adopted of providing in the middle of the street untrapped openings into the sewers. These openings must be endured until a better mode of ventilation shall be adopted, although the foul smells they emit are frequently very great nui- sances.”+ In addition to the cause assigned by the Re- ferees for the expulsion of the poisonous gases, I will mention the changes of atmospheric pressure. The fall of the barometer of one inch will of course occasion the liberation of one-thirtieth of the entire volume of gas. The smell so generally observed to arise in the neighbour- hood of drains before rain may probably be referred to this cause. It is also worthy of remark, that in winter the comparatively warm air of the sewers will have a tendency to rise. May not the greater mortality during that part of the year be partly attributed to this circumstance ? * “Tt is extremely undesirable in a sanitary point of view to cause sewage- water to be intermixed with sea-water.” — Messrs. Hawksley, Bidder and Bazalgette’s Report, p. 52. + Report of Government Referees, p. 40. 152 MR. J. P. JOULE ON THE UTILIZATION OF THE 6th. The proposed system must be considered a filthy one, as instead of removing sewage to the soil, which is the natural deodorizer, it will cause its accumulation in the bed of the river at a distance of only a few miles from the city. Even the liquid portion will remain for months near the spot where it is introduced, as is proved by the experi- ments of Mr. Forster, who found that a float put into the river at Barking advances only five miles in its course to- wards the sea in an entire fortnight.* The above are some of my reasons for believing that the proposed plan of the Board of Works will fail in pro- moting the object to attain which the promoters have sacrificed what ought to have been their first considera- tion. I enter not now on various points, such as the des- truction of fish in the river and in the wells of ships, and consequent interference with a useful trade; the formation of banks apprehended by some, and the consequent im- pediment to navigation; the expense; and other details which must be of minor importance in a question ulti- mately involving the life and subsistence of an entire population. The government Referees remark that the pollution of streams by sewage throughout the country is an evil which is increasing with improved house-drainage,t and they sought to place the outfalls as low as Sea Reach. The present is a plan which supplements and perpetuates the * Report of Government Referees, p. 172. Walter Crum Esq., F.R.S., has suggested to me as very possible that there may be times when greater quantity of water enters the Thames by flood than goes out by ebb-tide, owing to the large quantity of water taken from the river by evapora- tion in dry and hot weather.— See Dalton on Rain, Evaporation, &c., Manch. Memoirs, vol. v. p. 346. + The pollution of springs is a still more serious evil, in many instances involving the necessity of conveying the rain which falls on the moors through a long series of pipes. It is to be doubted whether such water is as good for drinking purposes as uncontaminated spring water. SEWAGE OF LONDON AND OTHER LARGE Towns. 153 evil of which they complain; a new patch is to be added to the old garment, and as a natural consequence the rent will be made worse. An evil ought to be honestly and fairly met, not merely slurred over and disguised. I now pass from the consideration of works which when completed will, there is every reason to believe, result in total failure, and will endeavour to show the practicability of realizing the first great object of removing sewage, viz., its utilization. Much has been said on this subject since Mr. John Martin in 1828 drew attention to the waste which was even then going on; but judging from the acts of public bodies, it would seem to be still doubtful in the minds of a great portion of the community whether the saving of the manure of cities is a matter of any consider- able importance. It is therefore desirable that the actual facts should be constantly brought under review. For this purpose I might bring the evidence of nearly every scientific chemist, but will content myself with quoting from Liebig, both on account of the great attention he has paid to the subject and the circumstance that, having been interested in artificial manures, he is not liable to be un- duly biassed in favour of natural ones. At page 177 of his Chemistry of Agriculture and Physiology this distin- guished philosopher says: “The mineral ingredients of food have been obtained from our fields, having been re- moved from them in the form of seeds, of roots, and of herbs. In the vital processes of animals the combustible elements of the food are converted into compounds of oxygen, while the urine and fceces contain the constituents of the soil abstracted from our fields; so that by incorpo- rating these excrements with our land we restore it to its original state of fertility. If they are given to a field deficient in ingredients necessary for the growth of plants, it will be rendered fertile for all kinds of crops. A part of a crop taken from a field is used in feeding and fat- 154 MR, J. P. JOULE ON THE UTILIZATION OF THE tening animals, which are afterwards consumed by man. Another part is used directly in the form of potatoes, meal, or vegetables; while a third part, consisting of the remnants of plants, is employed as litter in the form of straw, &e. It is evident that all the constituents of the fields, removed from it in the form of animals, corn and fruit, may again be obtained in the liquid and solid excre- ments of man, and in the bones and blood of slaughtered animals. It altogether depends upon us to keep our fields in a constant state of composition and fertility by the careful collection of these substances. We are able to calculate how much of the ingredients of the soil are re- moved by a sheep, by an ox, or in the milk of a cow, or how much we couvey from it in a bushel of barley, wheat, or potatoes. From the known composition of tle excre- ments of man, we are also able to calculate how much of them it is necessary to supply to a field to compensate for the loss that it has sustained.” Again, in page 181, he says: “In the solid and liquid excrements of man and of animals; we restore to our fields the ashes of the plants which served to nourish these animals. These ashes con- sist of certain soluble salts and insoluble earths which a fertile soil must yield, for they are indispensable to the growth of cultivated plants. It cannot admit of a doubt that, by introducing these excrements to the soil, wesgive to it the power of affording food to a new crop, or, in other words, we reinstate the equilibrium which had been dis- turbed. Now that we know that the constituents of the food pass over into the urine and excrements of the animal fed upon it, we can with great ease determine the different value of various kinds of manure. The solid and liquid excrements of an animal are of the highest value, as manure Sor those plants which furnished food to the animal.” From the above incontestable principles we may easily calculate the magnitude of the loss which is sustained by SEWAGE OF LONDON AND OTHER LARGE TOWNS. 155 the waste of sewage. If the excrements of an animal are not returned to the soil the food of that animal cannot be reproduced. Hence the amount of barrenness communi- cated to the soil by the system now endeavoured to be enforced in our towns may, considering no food or manure to be imported from other countries, be directly estimated by the food consumed by the inhabitants. Dalton, in the fifth volume of the Manchester Memoirs, 2nd series, has given the aggregate of the articles of food consumed by himself in fourteen days, his habits, daily occupations, and manner of living being exceedingly regular. They are — Bread’ 71..22..0080 163 oz. Milk... 43853 oz. Oatcake..,......... 79 ,, Beer... 230 ,, Oatmeal, «acs ievsk whe sd Depts, Gur 5 Butcher’s meat... 543 ,, Potatoes’ ......2. 130 ,, Pastnyern ses sees Bow Clicesal?.28.2392 32 4, otal: 2... 5251 ,, solids. 741% ,, fluids, Much more than the above quantities are consumed by the luxurious, much less by the aged and invalid. I think on the whole, and for our present purpose, that we may take them as the food of every man, woman and child in the metropolis. Hence we may infer that the 2,600,000 inhabitants of London consume every day provisions equi- valent to — 1,316 tons of bread. . 505,000 gallons of milk. 282 tons of butcher’s meat. 267,000 gallons of beer, 674 tous of potatoes. 88,000 gallons of tea. 285 tons of pastry. = 166 tons of cheese. Total,..2,723 tons of solid, and 860,000 gallons of liquid food. This is therefore the daily rate at which the productive power of the country suffers by the waste of one large town, and this is done in the face of a rapidly increasing VOL. XV. % 156 MR. J. P. JOULE ON THE UTILIZATION OF THE population. Yet there are many who treat the subject entirely as a commercial one, and if the cost of transit is such as to prevent sewage competing with guano in the market, they argue that it ought to be thrown away as refuse. But this is a fallacious, narrow-minded and selfish view of the subject. In order apparently to save ourselves a little money at the present moment it entails a heavy burden-on the inhabitants of the country in subsequent years. Guano will not last for ever. According to the Peruvian Survey the Chincha Islands can yield 18,200,000 tons. Of this quantity Great Britain alone consumed in 1857 no less than 288,362 tons, which, if we consider the entire waste of sewage in Great Britain to be double that of London, will almost exactly make up for it in money value. If the produce of the above islands, which afford the best guano, be reserved for the sole use of Great Bri- tain, it will last only sixty-three years at the present rate of consumption. JI am aware that other supplies have been found in various parts of the world, and that there is a trifling additional deposit each year. But when we con- sider the competition which will eventually take place on the part of other countries to secure so valuable a manure, and also the ever increasing difficulties of obtaining it, we cannot trust to our being able to import it in the quanti- ties we now do for even so long a period as that above named. In addition to the help derived from guano, the soil of Britain is relieved from the present effects of sewage waste by large importations of corn and cattle and of bone ma- nure. But such a supply can continue only so long as foreign governments remain in ignorance of the permanent injury sustained by their fields. Liebig, in his Agricul- tural Chemistry, complains that, if the exportation of bones continued on the then scale, the German soil would be- come gradually exhausted. SEWAGE OF LONDON AND OTHER LARGE TOWNS. 157 Besides, we ought not to be satisfied with merely keep- ing the productive power of our agriculture from decline. With a rapidly increasing population the wisest course would be to reserve such supplies of guano as we may be able to obtain for the purpose for which nature appears to have designed it, that of forming a fertile soil where sterility at present exists. In concluding this part of my subject I would urge the importance of recollecting that in the estimate of fertility, regard should be paid not only to the weight of a crop but also to its nutritive value, determined in each case by che- mical analysis. Liebig states that an increase of animal manure gives rise not only to an increase of the number of seeds, but also to a most remarkable increase in the pro- portion of those nitrogenous substances which are the most important constituents of food. Having endeavoured to show the imperative necessity, I will say a few words on the means, of putting a stop to the present waste. The first step I conceive should be to prohibit the introduction into the sewage of any organic matter which can be avoided. For instance, scavengers should be constantly employed in collecting and removing horse dung from the streets. The present system of send- ing carts round at long intervals of time allows by far the larger portion of this manure to be washed by rain into the sewers, thus forming a very serious addition to their impurity. Then why should slaughter-houses be tolerated? If only meat slaughtered in the country were admitted into the town, I submit that the meat would be cheaper in regard to its intrinsic nutritive value. The distress suf- fered by the animals in their passage from the field to the town slaughter-house destroys the richness and flavour of their meat, even if it do not render it positively unwhole- 158 MR. J. P. JOULE ON THE UTILIZATION OF THE some. By the present system a large quantity of offal and blood is removed from the country, where it would be a valuable manure, to the town, where it is a dangerous nuisance. I may mention in this place the subject of intramural interments, which even at the present day have not been entirely discontinued. Tens of thousands of human bodies in a disintegrated and decomposed state have floated down the sewers of London into the Thames. ‘The drainage of burial grounds into sewers is, in fact, enjoined by act of parliament. Now the body of any human being after death ought, in accordance with the Divine ordinance, to be permitted to return to the dust whence it came. For this purpose metallic coffins are unsuitable; and the body should be placed at a moderate depth below a soil on which there is a vegetable growth. I cannot enter into details on this highly important subject, but I am satisfied that the object of rapid conversion into vegetable life may be attained without in the least degree hurting, but rather subserving, those feelings of affection and reverence with which we regard the dead. After prohibiting the unnecessary introduction of or- ganic matter, the next step will be to deal with the sewage proper. And here we find at the outset that the enormous quantity of water mixed with it in the drains prevents the possibility of using it in that state for agricultural purposes. Messrs. Bidder, Hawksley and Bazalgette, among other objections, come to the following conclusions in their Report. First: “That the fertilizing properties of the organic matters contained in town refuse are for the most part destroyed by the long continued action of water.” Second: ‘‘That the cost and difficulties attending the ap- plication of liquid sewage in large quantities are absolutely prohibitory of its use.” Third: “‘ That liquid sewage can- not m general be used with advantage in this climate, ee ees SEWAGE OF LONDON AND OTHER LARGE TOWNS. 159 except in particular states of the weather, and in certain stages of the growth of the crops to which it is applied.’’”* The precipitation processes by lime, &c., even though at present commercially valueless, ought to be persisted in, if it is only in our power to deal with largely diluted sewage. But, according to Hofmann and Witt, not more than one-third of the fertilizing constituents can be thus separated. It is obvious, therefore, that we should deal with the sewage in a more concentrated form and before it is diluted with rain and other comparatively clear water. The separate system has been frequently advocated, but there is some doubt whether sewage in a concentrated form would flow through a long series of pipes of very mo- derate inclination. With these facts before me I see no alternative but a return to the cesspool system, to which I believe no inconvenience or nuisance attaches, except where it is attempted carelessly and with inefficient me- chanical and other appliances. The following is a plan which I venture to recommend in places where water- closets are generally used. I would place in the centre of the streets, cesspools having a capacity of about 1000 gallons. Each cesspool to be for the use of some 400 inhabitants, say 50 houses, and to collect water from urinals, cab stands, &c. The present sewers to be solely employed in carrying off rain and other comparatively clear water. A drain, or generally two drains of considerable inclina- tion extending from the cesspool up and down the street, to receive the water-closet pipes from the houses on both sides of the street. The total length of drain would be about 200 yards. A force-pump, permanently fixed in the cesspool, to be used every night for the purpose of pumping out the sew- age collected in it during the last 24 hours. * Report, p. 104. 160 MR. J. P. JOULE ON THE UTILIZATION, ETC. The sewage thus pumped to be discharged into tanks, and then conveyed to a railway to be carried to reservoirs situated at convenient localities in the country. Each tank might have a capacity of ten tons, and would then hold the contents of five cesspools. It might be drawn by a traction steam-engine, which also might be employed for the pumping. The discharge pipe of the force-pump, as well as its piston-rod, might rise to the level of the street, and the requisite connexions be screwed or clamped on when required. Immediately after empty- ing the cesspool a portion of McDougall and Smith’s dis- infecting powder* might be thrown in. This, acting on the sewage at an early period, would, Dr. Smith states, have the best effect in deodorizing, and in preserving the fertilizing property. I believe that in the above way Sicip sewage of London might be conveyed to the fields, and a very large annual profit realized, instead of the dead loss of three millions sterling, which must be incurred if the plan of the Metro- politan Board is carried out. The other advantages would consist in,— lst: Easy construction and repair. 2nd: Total prevention of infiltration of sewage, and the effects of accumulations of noxious gas. 3rd: Rapid removal of sewage before decomposition has had time to take place. 4th: An unpolluted river. * Sulphite of magnesia and lime, and carbolate of lime, 161 XII.— An Account of the Fall of Rain at Manchester, from the Year 1786 to 1857 inclusive. By Mr. Joun Cortis. Read November 16th, 1858. Turis paper has been prepared with a view of making the tables complete to the end of last year, and of showing at a glance the amount of rain which has fallen during the last 72 years. The account of the fall of rain from 1786 to 1793 inclusive is taken from the tables prepared by Mr. George Walker, and published in the Society’s Memoirs, vol. iv. pp. 584 and 585, old series. From 1794 to 1840 inclusive, from the tables prepared by Dr. Dalton, and pub- lished in the Society’s Memoirs, vol. v. part ii. p. 668, old series; vol. i. p. 496, and vol. vi. pp. 575 and 576, new series. From 184] to 1854 inclusive, from observations taken by Mr. Joseph Casartelli, of this city, who kindly communicated them to me. And from 1855 to 1857 in- clusive, from observations taken by myself in Plymouth Grove, Chorlton-upon-Medlock, on the south side of the town. The gauge I employ is a funnel 8} inches in dia- meter, with a perpendicular rim 5 inches high, and the top of the rim is 2 feet 3 inches above the ground, the fall of rain being registered daily at ten p.m. in a graduated glass cylinder. The gauge is situated in a garden perfectly free from surrounding objects, the nearest buildings being on the south side, and twenty to thirty yards distant; while it is more than thrice the above distance on the other sides from elevated objects. Dr. Dalton’s gauge was a funnel 10 inches in diameter, surrounded by a perpendicular rim 3 inches high, the top of which was a little more than two 162 MR. J. CURTIS ON THE FALL OF RAIN AT feet from the ground, and was situated in a garden on the south-east side of the town, and twenty yards distant from any house or elevated object.* Mr. Casartelli’s gauge was a funnel 5 inches in diameter, with a perpendicular rim 2 inches high, the top of which was between two and three feet above the ground, and was situated on the south-east side of the town, ten yards distant from a house on two sides, and the same from a wall nine feet high on the two other sides. It will thus be seen that the gauges employed in registering these observations were all of the same con- struction, with very little difference in their distance from the ground, and that the places where they were registered were within one mile from each other. They are also of the kind now recommended by Mr. Glaisher, of the British Meteorological Society, as the best for taking observa- tions on the fall of rain. It may be well here to state that Dr. Dalton found Mr. Walker’s returns to exceed his own by about four inches in the year, and that on inspecting Mr. Walker’s gauge he had reason to think that the method of measuring the rain employed was not susceptible of sufficient accuracy, and on his suggesting the same to Mr. Walker, the latter seemed to acquiesce.t For this reason I have given the fall of rain as collected by Mr. Walker in a separate table, so as to enable me to give two averages, the one with his observations included and the other without. In Dr. Dalton’s observations the fall of rain in the months of March and April 1807, December 1809, and January and February 1810, are not given. To fill up these blanks, and make the tables complete, I have in- serted the average fall of rain in those months, so that I am enabled to give the mean and total for each month and year, and also the mean and total for each month and year during the entire series of seventy-two years. * See vol. iii. p. 195, second series, of the Society’s Memoirs. + See Dr. Dalton’s remarks, vol. iii. p, 498, new series. MANCHESTER, FROM 1786 To 1857 INCLUSIVE. 165 d From the foregoing tables it will be seen that, includ- ing Mr. Walker’s returns, the average fall of rain during the 72 years is 36°3988; and, excluding his returns, the average for the remaining 64 years is 35°5620. Dr. Dal- ton’s average of 47 years is 35°523, showing the small dif- ference of ‘039 between the two latter averages. I think we may, therefore, assume that Dr. Dalton was right in supposing that Mr. Walker’s returns were in excess of the reality ; and, for the purpose of arriving at a correct aver- age, it will be safer to omit them and adopt 35°5620 as the average fall of rain for Manchester. As the situation of the rain-gauge employed, and the influence of surrounding objects on it will produce different results, the following table has been prepared to show the mean monthly and yearly fall of rain, as registered by each observer, of the whole period and of the 64 years, the average obtained during the latter period being that which I recommend being adopted as nearest the truth. Walker. Dalton. Casartelli. Curtis. Total 8 years. 47 years. 14 years. 3 years. 72 years. St years. 1786 to 1798 | 1784'to 1840 | 1841'to 1854 | 1855 to 1857 | 1786 to 1857 | 1794 to 1857 inclusive. | inclusive. | inclusive. | inclusive. | inclusive. inclusive. Inches. Inches, Inches, Inches, Inches. Inches, January .| 2°4685 2°257 33390 2-150 274890 2:4915 February 2°7499 | 2°443 2-437 1946 2°4556 2°4190 271145 2 308 2-261 1-526 272427 2°2588 2°3019 27114 1697 2/123 2°0536 2°0225 3°5103 2°4.46 27134 2°376 2°5008 23746 3°3000 2°691 3319 3°106 2°8985 2-8483 455832 3°706 3°793 3°663 3°8187 3°7231 47499 3°463 3°767 4356 3°7025 3°5715 452144, 3192 3°103 2356 3°2552 3'1353 4°5104 3°754 4: 262, 3°2238 39150 3°8404 November] 3°3019 3°712 3°416 2'020 3°5386 3°5682 December| 5°2873 3°437 37148 27012 3°5286 3°3088 Total... 43°0922 | 35°523 36672 _ 30°857 363988 | 35°5620 First 5 months 164451 | 14259 | 15°183 | 13°227 | 146402 | 14°4147 Second } G mos. 26°6471 | 21:264 | 21-489 | 17°630 | 21°7586 | 21°1473 The preceding table shows that, taking the average, April is the driest and October the wettest month in the VOL. XV. z 164 MR. J. CURTIS ON THE FALL OF RAIN AT year, and that the fall of rain in the first 6 months of the year is to that of the last 6 months as 2 to 8— that there is less difference between the two periods in the later returns than in the earlier ones; for Mr. Walker’s returns show a proportion of 16 to 26, Dr. Dalton’s of 14 to 21, Mr. Casartelli’s of 15-1 to 21:24, and my own of 13:2 to 17°6. Whether this progressive diminution of difference between the first and the last half of the year is owing to the later returns being for shorter periods than the former ones, or to some change in the fall of rain influenced by a change of circumstances, future returns will show. The following table shows the greatest and least amount of rain which has fallen in every month during the 72 years, with the year in which it fell, and the name of the collector. | Year. | Inches. || Year. ae | January. 1806 | 5°851 | Dalton. Ant 0320 | Dalton. February 1848 | 6565 | Casartelli. | 1800 | 0-440 | Dalton. March.. 1827 6:030 | Dalton. 0-180 | Dalton. | April... | 1791 | 4°750 | Walker. | 1842 | 0-160 | Casartelli. _ 1S 2) i=} 172) _~ May ... | 1792 | 8000 | Walker. || 1844 | 0-090 | Casartelli. June ... | 1830 | 7°055 | Dalton. 1826 | 0-200 | Dalton. July ... | 1828 11480 | Dalton. 1800 0-290 | Dalton. August .| 1799 | 8740 | Dalton. 1801 0-730 | Dalton. September) 1792 | 9°000 | Walker. | 0:240 | Dalton. October . | 1787 | 9:000 | Walker. 1817 | 0-604 | Dalton. November, 1825 | 7°375 | Dalton. || 1805 | 0-624 | Dalton. December | 1792 9°500 | Walker. 0-070 | Casartelli. i v2) oO rg i ios) sg Lol The largest amount of rain fell in 1792, and was 55°250 inches; the least amount of rain fell in 1826, and was 24-910 inches. The rain which fell from 1793 to 1814 was below the average of the 64 years; from 1815 to 1852 it was above ; and from 1853 to the present time it has again fallen be- low, leading to the inference that we have entered into a low series, and that, consequently, we may for some time expect the rain fall to remain below the average, though Did’ inch, by Mr. John Curtis, from the data of ¢ rclusive, and of Mr. John Curtis 0 t aa 1 @ mw pw wy Poms id Inches. John Curtis, from the data of and of Mr. John Curtis 4 inclusive ; 5 628 7 aie —— 2é81}— ++ = Set 9EST ical of Mr. Joseph Casartelli from 184% fo 18 from 1855 to 1857 inclusive, ‘e; | \ jaa eels | Dr. John Dalton, from 1794 to 1840 inclusi 1B ; Soil 096-18 s\p===SS555500c cr “| 008.8 ah i {Aa a ISIE “oe Bae OCR LE | howing the Fall of Rain in Manchester, to the scale of one-sixth of a standard inch, for one standard inch, by Mr. Diagram showing the Fall a, / ] | | | | T ! | | | | | — J e - | | T y nehes. 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THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 175 5. When n=5, assume X=H=X + G &y+ Az Ly + Ay Xy~+ A, 2X5, Xg=X1 + Ay Lat Ay X34 A, U4 Ay 2s, Xs=Ay + dy Xy+ Ay Wzy+ Ay Uy+ My Ws, Xy=X, + Ay e+ As Ly + Ay Ly+Q, Ws, and 1 4(#) 3X7 Kg KX, Ky. Then the condition necessary and sufficient for the sym- metry of the terms in 2* is p l=a, G7 a3 a; The corresponding conditions for the terms in 2}, and vv are Ya=F ay Ay A,= Aj Az Az + A, 2 y+ A, G2 y+ Ay Az a = Aj My Ag+ Gy Ay AZ 4+, As A} + a3 Ay Ay = Aj As Ay + A, Wy As + A, Ay Wj + Ay A My 5 and those for terms in #2 x are BA, Uy = Ay My Ag+ My A Ug + Ay G3 A+ Uy 3 A+ Gy G+ Gy a3 = 2a, dy ds a,+ af a2 + a} a3 + a3 a} + a2 a. Tt will not be necessary to exhibit the rest since the above are sufficient for the determination of a. The first and second indicate that a, a, a, and a, are the roots of an equation of the form a'— Aad’+ Ba?- Aa+1=0, and, combining them with the others above given, we find Sa Sa, dy A3= Ja? Ay dg +40, dy Az Ay; or A’?=38A+4; and A=4, or -1. Again : — i Yd, Mg = Ay Ay Us Oy + 4(3a+ Fay az) (= B(2G, )? + 2a — BA Ay As — 2A) Ay As A; or B=tB?-$A-2. Hence the three systems A=4, B=6, or, =4;,A=-1, B=I, give rise respectively to the relations (a—1)*=0, (a-1)*-—10a=0, and a+e@+a+a+1=0. 176 ~—- REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND The first renders X symmetric. The second does not enable us to satisfy the third equation of condition. But the roots of the third (which are, in fact, the unreal fifth roots of unity) satisfy them all. Let then i, ?, ® and # denote these roots, and Si) =44+1 MH4+P at? att 4;; then JPM=At? +H agti x+F x,, J@P)=Az +B dati Lett H+? as, SJ@/) HA +t at? H+? wy+t 23, and m(2) =fli) fl) fE) FO. It will be observed that, if from the above expressions we expunge the 2’s, the four horizontal rows read downwards are identical in value and order with the four vertical columus read from left to right, and that 7 and # lie in inverse symmetry upon, and 2 and @ around, diagonals. It does not appear to be possible to render 7,(z) more nearly symmetrical. A similar arrangement when x is prime, or a modification of it when x is odd, will probably be found available. "When all the divisors of m are even, we must adopt an arrangement analagous to that employed for bi- quadratics. (See Art. 4.) By actual development, we find fli) -f@)=Se +741) +79), +. f@) f@) =30? +762) +760), where T Ly Mgt Ly %gt+X3%y +X 2, +%5 2, 7 2 Ut Hy Ly tL; Lat Mo %yt+%yX, and p(t) =i+e ; +. ma) ={3a*+7$() +74@)} {34+ 74@) +79O} - Developing and bearing in mind the known properties of i, and the relation T+T =A ao, we find w(x) = (2a)? — Sa? Bay @— (Fa, #2)? +577 =(Sx)*-5( Sa)? Fa, v2.4 5( Za, x)? +577’. The function t7’ is unsymmetric relatively to x; conse- THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 177 quently, when n=5, 7,_,(v) cannot in general be rendered symmetric. 6. To show the bearing of this theory on the solution of equations. Let 2,, 2%, %3,-++x, represent the roots of an equation of the 2 th degree, xv" +ax" + ba” +--+» +sx+t=0, and consider the effect of supposing 7r,,_;(~) to vanish. For the quadratic xv +ar+b=0, we have (Art. 2) (x) = V (ay +2Xq)*— 4a, = V a — 4b, and the evanescence of 77,(z) gives b= ja’, *, @+axr+ia=0, or (#+4a)’?=0, which is immediately solvible. 7. For the cubic a+ azxr’+br+c=0, we have (Art. 3) T(x) = (Sx)? - 38a, %=a — 3b, and the evanescence of 7,(x) gives b=4a’, “. P+axr+4+ie¢ x+c=0, or (w+4a)*— (3a -c)=0, which admits of easy solution. 8. For the biquadratic a+az*+ ba’+cr+d=0, we have (Art. 4) T(x) = (Lx)> — 42a La, 2+ Bra, #2 Xs = -a>+4ab-8e, and the evanescence of 7;(”) gives b= i@ + cs a *, @+ae+ (3084 )a* + 00+ a=. a or («+ 4ar-£) - (4-4) =0, which also admits of being easily resolved. 178 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 9. Thus for the lower equations, the evanescence of the symmetric product conducts to a solution. If therefore the result of the elimination of 2 between w+ ax" 4 ba"? +--+ +8a4+t=0, and y —W(v)=0, where yf is rational, be represented by y+ Ay” *+ By”? + ---+8y+T=0, and yy be so constructed that 7,_,(y) may vanish, y, and consequently x, will become known. The expression 77,_,(p+) does not contain p, and the relation y=(v)=pte leads to illusory results; but if we assume y=p(v)=pet+ 2, we can reduce the solution of the general biquadratié to that of a cubic, and the solution of the general cubic to that of a quadratic. For since p only enters the product T,-.(px+a"*) to the degree which y attains in 7,_,(y), that is, to the (n—1)th degree, and 7,(pa2+a2?) and 7;(pe+a*) are symmetric with respect to 2, it follows that the equations 1(px+a2?)=0, and 7;(pa2+2*)=0, are respectively of the second and third degree in p, and that their solution gives p, a symmetric function of 2, in terms of a, 6, &c. 10. For quadratics this method must be modified. It is impossible to construct y so that m(y) may vanish without being led to nugatory results. But in this case the symmetric product yields an immediate solution. For (Art. 2) ™(#) =*(a- a) = V(x, +2)? — 4a, a; or a — B= > Va —Ab. and &@+%=- a, whence the usual solution THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 179 1l. Cubic Equations. The elimination of x between 2 +ax’+be+c=0= (2), and y= (a) =pa +2°, gives y+ Ay?+ By+C=0, where A=-y=- (p>#+ zz") =ap —(a— 26), B= 2y, Pr=P VA Lz + PU Xz + Taj A = bp’ — (ab—3c)p—2ac+ 6, and C=- 9, YoYs=— 2 Lyk, p> +p” LH + PLA, Lz + 2 Xp 2s) =c(p*— ap’ + bp —¢) =— ex(—p). Again: — The evanescence of the symmetric product gives 7(Y) =72( px + 2*)=0, or (a*—3b)p*-- (2a -7ab+9c)p+ at — 40? b+ 6ac+0?=0; and, if we make a@—3b=x, ab-9c=n, and 4’-3ac=p, this equation takes the form Kp’ — (2ak—r)p +a? «-ar+pu=0, Qax—r+ V2 — dep Pp = . 2K Consequently A, B and C are known; and, since (Art. 7) G+-)=@)-¢ 3f/ KR S ae y=-3+V(G)-C. Appropriating this root to y,, the corresponding formulz for y, and y; may be obtained by writing ¢ and é before the cubic radical; and, combining the two equations at the head of this article, we are conducted to (y+p*—ap+b)x—(p-a)y+c=0, pa _P-OYRe y+p(p—a)+6 All the roots of the proposed cubic may be obtained by substituting y,, y, and ys successively for y in this expres- sion. VOL, XV. BB whence and 180 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 12. If, in place of y= (wv) =pet 2, y=¥(2)=(pt+2)"; and eliminate z between this equation and the proposed one, there results an equation in y for which we assume 3p*—2ap+b A=- y= Bhs: Sop -3p+a B=> SS Nn Y2 x(—p) and C=-n y= 1/273 x(—p) The condition 77,(y)=0 now gives Kp’ —Ap+ w=0, rt V2 = 4A or SSS 2K Hence, as before, A, B, C and y are known, and the roots of the proposed cubic are 2=Y,'—P, %=Y_—p, and 2,=y;*—p. 13. Biquadratic Equations. If we eliminate w between a +aa*>+ ba’+cx+d=0, and x +px-y=0, there results y'+Ay?+By’+Cy+D=0, where A= y=—(pla+ 22°) =ap— (a — 26), B=Sy, =P UM, Ly +PUAj y+ VA @ = bp” — (ab—38c)p -2ac+ 6+ 2d, CH=-ly) Yo Yg=— (PVA, Wy Let pay Ly Wz +pBaj #3 xs + Sa} x3 23) = cp* — (ac —4d)p” — (8ad — bc)p + 2bd-c’, and D=YoYsYs=% Vp Xq@(P+PLL +P LM, Uy + PUA Ly Ly + 2 22 23 U4) = d( p* — ap* + bp” — cp+d). THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 181 The evanescence of the symmetric product gives my) =Ts(p2+a*)=0, or (a3 — 4ab + 8c)p* — (3a* — 14a? b + 20ac + 86° -82d)p” + (3a° — 16a* b+ 200° c+ 16ab? — 32ad —16bc)p — (a Gat b 4+ 8a® c+ 8a? b°— 8a? d—16abe + 8c”) =0, a cubic in p. ~ Again: — The biquadratic in y may be put under the form (Art. 8) Fett (y+4Ay-x) =(q)-D: or phe ieee ery Finally : — Combining the equations v+aa2+ba*?+cx+d=0, and vtpe—-y=0. by the method of the highest common divisor, we find (ay -—2py —p>+ap*-bp+ojaty + (P- ap+b)y+d=0, ya yt (r—apt+bjy+d (Qp—a)y+p*-ap*-+bp-e and, as in cubics, all the values of z may be evolved. 14. Of course y admits of an infinite variety of con- structions equaily available. For instance, if we give to y another form, and assume y=pet+a”; then, eliminating « between this equation and the pro- posed biquadratic, we are conducted to an equation in y for which or A=apt9, a pie, GO Ad ab B=bp'+ —{—Pt+@ Ce = Bee SG ; ae +7) 2bd — —h-2d P—2 1 deep 2 b ad gt b D=dp'- d a Pa and the evanescence of the symmetric product now gives 182 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND (a® — 4ab + 8c) p> — AG c+ 8ad — 4bc)p* + 5 (4abd — ac* — 8cd)p + i (8ad* — 4bed + c*) =0. Hence p may be determined; A, B, C, D and conse- quently y become known, as before; and z is given by a= — ft apy - Up + bpp y* + apy? + (bp —2)py + ep? — ap?’ a result obtained, like the corresponding one in the last article, by the aid of the common divisor. Secrion II. The Resolvent Product for Quintics, and a new Cyclical Symbol. 15. For the general equation of the fifth degree, xv + ax + ba + cx*+dx+e=0, the resolvent product is (Art. 5) 1 4(@) = (Dax)* — 5(Lax)? Day v.45 (La, 2)? +577 =a'—5@6+56?+ 517’. This product is six-valued. For SO=C a t+ia,t+Pat+ Patia;, and therefore f(i) has 1-2-3-4-5 or 120 values, which may be formed either by permuting the powers of é or the roots of the quintic in all possible ways. Let us call f(z) the first factor of 7,(#). Then each of the values of f(i) may be made a first factor, so that, at first sight, it might seem that 7,(#) has 120 values. But by the properties of 3, m2) =f) SE) L® fe) Whence it appears that when /(i), f(#), or f(#*), becomes the first factor, no new values of 7,(x) are introduced, but we are remitted to our former value. Thus the number is reduced one-fourth. Again: — Let THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 183 J (i) Sta, +P H+ P a+b a+? 2.=H7f(), then f'(#*)=?f(?), f(P)=#fE), and f@)=FFE)s ~LOSLE) LE) LO=SO FE) SE) FE). That is, when /’(i), or if(i), is made the first factor, no new values of 7,(x) are introduced, but we are remitted to our former value. A similar argument holds for 7/(), #f(i), and zf(2). So that the number of values is still fur- ther reduced one-fifth, and 7,(#) has only 120+4-5 or 6 values.* 16. The same result may be arrived at by considering the form of the product, without reference to its factors. For THB, Ly +My Lg t+ XU tH, Lz+ Xs Xy is a circular function, each term being derived from the preceding by advancing the roots a step in a certain cycle ++ Ly Vy Uy Ly Vs Ly Ly Vy Ly 5+ >> And, since the coefficients of the terms in the expression for tT are equal, we may, in forming its values, regard one of the roots, say 2,, as fixed, while the others are permuted inter se. Thus we shall have 1-2-3-4 cycles, giving rise to 24 corresponding expressions for 7; but since THX Ut Xo Ut %yUyt+ XX, 4232, HL Ly UX + Xe ky t Ly Xs t Ws Xs, these 24 expressions may be grouped in pairs, the members of each pair being equal. Hence 7 has only 12 values. Again : — Since T =H Let 23 ty Lat AU t+ U2, HL Ly t+ Ly Lyt LyX + UX + W522, the several values of 7 may be referred to the same 12 cycles that arise in the formation of the values of r. Con- * For this proof I am indebted to my friend Mr. Cockle; and I may add that, having from time to time sent him portions of my researches, he has been so good as to give me the benefit of his more extensive reading on the subject, to point out coincidences, to suggest modifications, and in various other ways to help me to improye my methods and abridge my calculations. 184 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND sequently, since 7 and 7’ are complementary to each other (for their sum=>2,7,=6, a one-valued function), it fol- lows that rr’, and therefore a —5@64+582+ 57’, or 74(#), is, as we have otherwise proved in the last article, a six-valued function. 17. The six values of rr’ may be exhibited in two ways. First, (@y Hy + Ly Xz ty Xp + a4 X3 +25 X,) + (Ly %yt+%y Xt %yX3+ LyX, +Xs Xo), (2 Ly + Hy Ly + Ly X3 + Us Ly t+ Xj) + (Lj Ay t+XyUst+ My Xy~+XLz%+AyX), (ay y+ Wy Uy + Hy Wy + Hy Xs + Ws Xj) + (# Hy + LyX; + @yL\ + X3%y+H;X%), (ay y+ Vy My + Hy U5 + Xs U3 t+ X3 HX) (@ Ly t @y Ly + UyXy +L %+Lz Xe), (ay Ly + Wy H+ Hl; y+ Ay Ly t ay Xy) + (¥, Vy + LyX + M5 Xj +X Le+Xy 2s), (ay yt ay X5+ Hs y+ Hy Ly + Ay X,) + (Up + Ly My + U3 Ly + Hy Lz+Xz 2s), corresponding respectively to the cycles XY a LX, Sees en Psy rebels LM peters ee mee era as ae aM , XN ¢ \ ‘ / ‘ / ‘ X,.f Xo Ly ‘. 1 bey oy eae: 2 % 1 \% H 1 I \ 1 1 1 1 ! 1 1 \ ! ‘ i \ ! \ / \ ' \ ui ‘ ’ \ / \ i N. ¢ AY 7 \ BS “, ‘s Aen XS ra preteen BL ee. % we Pa seas Ber ta Oe ie a f Sat as. y yer se. 2z,/ Ne A yg be ee \2. 3] \ tA 41 y os aT eat ! 1 ' ! 1 H 1 1 H I ' 1 ‘ 1 H \ ‘ \ i \ ‘ i 7 ‘ i ‘ a \ 7 ‘ a \ 7 ‘. 7 Lg SS ‘. Pd See aw oe 9 ™ —— | ae pet - S 4 pe ea X, orn or, secondly, (@y Lat Ly Uy t Ls V+ Le X3t Uy As) + (@, Le +43 Xz + %5 Lat Ly M+ UN), (@y y+ 24 X54 Uy Hy +Ly%yt a5 Ls) + (@ Ly Uy y+ Ly @y+ %yUzt+Xs%), (2 y+ 23 Ls + Wy Vy + Hy + Hy Hy) © (1 yt Hy Hy + Ly Xs +5 Uy+%, 2), (@y Ly + ay @5 4 Lz Ly + ay Lys as) + (LT, Lp +a, Xt 3 %y4+LyXs+H5X), (ay y+ Lg V+ Hy Ly + 4&2 +L5 Ls) + (ay 3+ a5 Xy+ Ly yt LL; +52) (2, y+ 4 Ag+ Hq H+ Hy H+ X54) + (X, Xp + ay H+ XyX3t+ XyUz+H5 2), THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 185 corresponding respectively to the cycles Ty x4 & i a eee - -_——— ae oe a ms, oe a *4 Me Re. my igs yy Ee Mi i ‘2 Si 4 ‘3 Ti ¥ 3 H Fi ! 1 8 : 9 ' 1 i 1 ! ' t \ , \ / \ : \ £ re y \ / x 7 Si oft x vs ‘2 Sak. SB Xs Z5 = eee c U4. Lz, we na Ly L Wy x woot aen wes pee t ’. a ~. - ~ Ny: ¥e KA s of xX / \ y <\ ‘ Wy 4 U3} U3 aie 4 1 I 1 1 1 Tho setitorce", Ve fare nyo heehee’ Py gt8 Sod ‘ ! 4 / \ Fi \ ‘s *y “f \ ff ‘ / 4 4 a XN. ve 8. oe - ~~. -- etn he X~ ———— ee AL Uy ~———— Xs X3 ----" Ig 18. New Cyclical Symbol. Let (0) be a function of a, and let x(qg) be derived from y(0), by advancing each of the roots contained in it g steps in the cycle r. And let =x (0) =x(0) +x(1) +x(2) +x(3) +x (4). Then 2, R= Da", >, vt (et +22) = sar zt, &e., and ‘ Pay ae = = er ta 1% = 472%, 10 %3= 247% #3, 2/2 2, Hy= Dg 2, Xe, 2, 2 U3= 2/5 2, 25, , ‘4 / B's By Pg D9 Ly Ls 5 i's Ly Lg= Dig 2 Bs y D4 Py Lg= DX, X25 why 24= > oP, X45 , / D's Ly Py= 2X Ls, D5 By Lg= DX Ws, Big ty Ba 2 lly oe Do fia el Lew The six values of r7’ may now be briefly expressed as follows : Ty T = By Ly BL ye= By Uy 7 Uy Ls=T, 7 Tz T g= Dy Vy Ly Dy Vy Le=D"g Ly Ly B'gL, Lz=Ty T's 737 g= 3X) Ly % gL Ly= Dg Ly Uy X'g Lj Lz=Ty T'g, TT gi" Hy Wy D4 Ly MH QV Wy 9X, He=TyT'0, TT p= D5 By Ly 5X, Ve= JV, My BX Le=TT yn, ToT GD GX Lyd GX, MHD py Hy Dy M=T yy. >’, must not be confounded with Mr. Cockle’s >’ of 186 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND epimetrics, being more general in its conception, and possessing peculiar working properties. It is a symbol of cyclical operation. 19. Since each root recurs at every fifth step in any given cycle, we have x(5)=x(0), and Bx) =x(1) +x(2) +x (3) +x(4) +x(5) =27-x0) 5 whence, by generalization and induction, 3 X(9) = 3x (0). That is, the circular function 3’,(0) is not affected in value by the simultaneous advancing or receding of the roots which it contains any number of steps (g) in the cycle 7, Consequently 3 Xi (0) 2, Xe Ms = 5’, [xi(0) £x2(0) + x2(1) +.X2(2) + x2(3) + x2(4) §] = 3%", [x2(0) 2x1 (0) +x1(1) + xn (2) + 20(8 Re )5]. This commutative property will be found of great prac- tical utility in dealing with circular functions. 20. For example. Taking the first cycle as a type of the rest, and omitting, for the sake of simplicity, the unit suffix, we have TT =D Ly Ly 2 Ly Bs =D" fay Hy(@y Vet Hy Uy +2 Xz +H, Xj +25 Xo) § = (af Ly y+ Ay WZ Mp4 Ly Ly Ly Ly + Uj Uy Vy +2, Ly 25). Now 32, #3 %;= D2, Xa #3 2,4, and applying the theorem XD =2"x0); we find >a, Br=>'22 7,23, and Xa, 23 43=2' Li Uys; "TT Hay Ly Ly Vy HD Tig y+ Le Xt Lg Xs + Hy Hs) ; but Daj ay P= I’ ai (ay Ly + Hy Uy t Ly Ls + Hy Uy +a%yXs+Xy 2s), *, TT! = Say Uy Wy Ly + YH} Uy Ly — BX} (Lp Ws +23 Xs) 5 and consequently (Art. 5) m4(x) = (Sa)*—5 (Fax)? Fay e+ 5( Bay Aq) + 53a, Ly ya +5 Say ay 25-53’ a}(%y Ws; + 23 Xs) =a' — 5a? b+ 5ac+58? — 15d — 532i (x2 %5+ 23%). THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 187 Results, corresponding to a certain extent with those above given, would have been obtained if, instead of con- fining ourselves to five, we had dealt with any number of symbols. 21. We now proceed to show that, when the resolvent product 7,(7) vanishes, the general equation of the fifth degree admits of finite algebraic solution. In establishing the analagous proposition for the lower equations (Art. 6, 7, 8), we supposed the product 7,_,(”) to vanish, with- out distinguishing its factors. But by assuming the evanescence of a factor we should of course be conducted to substantially the same results, and in dealing with the higher equations it will be found that the latter method possesses peculiar advantages over the former. In order to abridge the calculations, we shall suppose the given equation to be deprived of its second term; 7.e. that —@=24,+4,+43,+%,4+27;=0, a supposition that will not affect any of the values of f(z”). Next, if we make fli) =a, +i 4+? 2,48 v.47 25=0, f@=HtPxttati t+? 2=5A, f@®P =2,4 Pati attest? 27,=5f, and f=a2, +t ar,+8 a+? ati #,=5Ps, we shall have ®=P,+P.+Ps, 2=t Bi +t B,+? Bs, M=t B+? P,+tPs, =P B, +0 Bo+t Bs, =i B,+t B,+? Bs, and b= 32, #,=— 58; Py. Whence we find a°=158, 83+ 1563 8s, Yxt=2063 8, + 808? B3 + 208, 63, and = 3a°=5Pi+ 1506; A, Aj + 100P, 62 83+ 563+ 523. But by the method of the limiting equation, or otherwise, we obtain Se=- 8c, Fxt=2h-Ad, and Fa°=5(be-e); and by comparison with the above, VOL, XV. cc 188 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND —c=56,63+5f3 Ps, — d=5 6} 8, +58, B3- 36", and — e=f}+ 83+ B3— 6(8, B3— BBs). Or,* if we put —b=5P, —c=5Q, —d+10°=5S, e=H, Bz 8;=u, and B;=Puv, we shall have 0 =O abe « ates w+ Pyu— PSv=0......». . 2) —10P? v2 w+ P® v3 + P?(5PQ + E)v?+ Pov= 0. <8); ae elimination of v between (1) and (2) gives u’—Quv?+ P(P?+S)u-P?Q=0 . . (A), and between (1) and (3), u® — 8Qu?> — (1LP?— 3Q?)ut + (PE + 15P°Q — Q*)u3 — P?(QE + P+ 5PQ*)u?+ 2P°Qu—P°Q’?=0 . (5). Proceeding witb (4) and (5) by division, we obtain awv+bhut+e=O0 ... . (6), where a=—11P*+13P"S — 2PQ’+8’, b= — P(P?+S)E- Q(16P*+8P*S — PQ’+S"), and c= P?Q(PE+5P°Q+4+ QS). We might now carry on the process of division until we arrived at a linear equation in uw, but we shall be con- ducted to a more simple and elegant result by proceeding thus: — Arrange the terms in (4) and (5) according to ascending powers of wu, divide the latter equation by w’, and the result by the former, until there arises a biquad- ratic in wu. Arrange the terms of this equation according to descending powers of uw, and divide it by (4); there will result a quadratic in uv. In effect, we shall be conducted to * Mr. Cockle’s notation is here adopted, so far as my own objects will allow, in order to facilitate the comparison of our results. See a series of papers by that gentleman “On Equations of the Fifth Degree,” published in the appendices of the Lady’s and Gentleman’s Diary for 1848, 1851, 1856, 1857 and 1858. THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 189 Ree PO SOT TSO! (Zs where A=11P?+ 2PS — Q’, B= — PE — Q(16P°+3PS — Q’), and C=P°QE + P?(6PQ’+S8). By (6, 7); (a — AP)u? + (b — BP)w+ (¢e - CP) =0, or 7) on gh OO a ee ee (=) 3 where A’=11P*S — PQ?+8?, B’= — PSE - QS(5P?+8), and C’= — P*?(P?Q’+ PS? - Q’S). Combining (7) and (8), we find (A’ B- AB’)u+ (A’ C— AC’) =0, or P(mE+n)u — P(pE+q)=0, ote ya PEt mE+n’ where m= P?Q?+ PS? - Q’S, n= — Q(121P*S — 16P°Q’ + 28P?S? — 9PQ’S + Q*t+83), p= — PQ(11P?S — PQ?+8?), and g= — P(11 P°Q?+ 11 P*S?+ 57P°Q’S — 7P°Q*+ 13P*S* +2PQ°S’ + Q'S + §%). ; Moreover, multiplying (1) into PS, and (2) into Q, sub- tracting the former from the latter result, and dividing the difference by w, we obtain Quw?—-PSu+P(P?Q-S8v)=0 . . (9). Next, multiplying (1) into Q, and subtracting (9) from the result, we have (PS — Q?+ Qu)v — P(P?Q — Su) =0, o = E(PQ-Su) _P(pE+g) | PS — Q?+ Qu mE+n where m = —8(LOP*Q? — P*S?+-3PQ*S — Q‘), n' = — Q(11P°Q? + 132 P°S? — 80PQ’S + 9P*Q!‘ + 41 P88 — 35P°Q’S? + 11PQ‘S + 2PS* — Q*- Q’S'), 190 REV. R. HARLEY ON SYMMETRIC FRODUCTS, AND p =PQ(P°Q?+ 12P°S? — 2PQ’S + 8%), and q =— P(110P°Q?S — 16P!Q* — 11 P*S*— 29 P°Q*S* — 2P°Q'S — 13P2S!+ PQ®— PQ’S' — Q'S?- 8°). The quantities w and v being now known, Bi, Ba and BG are given by /p4 oe B= / zy = V Pv, and B,= a and the roots of the quintic a2 —5Pe-—5Q2*?-5Rer+E=0 . . (10), where R=-ic=S8-P’, are the values of the expression A wd 3 arya )* Pv + (1) = ee - 22. Substituting for w its value in (8), we obtain A’(pE+q)?+B'(pE +9) (mE +n) + C’(mE + )?=0, a relation which admits of simplification. In fact, if we restore the values of A’, B’; C’, m, n, p and gq in terms of P, Q and §, that relation takes the form A’m(PQSE?’ + «hk? + AE + p) =0, where x, X and yp are rational and integral functions of PQ and 8. ; +, PQSE’+ «E?+r’E+p=0; or, in effect, PQSES + § P!‘Q?+ P’S?+ 4P?Q*S + PQ! + PS’ + Q’S*} FE” + §121P°QS + 10P°Q? + 175 P!QS? — 88P*Q'S + 12 P*Q° +27P°QS? — 7PQ’S? + 2Q°S + 2QS*2E + 121 P°Q? 4+ 121 P8S?— 561P7Q?S + 102P°Q! + 286 PS + 227 P°Q’S? — 137P!Q!S + 191 P*S* + 17P°Q* — 30P2Q’S* + 8P*Q#S? + 26P°S' — 7PQ?S! — 7PQ'S + Q’+ 2Q'S?+S°=0 . (11), a relation due to Mr. Cockle, who first announced it in the Appendix to the Lady’s and Gentleman’s Diary for last year, p. 82. Of quintics, whose roots have, as yet, been exhibited,* this is the most comprehensive form. It * Mr. Cockle, in the Paper above referred to, does not ewhibit the roots THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 19] includes, as particular cases, the quadrinomials of De Moivre and Euler, and the ordinary binomial. 23. Thus comparing De Moivre’s equation 2 —5av+5a2r—-2Q2y=0 . . . (12) with (10), we see that P=a, Q=0, R=-a’, and E=-2y, “, P=-R, and S=P?+R=0. Now when Q@ and S vanish, the condition (11) is satisfied, and B, B,=P, B, B3+P383=9, Bi ;+8,63=90, and Bi + 68+ B3+ 4P(B; 5 — Bz Bs) =— E.* The second and third of these equations are satisfied by 8,=0, and the fourth becomes Bi + B=— E, which, combined with the first, gives Bi=-YE+ VIE PPayt Ve— ai, Gi=-4E- VEEP- Poy - Vop—a; and the roots of (12) are the five values of the expression 2 (1)? Vy + Vep— a+ (1)? Vy Vor = a, o a(1)3 V y+ VP = 7h 24. Next, when P vanishes, S=R, and (11) becomes (QRE + Q!+R*)?=0, or QRE-+ Q*-+ R?=0; Euler’s criterion. In this case (Art. 21) Bi B.=0, B, K3+R3P;=Q, BiB: +P2As=R, and 8} + P2+ Ps=— EH. of the quintic, but they are implicitly given. The value of v, which may be deduced from Mr. Cockle’s equations, will be found to differ in form from that given in the text (Art. 21), a circumstance arising from the dif- ference in our methods of eliminating v and of combining the eliminants. * It is to be noticed that these equations are employed in order to avoid the vanishing fractions which would arise in attempting to deduce the roots- immediately from the formule given in Art, 21. (1)? Vy + Vop= ai + 192 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND From the first equation we have §, or ®,=0, and the second and third give ; eh Oe 2 R 5 or (3: =—, and §: or B=—, 3 or £3 R and #3 or 8 Q which satisfy the fourth. For Q? R?2 Tasks alg or QRE+ Q*+ R'=0. Consequently, when this relation holds, the roots of the quintic a — 5Qa*-5Ra+EH=0, are all included in the formula $5, @ pnts, BR. Yt ea hal 25. Again: — When P, Q and R vanish, (11) is satis- fied, and ee te x=(1)°V -E, the solution of the ordinary binomial form. 26. To return to the resolvent product. Denote 7,(z) by @, and put DF (+ 2,24) =U ; then (Art. 20) TT = LX, Ly XX, + VX? 2X, x,—- U=ac—d-U, and 64+5U =a' -—5a@b+5ac+50?- 15d. Hence, for the equation a —5M2t* — 5P2* —- 5Q2*— 5Rr+E=0, we have 77 =5°MQ+38-5R-U, and 64+5U=5'M'+5*M*P+5°MQ+5°P?+3-5°R. Now it is known that the general equation of the fifth degree may be put under any one of the six following forms, viz., x — 5Pa -—5Q2*°-5Re7+E=0 .. . (a) 2-5Qe?-5R2a+E=0 ... (0) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 193 Riad h=07%, as. fe) e-5Q7 +E=0.... (2) g@=—5ee +H=0» .... We) # —5Ma*+EH=0°. . .-(f) (See Mr. Jerrard’s “‘ Mathematical Researches ;” Sir W. R. Hamilton’s “Inquiry into the Validity of Mr. Jerrard’s Method,” published in the Sixth Report of the British Association for the Advancement of Science; and M. Serret’s “Cours d’Algébre Supérieure,” Note V.) For (a), (@) and (ec), we have 7 =—-7=3-5R-U, or U=77+3-5R . (a); and the corresponding relation for (d), (e) and (f) is =-P=-U, or U=" . . ~(a). Again: — For (a), we have G5 Ui PPE Se bP Bet ye ets (8); the corresponding relation for (4) and (c) is 64-5 = 3 FR valine ot) (BS for (d), the relation is 6+5U=0 Sh gle (BE) and for (e) and (/), 2-bAU SRP appre droen(Be) and G2 Eau = pOME SI oe SI. i) respectively. It hence appears that (d) will be the most convenient form with which to deal in calculating the coefficients of the equation in @. 27. We know that, for (a), 0, 0, 0; 0, 0; 0g =k(PQSE’ + «E’+rE +p), where k is a numerical constant, and «, \ and yw have the same signification as in Art. 22. To determine f, let us take the particular equation a — 5° a’?=0; then, making 2,=0, 2,=0, #=5, %=5e, and #,=5e’, (e denoting, as in former articles, an unreal cube root of unity) we find 194 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND U = 21 ay (e223 +23 Ls) = 23 U4 05=5*e ; . by (B’), 0,=-5U,=—5’e; and similarly ,=6,=—5 6) G0; == 5, end 0;=-— 5; . 00503 0,0;,0,=5*. But in the particular case now under notice B=0;andp=Q'= 535: '3..0).050, 00:0, =k = BEES consequently Sk = br ero Generally, therefore, 0, O, 0; 0, 0; 0, = 5" (PQSEH? + «E?+2AE+ p), which gives for (d), by restoring the values of «x, A and pw and making P and § vanish, 0, 0, 0, 6, 6; 0g= 5% Q’. Section III. Direct Calculation of the Equation in @. 28. The six values of @ may be regarded as the roots of the equation +D,+D,64+D;4+D,0+D;0+D,=0 . (1). Suppose now that, by Mr. Jerrard’s method, or otherwise, the general equation of the fifth degree is reduced to the form v—-5Q2°+E=0 . . sepa then (Art. 26) 6=-5r=-—5U, and (Di = 20527 =—b>, Us We might, therefore, calculate this coefficient by writing out the six values of —0@, 57°, or 5U, and taking their sum; but the labour may be materially abridged by the following method. (See Section II., Art. 18, et seq.) TS Fa) — > ay wes, = 2 (TLR L, Wy Ly + Ly Ny Ly Ly +H, Ly Lys + Aj Ly Us) = Sy" (a? 23+ Qa} a, L;) +2DH, Wy Ly Xq3 or, since 22, #2 2, 2,=0, T= Dy (a H3 +277 a 2). THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 195 Now the symbol 3’ written without a suffix being under- stood to belong to the first cycle (which is taken as a type of the rest), we may obtain =7° without developing the _ function on the right hand side of this equation, or finding the values of 7° corresponding to the other cycles. The following theorem enables us to pass directly from 72” or U? to Sr" or [U". 29. Let T, or UF=2',(X,+X,++--+X,+X,), X being defined by Xn = Wut af ay a wet a a8 we vy ahacta’”’ &e., where a, 8, y, & and « are positive integers or (some of them) zero ; and m is the number of values of wv 2? aY a8 af. Then if we assume =X, = (a +a” +a” + &e.) Sat af vy x we, we shall have =r" or SU27= 3X, + osx, 7 Sri = ~SX,+ 3X, For, let > represent the sum of the 24 expressions for 7” or U", formed by applying all the cycles* (Art. 17) to any one of the values of r”” or U”. Then, since >’. X,, consists of 5, and consequently > >’,.X,, of 5-24, or 120, expres- sions of the form X,,, and since m= 120, or a sub-multiple of 120, it follows that Yr" or SU"= w0 eX. SX, + Oe “2 BX ,+ Ee But since 7” or U” is six-valued, we have Sr AS, and SU"=45U". Hence the theorem. 30. Applying this : — Since (Art. 28) =>’ (walt Qatan.2,), Pa — i a a+ ae Sat By Ls, = 32a} 73+ 2Z22 2, 4,=0. * In Art. 17 only 12 cycles are exhibited, but the remaining 12 are giyen by reading the symbols in a reverse order, VOL, XV. DD 196 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND Or, working with U, in place of 7’, UHL a7 (ty 5 +23 2s) 5 fe 6S SS (L421) Sat aap 2 at ay i), which confirms the value of =7?._ Consequeutly, Dj=52r=5>0=0: 31. Again: — By the commutative property established in Art. 19, T=! (aj ay + 2a} a X5)7; and developing 7 according to descending powers of 2, we have T= LF 02 + 27} ay Hs +22 02+ 2H, £3 y+ 2a, 2, 22+23 22 Fatty 3 Vy + U5 UE + 2lks a Ws + HG US 5 whence, multiplying by vj 23+ 2¢} 2, 2;, reducing by means of such relations as (Art. 19) Dv} Hg = DU} ty 3, BUX, UX, W{= DU} Wy L, X;, Se. and collecting similar terms, we find TA fa a+ Aa WL; + Uj Ly U5) + Oa ay v5 + BA} X3 as Ws + 4a} a9 a3 U5 + a @3 Ly U5) + Qay w3 3 wy + Bay Ly H3 &y He + ay U2 %5 Ui 25% 5 or, since Sw R= riage 7, end Dy! (Way W3 Wy Lz + Uj Hy Ly i Ls) = ZL, Vy XZA ys, r= 5 Sat af + 4(v} v3 v5 + x} vy v3) + Cat x5 03 + Bait x x, 5 + 4x} ay a3 15+ aX] a3 Wy V5) + Ay Was ay Ws$ + WBA} Wy Us WEA ADA ay 5 Ws Ws 5 o. Jt=85 24 8445 7 vi 7, +632} 23 224 8d ai avs, HAS a} Uy 03 y+ 12D ay wy V5 H+ BOR Xi Hwy 4 25. Or, working with U, W=3) (sie, 4-47 2 7), developing U, multiplying, reducing, &c., there results USB § (wh 08 a2 + af a8 03) + Qa ay my yg + (ah ah ya + ata ay a) +2 (ah ahaha, + wh ad aia + ah ay a2 a8 + 23 ay 0} 02) +2 (xt #3 03 a, y+ Ui Le Hs UEH5)t = D'S (wt av} 2 + vf x a}) +2 (ah 29 ary 5+ vf a9 as 5) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 197 + 2a] XZ x3 Ly + Uy Vy UE Ls + a} My Wy; +L} Vy Ly Xs) f + Qa ay Ly Ly Ly + 2DL} L313 Xs; . DUP = 2d 24 23 22 +1 2d 24 x, 20,73; + 4273 237,52, +4303 22 22 9,4 1222? 22 220,23. 32. By Newton’s theorem, aD, +D, 120+ +++ +D 3074+ 30= whence, making n=2, 3, 4 and 5 successively, and re- ducing by means of (Art. 30) D,=0;and,.50=0; we — =-135@, D,x=-436, Dy=-3(D.374+ 56) =} eps >), and D,=-43(D,27 + D363 + 36) =D,D, - 136°. 33. Applying the same theorem ft by Sire (2), Art. 28, we find i S37 =0, 2a°=5-5Q, F2*=0; 37° =-5E, F2°=3 -5°Q?,. Ja’=—0, Ja2=—8-5QH, Fa?=3-5°QS Sa — = Sat'=—11. Q7H,. Ja? =3-5'Q', Jv*®=13-5QH, Fr*=- 14.55, Fr"=3.5°Q'-—5ES, S28 =24.5°OE, Ja"=—17-54Q‘H, Saz¥®=38.5°Q8 —18-5QE’, Sv"=38-5°Q°E*?, and Ja%=—4.59Q°E+5E*, 34. We know that, when (a) and (8) are unequal, Su w= Sar Zax* — Sarre, and that Sot at = 44 (Sate Say, Or, dropping the subject x, and the symbol 3, we may, after Hirsch, write these relations thus: =[a][8]-[a+A], and (aa)=4§[a]*- [2a]}. Extending the method commonly employed to establish these relations we find the following : (ay) =[a](8) fv) - [a+4) fy] -[a +9] 06] - [a] [8 +71 +2[a+B8+y]; (a88) =} [a] [A}-2 [a+ 8] [8] - [a] [28] +2[a+ 26}}, (aaa) = 4} [a]* — 8[2a] [a] +2[8a]}?, (ay) = [a] (A) [7] (8) - [a] [8] fy +8] - [a] (6 +9] [8] 198 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND ~ [a] [6 +8] [y] + 2[a] [8-+7 +8] - [a+ 4 fv [8] ~ [a+ y] [6118] - [a+ 8][A][o] + [a+ A) fy +8] +2[a+7+8][6]+2[at+h+y] [6] + [a+9][B+y] +2[a+B+8][y]+ [a+y]}[8+8]-6la+8+y+6], (aBor”) = 44 [a] (8) Ly]? [2] [4] [27] - 2[a] 8+] fy] +2[a][8+2y] - [+A] [y]’-2[¢+ 7] [4] [7] + [a+] [2] +2[a+ 27] [6] +4[a+ 8+] [7 +2[a+y][B+y] -6[¢+8+2y]§, (aap) = 34 [a}*[A} - [a]*(28} - 4] [a +8] [6] +4[a][a+26] - [2a] [8]*+ [2a] [26] yelled [8B] +2[a+]*-6[2a+2P}}, (a888) =¥ [a] [8]* - 3[a] [8] [28] +2[a] (38) ~8[a+A][6]'+8[a +8] [28] + 6[a +28) [6] ~ ee t, and (aaaa) =3;} [a]*— 6[a]”[2a] + 8[a] [8a] + 3[ 2a}? — 6[4a]}. Hirsch denotes repeating exponents thus: Sze at by (a*), Sarat xh xf xf by (a*6"), &e but I have thought it conducive to clearness to express them as above. It will be observed that the crotchet [ | is employed to represent the sum of powers, and the parenthesis () that of other functions of the roots. By the aid of the formule in this article and the values of [1], [2], &c.... [20], given in the last, we may easily calculate the value of any symmetric function of x in terms of Q and E. 35. Thus taking the functions that occur in the value of oH va 31), we have * 4) = 44 [4]*-[8]}=430°-(-8-5QE){=4-5QE, =[s01)= [413101 - (7101 - ‘ [3] - [4° +2[8] —5QE, 5 (422) =41[4] (2]°-2 [6] [2] - [412+ 218)}=-8-5@, ¥(8311) =44 (3])°[1]?- [3)°[2] -4[8] (4) (1) +4(3] 5] * To prevent confusion, the symbol 3 is prefixed to the parenthesis (). THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 199 ~ [6]{1]° + [6] [2] +4(7] 1) +2[4]°-6(8]3 =-—3-5QE, S (8221) =4§ [3] (11[21*- (3) [1] [4] -2(8]°[2] +2(8) [5] ~ [4] [2]°-215] (1) (2) + [4° +2710) +46) (2) 42[5] [3] -6[8]}=-6-5QE, 5 (2222) =z f [2]*-6[2]°[4] +8[2] [6] + 3[4]*-6[8]} —2.5Q5, ¥ (22211) =-ES(111)=-F {1 }*-3[2] 21) +2(3]} =—5QE; - Sr#=8¥ (44) + 43431) + 63 (422)) + 83 (3311) 443 (3221) +123 (2222) + 365(22211) =— 4.5°QE. Also, (41111) =- E[3]=-3-5QH, and +, SU%= 23 (422) + 125 (41111) +45(3311) + 45(3221) 4123 (22211) =— 4.5°QE, which coincides, as it ought to do, with the value of 3'r*. 36. Now (Art. 32), D,=-}30=- 3 Srt=— 3 SU, and substituting for ¥7* or {U’, its value found in the last article, we obtain D,=2-5°QE. 37. The expression for r* (Art. 31) may be conveniently written thus : 7*= ’ [(44000) + 4§ (48001) + (41008) } + 6(42002) + 8(88101) + 44 (82102) + (32012) } + 4(22211)] + 23'(2222) + 4.3 (22211) ; or, arranging according to the strict descent of powers, and substituting for the symmetric functions, %(2222) and 5(22211), their values given in Art. 35, += &’§ (44000) + 4(43001) + 6(42002) + 4(41003) + 8(33101) + 4(32102) + 4(32012) + 4(22211) § Moreover (Art. 31), 200 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND = (22000) + 2(21001) + (20002) + 2(12100) + 2(10012) + (02200) + 2(01210) + (00220) + 2(00121) + (00022). Consequently 7°= 5 § (44000) + 4(43001) + 6(4:2002) + 4(41003) + 8(33101) + 4(32102) + 4(382012) + 4(22211)} + { (22000) + 2(21001) + (20002) + 2(12100) +2(10012) + (02200) + 2(00121) + (00022) }. Developing, reducing,* &c., we are conducted to 7°= >’ [ (66000) + 6§ (65001) + (61005)? + 15§ (64002) + (62004) t + 20(63003) + 24(55101) + $36(54102) +6(54012) + 6(52104) + 36(52014) } + 244 (53103) + (580138) } + 4 (44220) + 28(44202) + (44022) t + $20(44211) + 2(44121) + 20(44112)} + 84 (43203) + (43028) } + 40(431138) + $4(43221) + 24(43212) +12 (43122) + 12(42218) + 24(42123) + 4(41223) ¢ + 6(42222) + 16§ (33321) + (83312) } + 452(33222) + 8(382822)%]. Applying the theorem in Art. 29 we have ¥7°=383'(66) + 63'(651) + 153'(642) + 203'(633) + 24:3/(5511) + 213'(5421) + 245'(5331) + 803) (4422) + 423'(44.211) + 83'(4382) + 403 (43311) + 403 (48221) + 863'(42222) . + 963'(33321) + 303 (383222). By the aid of the formule in Art. 34, and the values of [1], [2], &c. in Art. 33, we find > (66)=8-5'Q4, 3(651)=— 3-5'Q*, B(642) =-3-5*Q' (633) =3-5'Q*, (5511) =0, 3(5421)=0, 5(5331)=0, > (4422) =0, 3(44211)=0, 3(4332)=0, =(43311)=0, * It may be convenient here to illustrate the principle of reduction pro- eeeded upon in this Paper in dealing with circular functions. According to the notation of the text, represent 3’eta® 2¥ > a= by 3/(aByde); and suppose (e.g.) that, of the five exponents, + is the greatest ; then =’ (aByée) must be replaced by its equivalent 3'(7dea8). Or, suppose that the greatest exponent (7) is repeated, and the function takes the form 3’(ayyde); this must be replaced by its equivalent =/(yydea). Andsoon. This mode of reduction is uniformly followed in the text. The greatest exponent leads. THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 201 3 (43221) =0, 3(42222)=0, 3(333821)=0, and > (33222) =0. Hence FP —=645'Q"; 38. Again (Art. 31):— U?= 2’ [4 (42002) + (40220)? + 2§ (83101) + (80311)? + 24 (82210) + (32021) + (81202) + (80122) ?}] +23(41111) +23(22211) ; or since (Art. 35) =(41111) =—3-5QE, and 3(22211)=-—5QE, U?=%’5 (42002) + (40220) + 2(33101) + 2(32210) + 2(82021) + 2(81202) + 2(30311) + 2(80122) —8-5QE. Next, U=’$ (21001) + (20110) } = (21001) + (20110) + (12100) + (11020) + (10201) + (10012) + (02011) + (01210) + (01102) + (00121)?. Nov, Us=>’§ (42002) + (40220) + 2(83101) + 2(82210) + 2(32021) + 2(81202) + 2(380311) + 2(80122)?U —8-5QEHU. Hence if, in the function affected by 5’, we substitute for U its value, develope, reduce, &c., as in the last article, we a obtain =’ [§ (63003) + (60330) } + § (62112) + (61221)? +34 (54102) + (52014) + (51240) + (50421)? + 8§ (53022) + (52320) + 52208) + (50232) } +4 (538211) + (52131) + (613812) + (51123)? +34 (44310) + (44013) + (41430) + (41403) } + 6§ (44121) + (42411)? + 64 (43302) + (43230) + (42033) + (40323) } + 4§ (43113) + (41331) } + § (43212) + 2(43122) + 2 (42821) + (42231) + 2(42213) + (42123) + (41322) + 2(41232) ? + 8(42222) + 45 (33321) + (83312) + (33231) + (83132) +24 (83222) + (32422)?] -8-5QEU ; 202 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND *, SU? =23(633) + 25 (62211) +33(5421) +62(5322) +83(53211) + 63(4431) + 123(44211) +123(4332) + 83(43311) + 63(43221) +483 (42222) + 24:3 (33321) + 123(83222) -—8-5QEXU. But 5(62211)=0, 3(5322) =0, 3(53211)=0, =(4431) =0, and SU=0; hence making these substitutions, and also for the other symmetric functions whose values are given in the last article, we obtain S'=6.500%, which confirms the value of =r°. 39. By Art. 82, D =-137°= Osa 3 sys . 3 3 3 3 ? and substituting for 27° or >U? its value (Art. 37 and 38), we have Dee. 40. Since (42222) = 2(42222) = E’[2] =0, and &’§ (83222) + (82822)} = 3(33222) = EH’ 3(11) = 0, the value of 7* (Art. 37) may be expressed thus : 7°= >’ [ (66000) + 6§ (65001) + (61005) } + 15464002) + (62004) } + 20(63003) + 24(55101) + $36(54102) + 6(54012) + 6(52104) + 36(52014)} + 244 (53103) + (58018)? + § (44220) + 28 (44202) + (44022) ¢ + 420(44211) + 2(44121) + 20(44112? + 8 (43203) + (43028) } + 40(43113) + $4(43221) + 24(43212) +12(43122) + 12(42213) + 24(42123) + 4(4] 223)? + 16§ (33321) + (83312) } + 44(33222)]. We might now arrange these functions according to de- sceuding powers, multiply by 7°, and proceed precisely as in Art. 37, but it will be more convenient to work with U. 41. Since 3 (42222)=0, 3”} (88321) + (83312) + (33231) + (83182)t = 3'(83321)=0, and 3'{(33222) + (82322)t = (33222) =0, we have (Art. 38) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 203 Us= 3" § (63003) + (62112) + (61221) + (60330) + 3(54102) + 4(53211) + 3(53022) +-3(52320) + 8(52203) +4(52131) + 3(52014) + 4(51312) + 3(51240) + 4(51123) +3(50421) + 8(50232) + 3(44310) + 6(44121) + 3(44013) + 6(43302) + 6(48230) + (43212) + 2(43122) + 4(43113) + 6(42411) + 2(42321) + (42231) + 2(42215) + (42125) + 6(42033) + 3(41430) + 3(41403) + 4(41331) + (41322) + 2(41232) + 6(40323)} —8-5QEU. Multiplying by U, and proceeding as in Art. 38, we find, after making the necessary reductions, &c., Ut=5" [4 (84004) + (80440) } + 24 (83113) + (81851) t + 2(82222) + 4§ (75103) + (73015) + (71350) + (70531)? + 44 (74023) + (73204) + (72480) + (70342)? + 84 (74212) + (72241) + (72124) + (71422)? + 8§ (73321) + (73132) + (72313) + (71233)t + 6§ (66202) + (60622) } + 43 (65014) + (64105) + (61540) + (60451)} + 103 (65311) + (63151) + (61513) + (61135)% + 12} (65122) + (62521) + (62215) + (61252) } + 6{ (64420) + (64042) + (62404) + (60244) } + 104 (64114) + (61441)} + 12§ (64303) + (63340) + (63034) + (60433) } + §20(64231) + 2(64213) + 6(64123) + 20(63412) + 6(63214) + 2(63124) + 6(62431) + 2(62341) + 20(62143) + 2(61452) + 6(61542) + 20(61324)} + {4(63322) + 4(63232) + 18(63223) + 18(62332) + 4(62323) + 4(62283)% + 124 (55402) + (55204) + (50542) + (50524)? + 124 (55141) » + (54511)} +124 (55330) + (55033) + (53530) + (53503)? + 8§ (55312) + (55213) + (51582) + (51523)} + 124 (55222) + (52522)} + $16(54421) + 6(54214) + 16(54142) + 6(54124) + 6(52441) + 16(52414) + 6(51442) + 16(51244)? + $6(54331) + 12(54313) + 16(54133) + 16(53431) + 6(53413) + 12(58341) + 16(53314) + 6(53143) + 12(53134) VOL, XV. EE 204 REY. R. HARLEY ON SYMMETRIC PRODUCTS, AND + 12(51433) + 16(51343) + 6(51334)} + $16(54322) + 4(54252) + 4(54223) + 4(53422) + 16(53242) + 4(53224) + 4(52432) + 16(52423) + 4(52342) + 4(52324) + 4(52243) + 16(52234)} + 6§ (53332) + (53323) + (53233) + (52335) } + 24(44440) +44 (44431) + (44413) + (44341) + (44148)? + 12§ (44422) + (44242)? + $10(44332) + 8(44323) + 10(442383) + 10(43432) + 10(43423) + 8(42433) + 4(43333)] —8-5QEU*. Applying the theorem in Art. 29 we have, LUt=23'(844) + 4.3)(83311) + 125'(82222) + 43)(7531) + 43'(7432) + 163 (74221) + 16373321) + 123)(6622) + 4.5'(6541) + 203'(65311) +2435 (65221) + 123'(6442) + 203'(64411) + 243'(6433) + 283'(64321) + 523'(63322) + 24.3; (5542) + 245/(55411) + 483'(5533) + 163 (55321) + 723)(55222) + 445'(54421) + 683'(54331) + 483 (54322) + 363 (53332) + 144.3'(4444) + 245'(44431) + 723) (44.422) + 563 (44332) + 243'(43333) — 8-5QES,U”. But 2 (844) =- 8-5°Q?E?, 3(83311)=11-5°Q?E?, 3(82222) =3-5°Q'E’, 3(7531)=3-5°Q?E?, 3 (7432) =4-5°Q7E’, 3(74221)=6-5°Q?E*, 3(73321)=6-5°Q?E*, 3 (6622) =11-5°Q?E?, 3 (6541) =-9-5°Q?E®, 3'(65311)=7- POE’, 2 (65221) =-3-5°Q@?*E®, 3 (6442) =—- 10-5°Q’E?, 364411) =2-5°Q°E’, 3(6433)=6-5°Q@’ EH, 3(64321) =6-5°Q°E* 3 (63322) =3-5°Q2E?, 3 (5542) =8-5°Q?E’, 355411) =-5°Q?E’, 3 (5533) =0, 3(55321)=-6-5°Q?E’, 3 (55222) =3-5°Q?H, 3(54421)=0, 3(54331)=6-5°Q?E? (54322) =-3-5°Q?H?, 3'(53332)=0, 3'(4444) =2-5°Q?E?, 3 (44431) =— 2-5°QPR*, 3/(44422)=5°Q2E*, 3/(44332)=0, (43333) =0, and (Art. 35) 2U?=- 4.5°QE. Consequently, DUt= 4. 5°Q?E? ; THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 205 and, therefore, (Art. 32) D,=}(2D}- 364) =4 (2D}- 5'3U") =} (2D3— 4. 5°@EY ; or, since (Art. 36) D,=2-5°QH, D,=5"Q2E?. 42. Since 5’ (82222) = 3'(82222)=3-5°Q2EY, 3” (63322) 4+ (63232) + (63223) + (62332) -+ (62323) + (62233)? = 5 (63322) =3-5°Q?E, 3’ § (55222) + (52522)? = 3 (55222) =3-5°Q?E?, 3’§(54331) + (54313) + (54133) + (53481) + (53413) + (53341) + (53314) + (53143) + (53134) + (51433) + (51343) + (51334)? = 354331) =6.5°Q2E?, S §{ (54322) + (54232) + (54223) + (53422) + (53242) 4 (53224) + (52432) + (52423) + (52342) + (52324) 4 (52243) + (52234)? = 3 (54322) —- 3. 5°Q2EX, 3§ (53332) + (53328) + (53233) + (52333)}—=3(53332)—0, 3’ (44440) = 3 (4444) =2-5°Q2EY, 3’ (44431) + (44413) + (44341) + (44148) = 3'(44431) =— 2. 5°Q@2E2, 35 (44422) + (44242)} = 5 (44422) = 5°22, S’f (44332) + (44323) 4+ (44233) + (43432) + (43423) + (42433)? = (44332) =0, and 3’ (43333) = 3(43333)=0, we have by substitution in the value of U* (Art. 41), Ut= > [f (84004) + (80440) } + 2§ (83113) + (81331)? +44 (75103) + (73015) + (71350) + (70531)? +4§ (74023) + (73204) + (72430) + (70342)? 4 8§ (74212) + (72241) + (72124) + (71422)? +8{ (73321) + (73182) + (72313) + (71233)? + 6§ (66202) + (60622) +44 (65014) + (64105) + (61540) + (60451)? + 10§ (65311) + (63151) + (61513) + 2 senha 65122) + (62521) + (62215) + (61252)? + 6§ (64420) + (64042) + (62404) + (60244)? + 10§ (64114) + (61441)? + 12§ (64303) + (63340) + (63034) + (60433)? + {20(64231) + 2(64213) + 6(64123) + 20(63412) + 6(63214) + 2(63124) + 6(62431) + 2(62341) + 20(62143) + 2(61432) + 6(61342) + 20(61324)? 206 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND +14 (63223) + (62832) } +124 (55402) + (55204) + (50542) + (50524)? + 12§ (55141) + (54511)? +125 (55330) + (55038) + (53530) + (53508)? + 8§ (55312) + (55213) + (51532) + (51528)? 4 §16(54421) + 6(54214) + 16(54142) + 6(541 24) + 6(52441) + 16(52414) + 6(51442) + 16(51244) ? + $6(54313) + 10(54133) + 10(53431) + 6(53341) +10(58314) + 6(53134) + 6(51433) + 10(51843)? 4.124 (54322) + (58242) + (52423) + (52234)? + 2§ (44332) + (44233) + (43432) + (43423) ¢] +26-5°Q’?E?- 8-5QEU?; or, arranging the terms affected by 3” according to the See of powers, )+ 6(64420) + 12(64303) + 20(64231) + 2(64218) 4123) + 10(64114) + 4(64105) + 6(64042) + 20(63412) + 12(63340) + 14(63223) + 6(63214) +10(63151) + 2(63124) + 12(63034) + 12(62521) + 6(62431) + 6(62404) + 2(62341) + 14(62332) 4 12(62215) + 20(62143) + 4(61540) + 10(61518) +10(61441) + 2(61432) + 6(61342) + 20(61324) 4 12(61252) + 10(61135) + 6(60622) + 4(60451) 4 12(60433) + 6(60244) + 12(55402) + 12(55330) + 8(55312) +8(55213) + 12(55204) + 12(55141) + 12(55033) + 12(54511) + 16(54421) + 12(54322) + 6(54313) + 6(54214) + 16(54142) + 10(54133) + 6(54124) + 12(53530) + 12(53503) + 10(53431) + 6(53341) + 10(53314) + 12(53242) + 6(53134) 4 6(52441) + 12(52423) + 16(5241 4) + 12(52234) + 8(51532) + 8(51523) + 6(51442) + 6(51433) — 5’§ (84004) + 2(83113) + 2(81331) + (80440) + 4(75103) + 8(74212) + 4(74023) + 8(73321) + 4(73204) + 8(73182) + 4(73015) + 4(72430) + 8(72318) + 8(72241) + 8(72124) + 8(71422) +4(713 + 8(71233) + 4(70581) + 4(70342) + 6(66202) + 10(65311) + 12(65122) + 4(65014) + 6( ) + 6(64123) ee ee THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 207 + 10(51343) + 16(51244) + 12(50542) + 12(50524) + 2(44332) + 2(44233) + 2(43432) + 2 (43423)? + 26.5°Q?E? — 8-5QEU?. Multiplying by U, and proceeding as before, we obtain U®= [§ (105005) + (100550)? +3§ (104114) + (101441)? + 2§ (103223) + (102332)? + 5§ (96104) + (94016) + (91460) + (90641)? +5§ (95024) + (94205) + (92540) + (90452)? + 144 (95213) + (93125) (92351) + (91532)? +144 (94133) + (93431) (93314) + (91343) + 16§ (94322) + (93242) (92423) + (92234)? + 10$ (87203) + (83027) (82370) + (80732)? +5§ (86015) + (85106) + (81650) + (80561)? + §24(8631 2) + 20(86123) + 24(83261) + 20(83216) + 20(82631) + 24(82136) + 24(81623) + 20(81362)? + 18§ (85115) + (81551)? + §20(85304) + 10(85043) + 10(84530) + 20(84035) + 20(83450) + 10(83405) + 20(80543) + 10(80354)? + §24(85421) + 3(85214) + 12(85124) + 12(84215) 4 24(84152) + 3(84125) + 12(82541) + 24(82514) + 3(82451) + 3(81542) + 12(81452) + 24(81245)} + 48§ (85232) + (83522) + (82325) + (82253)? + 48§ (84413) + (84341) + (83144) + (81434)? + 48 (84224) + (82442)? + §4(84323) + 6(84233) + 6(83432) + 4(83342) + 6(83324) + 4(83234) + 4(82433) + 6(82343)? + 60(83333) +185 (77411) + (77114) + (71741) + (71714)} + 30§ (77222) + (72722)? + §10(76520) + 20(76205) + 10(75062) + 20(75026) + 10(72605) + 20(72560) + 20(70652) + 10(70256)? + $30(76403) + 20(76034) + 30(74360) + 20(74306) + 20(73640) + 30(73046) + 30(70634) + 20(70463)? + {24(76214) + 30(76142) + 30/74621) + 24(74126) + 24(72461) + 30(72416) + 24(71642) + 30(71264)? + §54(76331) + 14(76313) + 54(73613) ( ) ( )¥ + + + = (ae s + 14(73361) + 54(73163) + 14(73136) + 14(71633) + 54(71336)? + 244 (76223) + (73226) + (72632) 208 REV. R. HARLEY ON SYMMETRIC FRODUCTS, AND + (72362)? + §£54(75512) + 54(75251) + 14(75215) + 14(75125) + 14(72551) + 54(72155) + 14(71552) + 54(71525)? + 30§(75440) + (74504) + (74054) + (70445)} + £32(75314) + 42(75143) + 44(75134) + 42(74531) + 44(74315) + 32(74135) + 44(73541) + 32(73451) + 42(73415) + 32(71543) + 44(71453) + 42(71354)? + {32(75422) + 6(75224) + 32(74252) + 6(74225) + 6(72542) + 32(72524) + 6(72452) + 32(72245) + {8(75332) + 68(75323) + 6(75233) + 6(73532) + 8(73523) + 68(73352) + 6(73325) + 8(73253) + 68(73235) + 68(72533) + 6(72353) + 8(72335)? + 6§ (74441) + (74414) + (74144) + (71444)? + {82(74432) + 14(74423) + 14(74342) + 20(74324) + 82(74243) + 20(74234) + 20(73442) + 82(73424) + 14(73244) + 20(72443) + 14(72434) + 82(72344)? + 8{(74333) + (73433) + (73343) + (73334)? + 20§ (66611) + (66161)? + 30§ (66350) + (66053) + (65630) + (65603)? + 48{ (66521) + (66125) + (62651) + (62615)? + § 12(66431) +34(66413) + 34(66314) + 12(66134) + 12(63641) + 12(63614) + 34(61643) + 34(61 634)? + {48(66422) + 72(66242) + 48(66224) + 72(64622) + 48(62642) + 48(62624)? + {30(66332) + 6(66323) + 30(66233) + 30(63632) + 30(63623) + 6(62633)} + 36§ (65405) + (65045) + (64550) + (60554)? + {40(65581) + 86(65518) + 86(65351) + 6(65315) + 40(65153) + 6(65135) + 6(63551) + 40(63515) + 86(63155) + 6(61553) + 36(61535) + 40(61355)? + $12(65522) + 12(65252) + 40(65225) + 40(62552) + 12(62525) + 12(62255)? + $52(65441) + 80(65414) + 12(65144) + 12(64541) + 52(64514) + 30(64451) +12(64415) +52(64154) + 30(64145) + 30(61544) + 12(61454) + 52(61445)? + §10(65482) + 28(65423) + 48(65342) + 18(65324) + 12(65243) + 56(65234) +12(64532) + 48(64523) + 28(64852) + 56(64325) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 209 + 10(64253) + 18(64235) + 56(63542) + 10(63524,) + 18(63452) + 12(63425) + 48(63254) + 28(63245) + 18(6254.3) + 28(62534) + 56(62453) + 48(62435) + 12(62354) + 10(62345)? + 465 (65333) + (68533) + (63353) + (63335)} + 20§ (64442) + (64424) + (64244) + (62444) } + {32(64433) + 32(64343) + 48 (64334) + 48(63443) + 32(63434) + 32(63344)? +184 (55541) + (55514) + (55451) + (65154)? +58} (55532) + (55523) + (55352) + (55253)? + $26(55442) + 40(55424) + 26(55244) + 26(54542) + 26(54.524) + 40(52544)? + §14(55433) + 24(55343) + 14(55334) + 24(54533) + 14(54543) + 14(53534)? +24} (54443) + (54434) + (54344) + (538444)? ] +26-5°Q°E?U — 8-5QEU*. The bar placed over the number 10 indicates that it is a single exponent, so that }’(105005) represents 3’2 v3 z°, 3’(104114) represents 3'2 wtx,a,a!, &c. Whence by the theorem in Art. 29, YU=23(1055) +65(104411) + 4.3 (103322) + 53 (9641) +52(9542) + 145(95321) + 282(94331) +822 (94322) + 102(8782) +52(8651) + 445 (86321) + 362 (85511) + 802(8543) +392 (85421) + 962(85322) + 96(84431) + 962 (84422) + 202(84332) + 8605 (83333) + 72=(77411) + 1803(77222) + 802(7652) + 502(7643) + 545(76421) + 1362(7633 1) + 482 (76322) + 1862(75521) + 605(7544) +1182(75431) + 76575422) + 1643(75332) + 862(74441) + 2325(74432) + 485 (74333) +1202(66611) + 605(6653) + 965(66521) +922(66431) + 83365 (66422) + 1325(66332) +722(6554) + 1645(65531) + 1285(65522) +1882 (65441) + 1725(65482) + 2763(65333) + 1205(64442) + 2243 (64433) + 1082(55541) + 3482 (55532) + 1842(55442) + 1043 (55433) 210 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND + 1443 (54443) + 26-5°@*E*SU — 8-5QEZU*. But (1055) =-5°Q°E+6-5E*, £(104411)=-3-5°Q°E +4 5E', 5 (103322)=5E', 3(9641)=4.5E4, ¥(9524)=-6-5°Q°E +4.5E', 5(95821)=—5E*, £(94831)=2-5E*, (94322) =2.5E!, 3(8782)=-6-5°Q°E+4.5E!, (8651) =4-5Et, 3 (86321) =4.5E', ¥(85511)=-3-5°QE+5E!, 5(8543) =4.5E, 3(85421)=-5E4, 3(85322)=2-5E, 2(84431) =2.5E4, 3(84422)—5E!, 3(84832)=-3-5E', (83333) =5Et, 3(77411)=-3-55Q°H+5E', 3(77222)=2-5Et, 3(7652)=-6-5°Q°E+4.-5E, 3(7643)=-5E!, 3(76421) =4.5E!, 3(76331)=2-5E!, 3(76322)=-3-5E*, 3(75521) =2.5E, 3(7544)=2-5E', 3(75431)=-5Et, 3(75422) =-3.5E!, 3(75332)=2-5E!, 3(74441)=-5E, 3(74432) =2.5E', 3(74333)=- 5B, =(66611)=-5°Q°E+2-5Ei, 3(6653)=2-5E*, 3(66521)=-3-5E', 3(66431)=—3-5E!, 3(66422)=5E*, 3(66332)—5E!, 3(6554)=-3-5Et, 3(65531)=2-5E!, 3(65522)=5Et, 3(65441)=2-5Hs, 3(65432)=-— 5E*, 3(65333)=-5Et, 2(64442)=— 5 Bt, 3(64433)=5Et, 3(55541)=-5E, 3(55532)=- 5H, 3(55442)=5E!, 3(55438)=5E', 3(54443)=- 5 Et, SU=0, and >U'=6-5'Q*. Hence, by substitution, we have SUF=— 158-5°Q5E+55E'; .. TAF=—5*2U! =158.5"Q°E — 5"Et, Now (Art. 32), DSH D tae: and (Art. 36 and 37) D,=2-5°QE, and D,=2-57Q!; consequently D,= —58-5°Q°E + 5"E!. 43. Again: =/(83333)=5(83333)=5E‘, 5'{(77222) + (72722)} =5(77222)=2-5E', &/§(75382) + (753823) + (75238) + (73532) + (73523) + (73352) + (73325) + (78258) + (73235) + (72533) + (72358) + (72335) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 211 ¥(75332)=2-5E*, &'$(74441) + (74414) + (74144) + 7444 t an 74441) =—! SE‘, 3 { (74432) + (74423) + (74342) + (74324) + (74243) + (74234) + (73442) + (73424) + (73244) + (72443) + (72434) + (72344) t =(74432)=2-5E', 2'4(74333) + (73433) + (73343) + (73334)? =¥(74833)=— 5E‘, >'{(66611) + (66161) (66611) =-5°Q°H+2-5E*, 3° (66422) + (66242). + (66224) + (64622) + (62642) + (62624)} = 2 (66422) =5E*, &$66332) + (66323) + (66233) + (63632) + (63623) + (62633)? = 3(66332)=5E*, 3'§(65531) + (65513) + (65351) + (65315) + (65153) + (65135) + (63551) + (63515) + (63155) + (61553) + (61535) + (61355) t =>(65531)=2-5E', 3” § (65522) + (65252) + (65225) + (62552) + (62525) + (62255)? = 3 (65522) = 5B, 2’ $ (65441) + (65414) + (65144) + (64541) + (64514) + (64451) + (64415) + (64154) + (64145) + (61544) + (61454) + (61445)? =>(65441)=2-.5E*, 34 (65432) + (65423) + (65342) + (65324) + (65243) + (65234) + (64532) + (64523) + (64352) + (64325) + (64253) + (64235) + (63542) + (63524) + (63452) + (63425) + (63254) + (63245) + (62543) + (62534) + (624538) + (62435) + (62354) + (62345)? = >(65432) =— 5Et, '§ (65333) + (63533) + (63353) + (63335) } = = (65333) =- 5, 35 (64442) + (64424) + (64244) + (62444)? = >(64442)=-—5E*, 5’§ (64433) + (64343) + (64334) + (63443) + (63434) + (63344)? = >(64433)=5E*, '$(55541) + (55514) + (55451) + (55154)? = >(55541) =-5E*, 37$(55532) + (55523) + (55352) + (55253)? =>(55532)=- 5E*, 3'$ (554.42) + (55424) + (55244) + (54542) + (54524) + (52544)} = 2 (55442) =5 Et, 3 § (5543) + (55343) + (55334) + (54533) + (53543) + (53534)} =>(55433)=5E*, and 3’ § (54443) + (54434) + (54344) + (53444) =>(54443)=-5E*. Hence, by sub- stitution in the value of U® given in the last article, we have VOL. XV. ry 212 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND U®='[§ (105005) + (100550)? + 3§ (104114) + (101441)? +2§ (103223) + (102332)? + 5{ (96104) + (94016) + (91460) + (90641) } +.5§ (95024) + (94205) (90452)? + 144 (95213) + (93125) (91582)? + 14§ (94133) + (93431) (91343)? + 16 (94322) + (93242) ( ) ee ) (80732) Re) ri) oo ot ese vo} ci) nN ci) oo 92234)? + 10§ (87203) + (83027 2 + 5$(86015) + (85106) (80561)? + {24(86312) + 20(86123) + 20(83216) + 20(82631) + 24(82136) 1 + 20(81362)? + 18 (85115) + (81551)? $20(85304) + 10(85048) + 10(84530) + 20(84035 ) ) ++ttt4t4 Bo [0] ey) ci) oo e ) 0(83450) + 10(83405) + 20(80543) + 10(80354) ? 24(85421) +8(85214) + 12(85124) + 12(84215 +24 (84152) + 3(84125) + 12($2541) + 24(82514) + 3(8245 1) +3(81542) + 12(81452) + 24(81245)? + 48§ (85232) + (83522) + (82325) + (82253) +48 § (84413) + (84341) + (83144) + (81434)} +48} (84224) + (82442) } + §4(84323) + 6(84233) + 6(83432) + 4(88342) + 6(83324) + 4(83234) + 4(82433) + 6(82343)? + 18§ (77411) + (77114) + (71741) + (71714) + {10(76520) + 20(76205) +10(75062) + 20(75026) + 10(72605) + 20(72560) + 20(70652) + 10(70256) + §30(76403) + 20(76034) + 80(74360) + 20(74806) + 20(73040) + 80(73046) + 30(70634) + 20(70463)? + §24(76214) + 30(76142) ) ) 13) ) He de she toe Ge dee 2s ae ee) ~ ise) “tI See ~a 0 ~™ ( ( + 80(74621) + 24(74126) + 24(72461) + 30(72416) + 24( ees +30(71264)? + {54(76331) + 14(76313) ( ( 4 54(73618) + 14(73361) +54(73163) + 14(73136) 414(7 as +54(71336)? + 24§ (76223) + (73226) + (72632) + (72362)? + {54(75512) + 54(75251) + 14(75215) + 14(75125) + 14(72551) +54(72155) nate ln 71525)? + 80{ (75440) + (74504) + (74054) + (70445)? + £82(75314) +.42(75143) ye oleae 4531) +44(74315) + 32(74135) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 215 dies. +32(73451) +.42(73415) + 32(71 44(71453) +.42(71354)? + $82(75422) + 6(7 ai 4252) + 6(74225) + 6(72542) + 32(7252 + 6(72452) + 82(72245)? + $2(75332) + 62(75 ) 294) ) 32 + 2(73523) + 62 (73352) + 2 (73253) + 62(73235) 24. ) 3 54. 5 4 3) + 62 (72533) + 2(72335)?} + {68(74432) + 6(74324 + 68(74243) + 6(74234) + 6(78442) + 68(73424 + 6(72443) + 68(72344) } + 3804 (66350) + (66053) + (65680) + (65603) } + 484 (66521) + (66125) + (62651) + (62615) } + $12(66431) + 34(66413) + 34(66314) + 12(66134) + 12(63641) + 12(63614) + 34(61643) + 34(61634) } + 244 (66242) + (64622) t +24 (66382) + (66233) + (63632) + (63623)? +36} (65405) +- (65045) + (64550) + (60554) ? + $34(65531) + 30(65513) + 30(65351) + 84(65153) + 34(63515) + 80(63155) + 30(61535) + 34(61355) } + 28} (65225) + (62552) } + {40(65441) + 18(65414) + 40(64514) + 18(64451) + 40(64154) + 18(64145) + 18(61544) + 40(61445) } + $18(65423) + 38(65342) + 8(65324) + 2(65243) + 46(65234) + 2(64532) + 88(64523) + 18(64352) + 46(64325) + 8(64235) + 46 (63542) + 8(63452) + 2(63425) + 38 (63254) + 18(63245) + 8(62543) + 18(62534) + 46 (62453) + 38 (624.35) + 2(62354)? +164 (64334) + ee + 14) (55424) + (52544) } + 104 (55343) + (54533) ¢) — 4.5°Q°E + 184. 5E* + 26-5°Q?E*U — 8. ox or, arranging according to descending powers, US= 35’ $ (105005) + 3(104114) + 2(103223) + 2(102332) + 8(101441) + (100550) + 5(96104) + 14(95213) +5(95024) + 16(94322) + 5(94205) + 14(941383) +5(94016) + 14(93431) + 14(93314) + 16(93242) + 14(93125) + 5(92540) + 16(92423) + 14(92351) + 16(92234) 4+ 14(91532) + 5(91460) + 14(91843) + 5(90641) + 5(90452) + 10(87203) + 24(86312) + 20(86123) + 5(86015) + 24(85421) + 20(85304) 214 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 4 48(85232) + 3(85214) + 12(85124) + 18(85115) + 5(85106) + 10(85043) + 10(84530) + 48(84413) + 48(84341) + 4(84323) + 6(84233) + 48 (84224) 4+12(84215) + 24(84152) + 3(84125) + 20(84035) 4 48(83522) + 20(83450) + 6(83432) + 10(83405) 4 4(83342) + 6(83324) + 24(83261) + 4(83234) + 20(83216) + 48(83144) + 10(83027) + 20(82631) 4 12(825411) + 24(82514) + 3(82451) +. 48(82442) 4 4(82433) + 10(82370) + 6(82348) + 48(82825) + 48(82258) + 24(82136) +5(81650) + 24(81623) 4 18(81551) +3(81542) + 12(81452) + 48(81434) 4 20(81362) + 24(81245) + 10(80732) + 5(80561) 4 20(80543) + 10(80354) + 18(77411) + 18(77114) + 10(76520) + 80(76403) + 54(76331) + 14(76313) + 24(76223) + 24(76214) + 20(76205) + 30(76142) + 20(76084) + 54(75512) + 30(75440) + 32(75422) + 2(75332) + 62(75328) + 32(75314) + 54(75251) + 6(75224) + 14(75215) + 42(75143) + 44(75134) + 14(75125) + 10(75062) + 20(75026) + 30(74621) + 42(74531) + 80(74504) + 68(74432) + 80(74360) + 6(74324) + 44(74315) + 20(74806) + 32(74252) + 68(74243) + 6(74284) + 6(74225) + 32(74135) + 24(74126) + 80(74054) + 20(73640) + 54(73613) + 44(73541) + 2(73523) + 82(78451) + 6(73442) + 68(73424) + 42(73415) + 14(73361) + 62(73352) + 2(78258) + 62(78235) + 24(73226) + 54(73163) + 14(73136) + 80(73046) + 24(72682) + 10(72605) + 20(72560) + 14(72551),+ 6(72542) + 62 (72533) + 82(72524) + 24(72461) + 6(72452) + 6(72443) + 30(72416) + 24(72362) + 68(72344) + 2(72335) 4 32(72245) + 54(72155) + 18(71741) +18(71714) 4 24(71642) + 14(71633) + 14(71552) + 32(71543) + 54(71525) + 44(71453) + 42(71354) + 54(71336) + 30(71264) + 20(70652) + 30(70634) + 20(70463) + 30(70445) + 10(70256) + 48(66521) + 12(66431) THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 215 + 34(66413) + 30(66350) + 24(66332) + 34(66314) + 24( 66242) + 24(66233) + 12(66134) + 48(66125) + 30(66053) + 30(65630) + 30(65603) + 34(65531) + 30(65513) + 40(65441) + 18(65423) + 18(65414) + 36(65405) + 30(65351) + 38( 65342) + 8(65324) + 2(65243) + 46(65234) + 28(65225) + 34(65153) + 36(65045) + 24(64622) + 36(64550) + 2(64532) + 38(64523) + 40(64514) + 18(64451) + 18(64352) + 16(64334) + 46(64525 ) + 8(64235) + 40(64154) + 18(64145) + 12(63641) + 24(63632) + 24(63623) 4.12(63614) + 46(63542) + 34(63515) +8(63452) + 16(63443) + 2(63425) + 38(63254) + 18(63245) + 30(63155) + 48(62651) + 48(62615) + 28(62552) + 8(62543) + 18(62534) + 46(62453) + 38( 62435) + 2(62354) + 34(61643) + 34(61634) + 18(61544) + 30(61535) + 40(61445) + 54(61355) 4+ 36(60554) + 14(55424) + 10(55343) + 10(54533) + 14(52544)} — 4.5°Q°E + 184. 5E* + 26-5°Q?E?U — 8-5QEU*. If now we were to multiply by U, and proceed as before, we should obtain the value of U*%; thence we might pass (Art. 29) to 2U°; and D, would be given by (Newton’s theorem) 6D,=2D2-+-6D,D)-- 3b? = = 6°" or 14.5 D,=5°(2-58Q8-4 aE ~SU9. But it is not necessary to calculate this coefficient, since it has been already found (see Sec. II., Art. 27) that Dg= 0, 0,05 0, 0;0,=54Q°. 44, The equation in @ is, therefore, + 2-5 QEH + 2-57 Q'H? + 5° QE? @? — 5" (58 Q* — E*) EO +54O8—0, which coincides in every point with the result obtained by Mr. Cockle. (See § 12 of his “ Researches in the Higher Algebra,” published in the present volume.) 45. The foregoing calculations are laborious, but they 216 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND have been conducted with the greatest care, and verified in a variety of ways. I believe that the equation in @ will be found to have an important bearing on the theory of quintics; and therefore I have thought it desirable to place its accuracy beyond all reasonable question. 46. We have seen how steadily the Method of Sym- metric Products mounts up to the higher equations, and in the course of our investigations we have been con- ducted to most significant results. Guided therefore by the analogy afforded by the lower equations, we might next consider the application of the method to quintics. But my professional occupations will not permit me at present to enter into this interesting discussion. I may however remark that even if the method fail to achieve the solution of the general problem, it will probably help to settle a controversy in which mathematicians of the greatest eminence have taken opposite sides, and to throw light upon the question, respecting which so much has been written, of the possibility or impossibility of express- ing a root of the general equation of the fifth degree by a finite combination of radicals and rational functions. ADDENDUM. The resolvent product (@) for a quintic equation is immediately connected with the last coefficient of the equation of the fourth degree which occurs in Lagrange’s theory, the coefficient in question being by the present theory determined as the root of an equation of the sixth degree. And, since it is part and parcel of Lagrange’s theory, that when any one root of the equation is known (ex. gr. if there is a root zero, which is the case considered in the second section), the roots of the proposed quintic can be determined by radicals, the form of the results ~~ THELR APPLICATION TO THE SOLUTION OF EQUATIONS. 217 obtained in the second section may be considered as given d priori by Lagrange’s theory. Lagrange arrives at a reducing equation which may be written S$-TS+4+US -X34+Y=0, where $=$§/(2”")??, and the coefficients T, U, &c. are, each of them, susceptible of only six different values, and Y is obviously the fifth power of the expression denoted in the above Paper by 6. Now if, in seeking these coefficients, we commence with T, we shall be conducted to functions of five dimensions with respect to z. Thus, developing S={fOPH=(Mti mtr Pate ate as) according to powers of 7, we have S=8 461+ &0 +E P+ &%, where, employing (for convenience) the new cyclical symbol, £=Sa° + 10S! {204 (wy +24 2,) + Bxy(a}a?-+23.22)} +1202; 2423 2,2, £,=5> (a4 x, + 2a} at + 4a} wy v7, + Gaj xy a5+ 12a} x, x52), E,= 5D (aj wy + 2a} ay t Aaj vy 25+ Gary, &, a+ 12x L2 %5 45), E,= 5D! (vf vy + 2ay v3 + Ar} ay Hy + Ox} Hj Xs + 12a} x, Hy 25), E52 (vt x; + Qa} a3t 4a} vy X14 Gay a3 x34 12axj 7, x, x5); and if we distinguish this value of 3 by 3(7), the others will be denoted by $(#”), S(#) and 3(2*). Consequently TH=SS=46,4+(&+&+84+8&)G+P +8 +H) =58, — Zé; or, since DE=Lx’ +5ret 2.4 lOXa} x} + Wea} x, 7, + 80Le7j a} x, + GOD 27 ay #3 +1202, 2 ¥3 2,25, T= 42a — 5d 2% x, — 10223 23 — 202.2% x, x3 — 30227} 43 25 — GOD a? ay 5%, + 50S’ § Qa} (ay v; + a3 %4) + 8a, (a3 v2 + £307) t +4800, Ly %3 Vy 5. Applying the =’ process (Sec. II., Art. 19), LS! a}( &ya5 + & gy) = Ef} (dats + yy) (LH — Ly— Ly— Xy— Xs) = La’ ai (ayes + aya) — Lay (yxy + VyHs + hag t+ U5; H Wy Ughy + UyV Ly + LyX Xs + Vy) 218 REV. R. HARLEY ON SYMMETRIC FRODUCTS, AND = Lard! a? (a,2; +232.) — Sxiaia,t dx, (222+ x32?) — Dar7, X32. Consequently T=42 2° —5iatex, —10Dx322 — 20D 232,27, — 130222232, — 1602 a2727,7,4 4802, 227,7,27;+ 00D 72277 (a2; + 2424) + 2503/2, (23a? + x22). But by commencing the calculations with Y (in place of T) we should have been conducted to the fifth power of the four-dimensioned function discussed in the above Essay. It is the form of that function which has enabled me to introduce 7 with such effect into the calculations ; and it is the introduction of 7 which has enabled Mr. Cockle (Phil. Mag., May 1859) to give a simpler form of root than the expression ia, +7™X + P?"™X, 4+ 8"X5-+ ea | to which he would otherwise have had recourse. The advantage of Lagrange’s and Vandermonde’s pro- cess is that the solution of their sextics would lead directly to that of the quintic. The disadvantages are the high dimensions of the symmetric functions, and the consequent difficulties of calculation and verification. The Method cf Symmetric Products, considered independently, leads only indirectly to the solution of a quintic. But it has the ad- vantage of lower dimensioned functions and of greater facilities for calculation and verification, >/aj(#.7;+232,) being a more manageable function than ¥’a#}(a2a,+ 43a) or 2x, (a3a2+x3v}). The unsymmetric expression for the roots of a quintic given by Lagrange and Vandermonde is undoubtedly an element of the theory of quintics as treated in the foregoing Essay, but the progress made will be tested by the simplicity of the results to which the Method of Symmetric Products has conducted us. If the difficulties of dealing with symmetric functions increase in more than arithmetical proportion to their dimensions, it THEIR APPLICATION TO THE SCLUTION OF EQUATIONS. 219 is a moderate expression to say that, by making pro- ducts instead of powers of linear functions of the roots the basis of the theory, the difficulties of the calculation are diminished by one-fifth. The cyclical symbol (%’), again, combining five terms in one, diminishes by four- fifths the four-fifths that remain. So that altogether more than five-sixths of the labour disappears. And since there is no six-valued function of five letters of less than four dimensions, it is improbable that any view of the subject can lead to a simpler reducing sextic than that in 0. VOL, XV. GQ 220 REV. T. P. KIRKMAN ON THE PARTITIONS XV.— On the Partitions and Reticulations of the r-gon. By THE Rev. Tuo. P. Kirkman, A.M., F.R.S., Rector of Croft-with-Southworth, and Honorary Member of the Literary and Philosophical Society of Manchester. Read March 22nd, 1859. 1. AttHoven I have given the formule of an inductive method (Philosophical Transactions, 1857, “On the parti- tions of the r-gon and the r-ace”’), whereby the partitions of the 7-gon made by any & diagonals, none crossing another, may be enumerated, much remains to be done before they are expressed directly in terms of 7 and &. The case of kK=r—3, that of the triangular partitions, has been discussed in a previous part of the present volume (p. 43), in which this direct expression has been investi- gated. The most convenient classification seems to be that according to the number of marginal faces; it is, at least, essential in the theory of the polyedra. In this Paper will be given the required expressions of the partitions of the 7-gon by *+1 diagonals, in which there are two marginal faces. 2. Such a partition, F, will always have 2+2 angles unoccupied by a diagonal. Let w of these —all except one in each marginal face -- vanish, that is, become angles of 180°. F has now become IF’, a (k+ 2)-partitioned 7’-gon (r’=r-—«), having two marginal triangles, and no angle, besides their two vertices, unoccupied by a diagonal. No face, either of F or of F’, has more than two diagonals in AND RETICULATIONS OF THE R-GON, PP # its circuit; for if any face has three, sections along them will cut off three polygons having each at least one angle outside its section unoccupied by a diagonal, because every polygon has at least one such angle above its base. But F and F’ have each only two such angles ; wherefore no face of either has more than two diagonals, and conse- quently no face of F’ is ampler than a quadrilateral. 3. If we pare away all the margin of I’, all its faces, except the two marginal triangles, will become quadri- laterals. F’ is now F”, a (2k+4)-gon, having / quadri- laterals, two triangles, and k+1 diagonals. Let all the & consecutive edges ab, bc, cd---- of quadrilaterals vanish except y, and in the y remaining quadrilaterals let all the opposite edges a’b’, U’c’, c'd’.-++ vanish except z; there are now only z quadrilaterals. This can be done in Ba is different wa yit % yelr ye. In order that F’ should be one of the resulting figures, it is necessary that the spared edges, together with the four in the marginal triangles, should be 7’. That is, k+z=r —4, or z=r —k-A, Wherefore F’ is one among kia yi Y qyll ” [r—k-4)1 figures, (y>09). 4. To find this sum, let 7’ —-k-4=0@; then all terms vanish in which y <0, and the series is Kol—-2 Geil orti-1_ (9 41)81-2 KeHAI (94 2) -—— + yet. ei yoraia eta yoeeiaz yet “Fase Ovi. 94] eel (+2)?! TO aes Ce eee yer2ii pu Ko+3i-l (9 4.8)314 yo+3i1 pot et BIA k= 8 O41, (K-O)(R-O-1) +1042) ea a+) yk (9+ 1)(0+2) 1-2 pin yr |-l Ark 1 ee ee Oe ee Yqyll © yr-k-4i1 yr—k-4 11 ? is the number of figures F’, which all differ at least in their position, all being placed so as to have their two marginal triangles on either hand. 5. The r-gon F is made from the 7”-gon F” by depositing z points (7’+2=r) in any one or more of 7” —2 positions, viz., about the vertices of the two marginal triangles of F’, and on the »” —4 other edges of I’. This can be done in (w+1)"-311 at 7-3" 3314 rsa poe different ways ; and the product of this number into &, just found, taken for every value of r=r-—7", is the entire number of r-gons F. That is,* if R?(r,4+1) denote the number of these r-gons which have k+1 diagonals and two axes of rever- sion, R(r,*+1) that of the 7-gons F which have one axis of reversion, and [?(r,k+1) that of those which are doubly irreversible, that is, which have no axis of reversion, but have a sequence of configuration repeated in the circuit; . and if I(r, +1) denote the number of these F which have no symmetry, but are irreversible, i.e. have their lower face different from their upper, we obtain, by writing r-—wx for 7’, * An example of the class R?(7,2) is formed by placing two triangles on opposite sides either of a square or of a hexagon. An example of R%(r,2) is got by placing two triangles about a hexagon, so that one side of it shall stand betweenthem. One of R% (7,2) is made by placing two triangles about a hexagon, so that no side shall be between them. Two triangles placed on any two sides of a pentagon form an example of R™°(r,2); and ifa diameter be drawn which is not one of the two axes in any of the class R%(r,2), it becomes one of I?(r,3). — Vid. Art. (8, 9,10, 11). The sym- metry of these figures is fully discussed in the memoir referred to in the first article, in the Philosophical Transactions. cw) ~ ie) AND RETICULATIONS OF THE R-GON. {R4+2(R+1P) +41} (7,441) kr—-*-k-4 {—1 (r est 3) r—«—3 | —1 ® r-a—k-4 |1 4 yr-73 2 » Qurtats , where z has every value that gives a result, namely, at0O, etr—-2k-4, wpr—-k—-A. For each one of the doubly reversible will be once only constructed, having but one possible posture about either axis; each one of the R will be twice made, once with either extremity of its axis uppermost; each of I? will also be twice constructed, to show both surfaces; and each of the I will be four times made, with either marginal face on the right hand, and with either surface uppermost. I cannot find that this sum 2, has any simpler expres- sion than the finite series which it represents. By this formula, when R*(r,k+1), R(r,k+1) and IP’(r,k+1) have been otherwise determined, we easily obtain the number of the largest class, the class I(r, +1) of these (k+2)-partitions of the r-gon which have two marginal faces. 6. Let the r-gon E be one of the number R?(r,k+1). This E has a certain number 2+2 of angles unoccupied by a diagonal; and, by the vanishing of x of these, E becomes the 7’-gon E’ (7’=r-—wz), having two marginal triangles. As the 2 points were symmetrically placed with respect to two axes of symmetry, their disappearance has not disturbed the symmetry, and E’ has the same two axes of reversion. All the k+1 diagonals of E’ are pa- rallel to one and perpendicular to the other of these axes, and EH’ is made up of & quadrilaterals and two marginal triangles, and 7’ =2k+4. When £ is odd, 7’ =4i +6, and there is a central quadri- lateral, on each side of which stand }(k—1) others. We turn E’ into E by adding @ points in the circuit so as to preserve the symmetry about the two axes. When v=4j+2, two of these will be placed at opposite mid- 224 REV. T. P. KIRKMAN ON THE PARTITIONS edges of the central quadrilateral, and the rest symmetri- cally about the two axes on the four half-edges of it, and on 4¢+4 other edges of HE’. That is, }(v-—2) points are to be laid in one or more groups in 7+2 positions in the fourth part of the circuit of E’, which can be done in e+o\n0 r—2k-2 ae. Kcapdion lstihior aa? - {1 a ways. When x#=4y, there are no points at the mid-edges of the central quadrilateral, but 42 points are deposited im the i+2 positions before mentioned, in some one of m2 pee L+4y 4 r—Qky 4 zt sa, - = ——____—_—- ways. eh pep When & is even, ”=4é+4, there is no central quadri- lateral, and v=4j. In this case 7 points are deposited on one-fourth part of E’ on i+ 1=37" positions, in some one of meee ie r—2Qky2" ——_ = ——__———__ ways. eee The number R? of r-gons E that can be thus constructed on the 7’-gon E’, to have two axes of reversion, may be thus expressed, k+1|4 k+1 1) Ki, Gate) 2 \ gas 2 (ie 2 4 h 4 R= - : Seaatie ae = > —— J2!? if we define that every irreducible fraction containing r or k shall be accounted zero. Of these three terms in R? either two or all will always thus vanish. 7. We take up next an 7-gon of the number R(r, +1), having one axis of reversion, two marginal faces, and k+1 AND RETICULATIONS OF THE R-GON. 225 ra diagonals. This axis, if it lies, of course an undrawn diameter, through two mid-edges of a central face, is an agonal axis; if it is drawn or drawable through two angles of G, it is a diagonal axis; if it lies, of course undrawn, through one angle and an opposite mid-edge, it is a mono- gonal axis. 8. Let the 7-gon G have an agonal axis of reversion. By the erasure, as before, of w angles not occupied by a diagonal, G becomes the 7’-gon G’, (7) =r-—z), having the same axis, and no unoccupied angle except the two vertices of the marginal triangles; and G’ becomes the (2k+4)-gon G”, if we pare its margin, having k=2i+1 quadrilaterals and two triangles. Of these k, let all the consecutive edges vanish except y, and in these y quadri- laterals let all the opposite edges vanish except z, whereby the & quadrilaterals are reduced to z. This can be done in wi gels (4(k- 1D) ee ye en LaLa wi ie so that the reduction of G” shall be effected symmetrically on both sides of the agonal axis. One of our results will be G’, if the spared edges ways, wherefore the number of figures obtained from G” is 5 (4(k-1))y"'7 pret ‘—k—5) | —1 y yyii "ker SS (y > 9). This series is easily proved, by the reasoning of Art, 4, ‘a Ly iy to be Gieatyyerasnin [i@r—k—) | 1 S Qk—-w'42 which is the number of r’-gons G’, to one of which we reduced G. This one becomes G by the addition of 4x points in 47’ positions, viz., about the vertex of a marginal triangle, on two opposite half-edges of the central quadri- lateral, and on }(r’—6) other edges of G’ all on the same 226 REV. T. P. KIRKMAN ON THE PARTITIONS side of the axis, and by the repetition of the operation in order reversed on the other side of it. The number of figures G thus obtained from G’ is (jet 12 _ Ge’ +a)-)eO4 Ler—pn pean 2 and the product of this number into S just found is the entire number of 7-gons G that have an agonal axis of reversion placed in every position about that axis. Among them will be ouce constructed every doubly reversible R? which has an agonal axis perpendicular to the axis through its marginal faces, and every singly reversible R” twice, once with either extremity of its axis upwards. This pro- duct then, summed for all values of v=r-—-7’, is (R?+2R)(r,k+1), wherefore putting for R? its value from Art. 6, when & is odd, and 7 - 2k=4m, Rr, kK+1) a! (ar As lj |—1 (4 (k 7) 1) )#e—#-#-8) ia Poe fi mall ee) a Rome) 11 +o ben are k+1 (a(r-2h)) ? a ja 1 ee where <0, wr =2h+ 4 (R?+2R”)(r, £+1) (ar 2s Yi |—1 (4(& bos 1))i#-*-) ht — 2 a ; e , Dk—h(r—2x) +2 =2, Jr) 11 [30-7—k5) | 1 2 =R”(r-1,k+1); wherefore R”™(r, &+1) QE= DPI GG) =2, pea % [20-2 11 5 where x+0, e<+tr-2h-8, xpr-—k-4, and fractions irre- ducible are counted zeros. That is, & is odd, 7 is odd, and # is even. 10. We are next to enumerate the 7-gons R“(r, £+1), which have £+1 diagonals, a diagonal axis, and two mar- ginal faces, which axis may be either drawn or undrawn. Let H be one of these figures having a drawn diagonal axis. It will have v+2 angles not occupied by a dia- gonal, by the vanishing of which H becomes H’ an 7’-gon (r’=r-—~) having the same axis and two marginal tri- angles. By removal of the margin, H’ becomes H”, a (24+4)-gon having & quadrilaterals and two marginal triangles. Of the 4% edges of these quadrilaterals that make one-fourth of the circuit of H’”, excluding the tri- angles, let all but y vanish, and in these y quadrilaterals let all the opposite edges vanish except z. The same changes being made in the other half of H”, so as to maintain the symmetry about the axis, H’ will be one of the resulting figures, if the spared edges VOL. Xv. HH 228 REV. T. P, KIRKMAN ON.THE PARTITIONS sk+2=43(7 -4), or z=43(7 -—k-4). The number of figures H’ to which H” is thus reducible is (Gk)Yi yeia (4h)Yi—2 yh) | 4 Wi yet 49 qe ee? 2 a sum which is shown by the reasoning of Art. 4 to be (4h) -** i: k—hr'+2 =a) 11 “2 ; Any one of these S r'-gons H’ becomes an r-gon H, if 4x points be placed in any of (37’—1) positions on half the 7’-gon, namely about the vertex of a marginal triangle, and on 3(7’—4) other edges of H’ on the same side of the axis, and if the operations be repeated in reverse order on the other side of it. That is, every figure H’ gives (doF1) 2 _ (Ar 42-2) eI Bein pen figures H. The product of this number into 8, summed for all values of 2=r-r', is the entire number of 7-gons having +1 diagonals and reversible about a drawn diagonal axis in every position about it. In other words, this product is, putting R“(r,4#+1)’ for those having one axis only, 2R%(7,£+1)/+R(r,4+1), where for R? is to be put its value when & is even, and r—2k=4m; for each of R” has two configurations about the axis, and each of R?* has only one about the axis which does not pass through the marginal faces. Where- fore, putting 7’ =r- «2, we obtain R“(r, k+1)' (4(r— 2) 14 (3 Bisset 1 yt. 12,4 - era) }21 ‘ pare 22 El (F(r - 24)? ams ; é el where # #4 yi wen among which L’ will be twice constructed, i.e. with either face uppermost, if the spared edges, 4(4-1)+z=4(7'- 6), or z=4(r'-k-5). If now, in each of these k-Vyyi3 yer 1a wil her) 17 figures, = r'-gons, we add 3a points in 3(7’—2) positions, viz. about the vertex of a marginal triangle, on a side of the central quadrilateral, and on the 4(r’—6) edges lying between those two positions; and if we repeat the distribution of 4a points in order round the circuit on the remaining half of the r’-gon, we shall complete upon each 7'-gon (de + 1)! dr 4-9-4) I9 Sa nS Eco figures, which will all have a sequence repeated in the circuit. If the 7/-gon on which we operate has not two axes of reversion, that is, if z<4(4—1) or 7’ <2k+4, all our re- sults will be of the class I?(7,4+1), for the addition of the 2 points has added no axis of reversion to the figure ; but if the 7’-gon should have two axes of reversion, 7.e. if r=2k+4, the a2 points, being distributed in every way, may be so arranged as to preserve the symmetry about both axes, while they still exhibit a sequence repeated in the circuit. That is, we shall have constructed both twice every one L’ of the class °(7,4+1), which can be reduced by the vanishing of a points to the (24+4)-gon, and also once every one of the class R?(r,4#+1) which has a central face; that is, the product of the last written function of AND RETICULATIONS OF THE R-GON. 231 r' and x into the preceding sum 3, taken for all values of y and of e=r-—r', is the number 22 (r,k£+1)+R*(7,4+1), when £ is odd. The sum 3, by the reasoning of Art. 4 is seen to be A(k-1)YI2 yk IAR-1)) IA =e SS C=», Dir $2 y yyvit [2@—k—-5) | 1 tr" —k—5) [1 wherefore, when & is odd, by Art. 6, P(r, 4+1) =1|3, Se nena ee el) haneeilaaer ere aie jie) |1 ]k0—#—k-5) | me) k+1)5 gate onze Bok z | 4. A ) he ee wee sp in? bof where #<{0, x GS EE EEL hee gk-A(r—2—2) , mitiist p02) 1 ; 3-#—-k-4) | 1 ji (Art. 9), where & is odd, r is odd, and 2 is even, 20, atr—2k-3, wpr-k-4. This is the number of r-gous partitioned by £+1 dia- gonals so as to have two marginal faces and a monogonal axis. R“(r,k+1)! (A(r—2) 1 (Ga) IA 2 7 3,4 jie a ala tl Ad hyy _ Gr-24))P an 1? where r, & and « are all even, #40, x0, there are 7 points within the 7-gon, in each of which three or more of the partitioning lines meet — some of these lines being diago- nals, some passing through a node and an angle of the r-gon, and others through two nodes. The & reticulations are & smaller polygons, all having only angles of less than 180°, into which the 7-gon is divided. I have succeeded in the discovery of a general inductive method of enumera- VOL. XV. II 236 REV. T. P. KIRKMAN ON THE PARTITIONS ting the j-nodal /-reticulations of the 7-gon, which involves no tentative process, nor any reference to figures. The subject is far too extensive to be here discussed, and the process of computation is very tedious, owing to the great number of formule due to the various symmetry and ge- neration of the results, which, for small values of 7, 7, and k, are collected in but small instalments. I shall content myself here with giving a complete account of the 7-reti- culations of the pentagon; and I think it highly probable that any mathematician, who may verify my propositions, will fully satisfy himself as to how far I am in possession of the whole theory, 1. The entire number of 7-reticulations of the pen- tagon is 7778, of which 413 are symmetrical. 2, Of these, the 4-nodal 7-reticulations are 62 sym- metrical and 1010 unsymmetrical. 3. The 5-nodal 7-reticulations are 85 symmetrical and 2000 unsymmetrical. 4. The 6-nodal 7-reticulations are 99 symmetrical and and 2282 unsymmetrical. 5. The 7-nodal 7-reticulations are 69 symmetrical and 1340 unsymmetrical. 6. The remainder have more than 7 nodes, or less than 4. In this enumeration no figure is counted which is either the repetition or the reflected image of any other. 18. I have to correct two oversights in the sixteenth and twenty-second articles of my Paper “ On the triedral partitions of the xv-ace,” in the present volume. In the sixteenth article, not only doubly irreversibles but also the doubly reversibles of Art. 15 are constructed, so that these ought to be subtracted before dividing by two. in the twenty-second article, the triply reversibles of Art. 20, as well as triply irreversibles, are constructed ; it AND RETICULATIONS OF THE R-GON. 208 is therefore necessary to substract those triply reversibles, and then to divide by two, for the reason given in the six- teenth article. The formule which have to be corrected in consequence in Art. 25 are (16), (19), (22) and (23). The reader will kindly alter them to the following : (16). PR?(4v+m+1,2a) = R?(4x + 1,22) oe 27-21 ie a—-2|1) x1" z+1) oe coe) a [zn =" ‘ j|1,,__ (19). IR?(40 + m+ 1,2v) = R*(42 + 1,22) m x—2|1 x gn-2 Coach < (G+1) 2Qr—2|1 m 2e—3 | 1 a (F+1 ) nm xt) pease to A oe (a 1 Qx—2 $1 or Gg j i ee (22). I3R3(67 +m +1,82) = R4(6x + 1,32) 2x2 | I v—2|1 GH) ee GH) weed er a (23). IR%(6x7+m+1,3x) =4R5(6x + 1,32) ™m ™ =+l1 x a i (m+ 1)%-#! 495 (é ) [est oy ae m 3a—3 | 1 m 2—2 | 1 ae Gay as GH) ; Co a ae 5 5 The formule of Art. 25 thus corrected, together with those of Art. 26, are all that are required for the enumera- tion of the triangular partitions of the 7-gon for all values of *. 238 DR. ROBERTS ON THE DIURNAL XVI.—A Contribution to Urology, embracing Observations on the Diurnal Variations in the Acidity of the Urine, chiefly in relation to Food. By Witi14M Roserts, B.A., M.D., Physician to the Manchester Royal Infirmary. Read May 3rd, 1859, Ir has been the universal belief, until recently, that the reaction of the human urine was, in health, invariably acid; and that a neutral or alkaline condition of it was either a sign of disease, or the consequence of partaking of alkaline substances, or of subacid fruits, the vegetable salts of which, being broken up in the blood, appeared in the urine as alkaline carbonates. In 1845 Dr. Bence Jones* called this belief in question, and announced that he found that the urine became not unfrequently alkaline a few hours after taking food. Three years later Dr. Jones investigated the subject comprehensively ; and the result of his inquiries were published in the Philosophical Transactions for 1849. He conducted the inquiry as follows : — The urine was passed immediately before breakfast (about nine), and the acidity per 1000 grains estimated by a test solution of carbonate of soda. After breakfast the urine was passed about every hour or hour-and-half, and the products separately ex- amined by the test solution. This went on till dinner (about six p.m). After dinner the observations were less frequent and regular. Generally the first examination was * Phil. Trans. 1845, p. 349. VARIATIONS IN THE ACIDITY OF THE URINE, 239 made about two hours after the meal, and the next about three hours later. The urine of sleep and of the following morning were also separately examined. As a rule, about twelve distinct observations were made in each period of twenty-four hours. The effect of ordinary mixed food, of purely animal and of purely vegetable diet, was investi- gated during separate days on which one or other class of diet was exclusively used. He sums up his results in the following propositions : — I. “As regards the variations of the acidity of the urine for three days on mixed diet. The acidity soon after food was found to decrease and to attain its lowest limit from three to five hours after breakfast and dinner; sooner however after breakfast than after dinner. The acidity then gradually increased and attained its highest limit just before food. If no food was taken the acidity of the urine did not decrease, but remained nearly the same for twelve hours. It fell immediately after food was taken. Il. “ When animal food only was taken, the diminution of the acidity after food was more marked and more last- ing than when a mixed diet was taken; and the acidity before food rose rather higher with a mixed diet than it did with animal food. Ill. “ When vegetable food only was taken the contrast with animal food was very marked. The urine did not decrease in acidity to the same degree; though it became neutral it did not become highly alkaline. The increase in the acidity of the urine was by no means so marked as the decrease of the alkalescence. The acidity of the urine was rather higher with the vegetable food than it was with animal food. IV. “The result of these experiments is, that the acidity of the urine is always changing, and that the changes de- pend on the state of the stomach. When much acid is in the stomach the acidity is then diminished. As the acid 240 DR. ROBERTS ON THE DIURNAL returns from the stomach the acidity of the urine increases, and generally reaches its highest limit before food is again taken.”’* These views have not obtained universal assent. It has been objected to them that they are based on observations on a single individual,t and cannot therefore be safely ap- plied generally. And not only have no confirmatory facts been advanced, but subsequent observations have tended to re-establish the old opinion, that a neutral or alkaline state of the urine is an abnormal one. Dr. Beneket ex- amined his own urine on twenty-three days, and failed to detect any depression of the acidity as a result of taking food. In addition, he made not less than a hundred day- observations on different sick and healthy persons with this result — that, although the urine did occasionally show a depressed acidity and even an alkaline state after a meal, and especially after breakfast, yet this was very far from being constantly the case. Dr. Julius Vogel§ rejects Dr. Jones’s experiments on * Phil. Trans., vol. lvi. p. 244. + Although the experiments detailed in Dr. Jones’s Paper were per- formed on a single individual, additional evidence of the effect of food on the reaction of the urine was advanced by him in the Philosophical Tran- sactions for 1845, p. 345. He there says: ‘‘ Dr. Andrews of Belfast stated to me, that having observed a case otherwise in perfect health, in which the urine was almost invariably alkaline about two hours after breakfast, so much so as frequently to be loaded with a deposition of phosphates whilst still in the bladder, he was led to observe the urine of about fifteen students in good health immediately after it was voided about noon. He found it to be alkaline in about two-thirds of the cases.” “At the present time I know five physicians in whom the above phenomena at this period of the day are more or less frequently visible in a greater or lesser degree, and in London this alkalescence will be found in those who are considered gene- rally healthy much oftener than is imagined.” t Archiv des Vereins fiir gem. Arb. zur Ford. der Wissensch. Heilkunde, vol. i. p. 438. § Neubauer und Vogel, Anleitung zur Analyse des Harns. 2nd edition, p- 175. VARIATIONS IN THE ACIDITY OF THE URINE. 241 the ground that all his determinations were reckoned for 1000 parts instead of per hour. This objection would have had weight if the density of each specimen had not been recorded. Great dilution of the urine from abundant potation would no doubt reduce the degree of acidity per 1000 parts, even when the quantity of acid discharged per hour remained constaut; but this source of fallacy was guarded against in the observations of Dr. Jones by taking the density of the secretion — this being a sufficient mea- sure of its concentration. Dr. Vogel goes on to say: “Researches undertaken partly by myself, and partly by others under my direction, showed uniformly that the greatest quantity of acid secreted per hour by the kidneys occurred during the night, the least in the forenoon, while a medium quantity was discharged in the afternoon after the principal meal. These results therefore are unfavour- able to the conclusions of Dr. Bence Jones, but do not tell conclusively against them, inasmuch as other circum- stances may have had an influence on the amount of acidity.” Dr. Sellers, in the Edinburgh Medical Journal for Janu- ary 1859, states that in a good many trials he has not been able to satisfy himself “that the rule, as laid down by Dr. B. Jones, is generally applicable in Edinburgh ; certainly not to the extent that the urine loses entirely its acid character, or that it becomes alkaline.” Nevertheless it has seemed to him “that the variations in the degree of its acidity are in some measure governed by the existing states of the stomach.” Dr. Delavaud (Gaz. Médicale, 1851, No. 44) found the urine becoming neutral or alkaline after breakfast, but not after dinner. Seeing this discrepancy in the results obtained by difler- ent observers, it seemed not undesirable to seek additional and exact information on the effect of food on the reaction 242 DR. ROBERTS ON THE DIURNAL of the urine, and to find out some means, if it were pos- sible, of reconciling the conflicting facts. The following experiments were undertaken with that purpose; and in a subsequent portion of the Paper some considerations are advanced which go far to account for this want of agree- ment. All the experiments herein detailed concerned a single individual. He was a healthy man, twenty-eight years of age, taking moderate exercise, living in most favourable hygienic conditions, and weighing 144 lbs. In order to ascertain the exact amount of change under- gone by the urine after food, it was thought essential to collect the secretion at each hourly period succeeding a meal, and by measuring its quantity and saturating power to obtain data from which the precise amount of free acid or free alkali separated per hour by the kidneys could be estimated. At periods more remote from meal-times the urine was usually collected every two hours. As compared with Dr. Jones’s method of merely ascertaining the de- gree of acidity or alkalescence per 1000 grains, it had this important advantage, that it eliminated the inaccuracies - consequent on the great inequality of concentration to which the urine is subject from food, drink, exercise and sleep, whereby its quantity and aqueousness rise or fall immensely, and with great suddenness. The following particulars were taken of each urine, and arranged in a tabular form : — I. The Time of Day during which it was secreted. II. The Quantity. This was estimated in a glass vessel graduated on the scale of the 1000-grain measure. If the period of secretion exceeded or fell short of an hour by five, ten, or fifteen minutes, or if the period was two or more hours, as during sleep, the hourly rate of secretion was exactly calculated from the quantity and the interval. The results are arranged in the second columns of the VARIATIONS IN THE ACIDITY OF THE URINE. 243 tables, and show in grain-measures the hourly rate of flow of the urine. From this volume-measure and the density the weight of urine per hour cau be readily calculated. III. The Density. A gravimeter of tried accuracy was usually employed. When the quantity was too scanty for the instrument, a 250-grain specific gravity bottle was substituted; and in one set of experiments of seven days (Table III.) the bottle was exclusively used. IV. The Solid Matters. The amount of solid residue per 1000 grains was calculated according to Christison’s formula.* A second calculation from this and the hourly quantity gave, in grains, the solids separated per hour. These are arranged in the fourth columns. It is not pre- tended that these figures represent with accuracy the actual amounts, but as relative values they may be fairly assumed as near the truth. The results obtained are ex- ceedingly uniform, considering the somewhat complicated, and avowedly uncertain, calculation on which they are based; and the remarkable conclusion to which they point is indicated with great distinctness. V. The Reaction. The degree of acidity or alkalinity was ascertained by a test solution after the usual method in volumetrical analyses. For the former a solution of caustic soda was employed, and for the latter dilute sul- phuric acid. The two solutions were made of equal satu- rating power; cach 100 grain measures being equivalent to 1 grain of dried carbonate of soda. The results are arranged in double columns, showing separately, in grains of dried carbonate of soda, the degree of acidity or alka- lescence per 1000 grain-measures, and the amounts per hour. 500 grains of urine were usually operated upon; but if the urine was very dilute 1000 grains; and if very concentrated and scanty 250 grains were employed. VI. The Appearance of the Urine. The condition of the * See table in Bird’s Urinary Deposits, 5th edition, p. 60. VOL. XV. KK 244 DR. ROBERTS ON THE DIURNAL urine as to clearness or turbidity, when passed, and the colour, were also recorded. Frequently, too, the appear- ance on cooling or standing was noted, but the results do not appear in the following tables. To complete the record, the times of the meals and their nature were chronicled. The condition of the body as to exercise, occupation and sleep was also notified, but inasmuch as a single statement will suffice for all the particulars, the details are not here recorded, so as not too greatly to complicate the tables. The mode of life of the subject of experiment was kept as nearly as possible uniform during the time of observa- tion. He usually rose at seven, breakfasted at eight, dined at two, sometimes at four; and took no further solid food until breakfast next morning. He retired to rest at one in the morning; so that when the days of observation were successive there were but six hours of sleep. As to occupation and exercise there were neces- sarily some variations, but these were reduced to a mini- mum. Variously engaged in-doors until ten or eleven in the morning, moderate out-of-door exercise was after- wards taken until one or two. After dinner occupation was sedentary for two or three hours; then moderate out- of-door exercise was taken for one or more hours. Care was taken to avoid any violent or protracted exertion on the one hand or a complete inactivity on the other. Walking exercise for an hour or two did not produce any perceptible effect on the results, The experiments are divided into six sets; each set embracing from three to seven days; and the results for the separate days are collected into a single table of averages, which shows the mean quantities for the days composing the set. The observations on the reaction, however, are given in detail for each separate day, and form a companion table to each table of averages. VARIATIONS IN THE ACIDITV OF THE URINE. 245 The first observations were made on ordinary mixed food. They form two sets of four and seven days re- spectively, and there are added some odd days, which could not be united together to form a distinct set. Effect of Mixed Diet. The first set embraces four days, on which hourly ob- servations were made from seven a.m, to eight a.m. next morning (except of course the hours of sleep, which form a single observation), a period of twenty-five hours. DR. ROBERTS ON THE DIURNAL 246 OF a, saasiaieiicmeiiemiaat = ws 2 ‘ oS SS TES Tae i a *(skep oa, “ToqUAe Yorr “1waty “1 66.0 | PLL || G62 | O-zcOL] ose | SL *(deatsy) ‘roquie Ystppat ‘avai || "| “* || P40 | O&-T | 4-6 | O-STOT | 189 | L-T *(skep omy.) ‘aoquie ojed ‘reat || ** | ** |] 68.0 | 44-1 |} Zo-8z | F-SZ0L | OLS | T-2T *(skep oa) ‘moquie oped ‘away | “" | “* || ILL | 88-1 |] 84-62 | Z-1Z0L | $09 | SI-IT *(shup omy) ‘roquie eyed ‘aveqQ | *** | “* | 90-1 | G80 || 10-8 | 9-PIOL | GOST | TL-OT *mvelo 10 Jesns | ‘pro’ cory ‘oulTey[e i sachets qmoy gta ‘aazjoo 90u0 Teaqneu 2009, tq, [8 . 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In constructing this and the succeeding tables of means a difficulty arose as to how to deal with those hours which on some of the days exhibited an acid, and on others an alkaline urine, such for example as from ten to eleven and eleven to twelve. The plan adopted has been to take a separate mean for the acid days and a separate mean for the alkaline days. This will explain why at these hours the urine is made to appear both acid and alkaline. Any obscurity or inconvenience arising from this will be obviated by a study of the companion tables, which give in ewtenso the daily results for the acid- ity and alkalinity. TABLE II. exhibits the hourly Variation of Reaction for the days composing Table I. The Plus sign is prefixed when the urine was Acid, and the Minus sign when Alkaline. Sept. 30, Oct. 4, Oct, 13, Oct. 21 First day Second day Third day Fourth day Fa | ee eens om | fae Se NM Lao Diet Per 1000 | °F. | Per 1000 | ,Per | Per 1000] jPer | Per 1000 | = eee) Seealie ay 7-8 | +1°88}+0°90]) +2°16/+0:81)| + 1:20] + 0°83]| + 1:56) + 0°80] Breakfast at 8. Tea with sugar +0°90}+0°77|| +1:46] + 0°94'| + 0:28] + 0°21|/ + 1:00) + 0°74)! and cream - — 1-00] -0°95|| — 0°64) — 0:55!| — 0°84! -0°81|| 0-00} 0°00) grica toast -1°32)-1°45]| + 0-20} + 0:21) — 0°42) — 0°44|| + 0:24. : with broiled — 0°38}—0'55|| + 0°68) + 0°45)| + 0:09] + 0°18/| + 0°42) + 0°42)! bacon, or eggs, 000} 0:00|| +0:94) + 0°52)| + 0°16) + 0°15]| + 0°72) +.0°75]| op pork chop. +0°41|+0°28]} +1-18] + 0:62) + 0°64) + 0°81|| + 0-42} +0: +0'22|+0:15|| +228] + 1-03)) + 0:88) + 0°68]| + 0°62 +0°32|+0°30|| + 2°52) +0°76) + 1:40! + 0°77] + 0°72 +128] +0°66|| + 2°36) + 0°83!) + 0°80) + 0°40) + 1°56] + 0°83) Dinner at 4. + 0°06] + 0°28)| + 0:26] + 0:33/| — 0:58} — 1:37]| + 0:28} + 0:59} Meat, potatoes, — 0°84) —0°88]| - 0°20) -— 0°15)| — 1-48] — 1°30) — 1°46} — 1°68) bread, cheese, — 1°30) -1°79|, - 0°76] - 0°58)| — 1:00) — 0:97)| — 0:25] - 0°14) beer. — 0°62) — 0°86]; -— 0°78} — 0°'78)| -- 0°56) — 0°58)| — 0°18] — 0°15) At 9 two cups + 0'36|+1°02)| + 0°86) + 0°45||-0:26]-0°58|| 0°00) 0:00)) of coffee with- +0°60)+1°77|| +1:22)+ 1:05) + 1:20) + 1:27) + 0°44) + 0°86|| out sugar or + 0°54) + 2°02)! +1-00) + 1°55)| + 2°28) + 1:27|| + 1°48} + 0'96)| cream. +0:96) + 1:39)) | + 1:62| + 0'89]| + 1:92] + 0-92! 1-7 oon |¢ + 1°70) + 1:18] + 0°74, + 0°69) + 1°86) + 0:80 7-6 |} +116] +097) (+124! +0511] + 2:24] + 0°67) 248 DR. ROBERTS ON THE DIURNAL On the first day a slight luncheon, consisting of two thin pieces of bread and butter and a glass of water was taken at two. This accounts for the slow degrees by which the acidity recovered its ordinary level after break- fast. About a pint of porter was taken between nine and eleven on the first night, and some bread and butter with tea between nine and ten on the second night. For this reason the urines passed after ten o’clock, on the first and second nights, are not reckoned in the table of means. Previous to these four days the condition of the urine had been examined at short intervals on five other days after breakfast, and on four days after dinner. On each day the urime became alkaline after dinner. After breakfast it became alkaline three times; but remained acid, in a diminished degree however, on the remaining two days. The times of change from acid to alkaline, and back again from alkaline to acid, having now been ascertained with tolerable certainty, it was thought unnecessary to carry out so rigidly the very laborious plan of hourly ob- servation throughout the entire day. In the succeeding experiments, therefore, hourly observations were only made at the critical periods, when the reaction was oscil- lating; but when the acidity was steadily rising, or had attained its usual level, observations were made every two hours. By this modification the experiments could be carried on with comparative ease, and continued for several consecutive days. The effect of mixed food was again subjected to obser- vation during seven days, all consecutive but one. Two meals a day were taken, and no alcoholic drinks. 249 VARIATIONS IN THE ACIDITY OF THE URINE. “TOU 109]B ‘pmbry 10 pros azoyyto ‘FUIYJOU SSUINAAD OdAT} pus ‘dno ouo Suruoso Joyous £40jva Jo sdno dat] SsuItm9AA OMA UC *1OJOM ‘asaayo “pworq ‘800984 -od “quay *Z 4 aouur(y *1999nq pue prarq ‘va4 10 aayzoo ‘Quay “S98 ysuyyroag 10 = *(sdup oatiq) “LOQUIV TOILE “wea GL-L1T | OS-PZOT 8-Z ‘doasy “Toque TSIpped “woyp) &&-ST | $0-860T L-1 gest ea (0) Nha €-8Z | 0-220T I-Il *taqure “B31 || q 99-48 | 9F-T6OT IL-6 "roquae “189[0) % PPLV | LL-ZZOT 6-4 *[eaqnou oto : antler sour} sig “IBoTO 9OUO “prqiny sautTy xTg || 8S-T | FS.T 10-67 | 46-E20T Np, “prow a0u0 *TBeTD doo “TsTMOTTAL ? = _ ‘ouTpeyye sowuty eat ce prqang £104 soury a 4V-T | 681 60-1 | 68- dons ee ere Rapist ‘xeaqo sot seated 09-0 | 34-0 || Z%-0 | $8.0 | F$-e¢ | TLSLOT | 026 : ; ‘BaIyooo uo soyeyyTy SuyyIs ; piano tema -odspiticiyo: Sabquiw reer f 29.0 | S&T | Fe.92 | 02-920 | 08F (utvor0 ow pus “duITOOD TO ZedNs TTA) ozo.) soqeyyty Surjisodep shea ; : GS-0 | 68-T | S621 | 0@-6Z0T | PLZ ‘S48 4euyyRorg -[8 Ajqvou § zoquae ‘avaqD 39210 syremay aouvivaddy aay 0001 ted nave 0001 tod ae Ajisuay hice Ayarreyry P2PIIIINS OM} 49YJ0 IY} PUY “ARIYNIASUOI a4anr asays fo aa4yg : sKop aay fo uvazy ‘Azuo poof 219Q0}252.4 TX ATIVL ‘OMT, 70 aauuip GYysig yw ysofywarg ‘Suiusoue yxou Jun usyD] spar poof piyos ou ynym 4ayfv “ud om, 10 uayv) uaeg poy cauurp ynyar uo arp poxrue fo sop oy MM VOL, XV. DR. ROBERTS ON THE DIURNAL 260 ‘on0T aod 08.0 Jo cotdap oy 07 OUTTVY[e SBA OAT OF INOJ WOAT qu, *spryos ou yng ‘rayeM Jo sdno 9e.1q4 10 omy TT pue J veanyoq ‘1ayeM “SJOLTvd LO $904 -vjod ‘prorq ‘pyres ‘dnos giqvyesoA “Z ye JoUUI *“pearq pue “uree10 ou yng ‘aeSns [ITAL ao °S 4B 4ysvpyeorg 66:0 + &V.0+ 69-0 + 89-0 + 09-0 + SLO+ €3-:0 + *9[qe} OY} UI poytesoador se uoTjovoI plow we aAvs sjonpoad poytun ou} yer} os fanoy aod ZF.O pue ‘QOOT aod 94.0 JO daadop 0} 04 plow’ oMMODaq PLY YI XIS puv oAy WoaMmyoq ynq f AnoYy rod 1Z.9 pue 63-6 + 9L6+ 06-0 + 9LT+ ¢ 0$:0 + OF-0- 86-0 — LV. ~ 6v-0 — Gh..+ d 88.0 + 9L-T + Vv-0+ 00-0 09:0 - 06-1 + GP-0 + G LL-0+ 6LTL+ €0-T + 00-0 18:0 - 66-1 + 69-0 + 0¢-0 + €4.0+ @LL+ 86.0 + GL.0 + 16-0 + 66-0 + *suo1qaod OM} UI passed svM XIS 0} INO MOI oUTIN OT, % UST+ Le..+ 94-0 + OV-L + 08-0 + 96-0 + 8:0+ 09.1 + 89-0 + GL:0+ 82-0 + ¢1-0- 82:0 + PrO+ PIC ammoy RCEe Ne | 0001 49d | — | Aep Uta ‘OL Wore — Aep yymog amoy Id % ‘dd | 000 49a ~ 80-1 + LL-0+ SP-0- 69-0 + 18-0 + oLo+ { 98-0 - 9L-T+ 64-0 + 62-1 + 64-1 + 49.0 + 00-0 96:0 + 64.0 + 08-0 + 00:0 08-L+ 681+ 90-.T + 98.0 + SPO + 00-T + 89.0 + 94-0 + PoL+ 01-0 + Pet GV-O+ amoy rod 0001 19d $$ rer _ Aup pau, ‘9 ‘qad Gh + 69:0 + anoy req. | 0001 ted ———— _ Anp puosag *¢ "Qa ‘(as0f2g sv pasn susis snuipy puv sng 24. ) TX 4qnZ Sursoguos stop anol ayy of Mppip ay) ur suoywLADYA ays SpQIYxd ‘TTX ATA VL ~o-T+ 040+ moy Jog Aep 4st “py “qed G66 + O00T 19d =) VARIATIONS IN THE ACIDITY OF THE URINE. 261 On the first day of this set the acidity fell in a very marked degree both after breakfast and dinner; on the second, the urine became neutral after breakfast for an hour, and alkaline and cloudy for two hours after dinner, and continued neutral yet another hour; on the third day it was alkaline after both meals; and on the fifth morning a breakfast of four pieces of dried toast and a cup of coffee without milk made the water passed two hours after turbid from phosphates, and highly alkaline ; it lost its acid re- action after dinner also on this day. Here lies abundant evidence that vegetable food is able to depress the reaction of the urine equally with mixed diet. 3. Effect of purely Animal Food. As in the case of vegetable food, two sets of experi- ments were made to ascertain the power of purely animal food to depress the acidity of the urine. The first set did not embrace the twenty-four hours, but began at seven in the morning and ceased about ten in the evening. Four days are included in this set; three of them alternated with days on which vegetable food only was taken at breakfast and dinner, but a supper of mixed food with ale or porter was taken at night. DR. ROBERTS ON THE DIURNAL 262 *1oyea “PLOT ‘gsaoto “quay “p ye touul “194844 PUB y[tur ‘sda “quay *g qe 4svyyeorg 910 *(sdup omy) “prow 300 ‘QuTTey[e 9OTA 7, *(sdup cory.) ‘(skep ser) ‘AUTTBY[B 9010 ‘plow sour} aoIt[y, “eet hd 901g “plow edIMT, syIeU19y “toquie ‘1wafD) || *"* ‘|| ¥40 | 49-0 || 08-69 | 0¢-9Z0T | OZIT | OT-6 *Ivo[O eoUO “prIqiny aorary, || 69-0 | 99-0 |] 6O-T | 0S-0 || 99-14 | 00.0201 | OFIT | 6-4 *Iwoyo somo ‘ystmorjes : “(efop eorgt) |{ paw prqana f4oa oor} || 060 | 620“ | || F024 | oozzor | ozrT | 4-9 “sequie fed ‘away || *** | ** |] €¢.0 | 880 || ST-€9 | 99-FT0T | OL4T | 9-S “Toque “IwalD) ||" “* | 10.1 | 8&1 || 96-8> | e8-ezot | OFZ | S-F “raqure “aBayO || * SF-T | 20-2 || 10-1F | e-rzor | 098 | F-2% ‘Toque “1ve[O) “* 1 6ZT | P&T || 10-8h | z9-ezor | OF6 | 3-2 ‘require “Iva || “** | *** |} 18.0 | $4.0 || 68-0S | 00-eZoT | OSOT | SI-IT “prqany 2 E e { souo ‘avazo sonny easy s|| 20T | 96T |] 29-0 | TH-0 || 00-29 | gz.reor | OTTT | TT-OT ‘Toque “IBID | 9Z-0 | SE-0 |] €O-T | 79-0 || 90-19 | o4.ezoT | O9TL | OT-6 ‘zoquae daop ‘twofQ |} “* | “* || 98.0 | oT || 69-8F | OO.gzot | 034 | 6-8 *BUITOOD WO sayeqyy Pac s 2 rine ahe sara anya 88.0 | 2&2 |] 69-12 | og.ezor | 088 | S-L | Aep moq ammoy | |}anoy tad} | MOY aouvivaddy zg puma zo, 1 d\!"spyos | Ausuoa | sumo | gine ‘gaynIasuo) you ‘stop Anof fo uvapy “Ano jv sauusp QYysiy yo ysvfypaag poof pouaun Maing ‘TIX ATAVL VARIATIONS IN THE ACIDITY OF THE URINE. 263 TABLE XIV. exhibits the Variations in the Acidity for the days composing Table XIII, (The Plus and Minus signs used as before). Jan, 24, Jan, 26, Jan, 28, Jan. 19, First day Second day Third day Fourth day - oS || — || -— —J II -- a Diet | Per 1000 | jot |/Per 1000] ,Fer. |! Per 100] :Per || Periooo | Per. + 3°14 | +0°77|| + 1°88) +115) +1:56)+0.78)| +3°52|+0:84/| Breakfast at 8. | Eggs and meat, +1°76 | + 0°68|| + 0°97/+1°07| +096] +0-91]) +2:40|+0-66) With milk and — 0°32 | - 0:22|| + 0°68| + 1-10] +060] +0-96|| —0-44|-0°31]) water. + 0°20 | + 0°14!) + 0°52) + 0°68) +0°52)+0°84)| -- 1:36 |—1:05 + 0°88 | + 1°23]| + 0°70} + 0°80) + 0°96|+0-91|| +0:48}+0:29 + 0°86 | + 1-19] + 1:28) + 1-11] +1-20)+1-24]| +2°84)+1-62 +0°98 | + 1:47|| + 2:00) + 1°36) +1°74|+1-44]] +3:30]}+1-55 + 1:23 | + 0:99]| + 1:24) + 0:94)! + 1°68}+1-10|] +0:38|+0°68 | Dinner at 4. + 0°90 | + 1:03] + 0°10, + 0°16) + 0°16 +0-41|| -0-62|-0-62)| Meat, cheese, — 0°15 | - 0°17|| — 0°88) — 1-28] — 0°73) -1-25]| +1:58|+1-39] milk and water. 1 | + 1-99|| — 9°94 - 1:32|| - 0°95] - 1-54 ; nal J +051} +1091! ~ 5.74) _ o-16l| 0-62| —o-zall | +188 [+193 > ma + 1°03) + 0°40) + 0°46 +0°94) + °94) The urine became alkaline each day after dinner, and twice after breakfast. On the third day, after breakfast, the urine sustained its acidity with a scarcely perceptible diminution ; though, as compared with the height to which it rose after the supposed alkaline tide had passed off, the depression was sufficiently marked, and still more marked if the hourly excretion of solids be taken into account, at the contrasted periods. In order to observe the effect of a longer continuance of an animal diet, the supper of mixed food on the previ- ous night was discontinued, and for three successive days animal food alone was taken. The following tables exhibit the results obtained. DR. ROBERTS ON THE DIURNAL 264 ‘TL pur G TWaaMyoq I9}VA JO SSUES VY “TOVVM SY TTUL ‘asoayo ‘yvaytT ‘@ ye qzouUuIg *IOVEM “YTLUT ‘s850 pue 4vaTtl *g 4B ysvyyvorg PIC “1aquIe Yor *Teal[O ‘Taq ULB YSippad “rva[O “MRI4S “IBa[O ‘aqme “ava[O “dequie ‘ava[O op 1) 3) 90TMy “ystaMorfos pue Appnut edt. 7, "ystnorped “Appnyy “raqurie ‘1Ba[D *oulpeyye 29TM4 “plow OTM T, LL-0 | 69-6 ¢ d Ors ‘Ioquie ‘IBa[D *roquie oyed “xea[Q ‘oUTTRY]e 9010 1 pue yeaynou 9000 ‘plow OI T, ‘oul[BY[B aou0 pue prow couo Terynau ddA y, ‘ *kpnoyjo souo } j \aeapo sour} ear, ‘raquae “rRo[D ‘raquae SIRaTO “Taquie “IBe[O SYIVUIY aouvseaddy ‘OM, yw dauuap ysig yo ysofyoarg ‘anynrasuo) asann ynym fo sacyy ‘shop snot fo uvapy ‘Mquo poof joumuup ‘AX ATAVL / i 79:0 | €P-L || ¥2-6o | SL-IZOL | O8P L-T = ¥ 16.0 | 19-0 || 248-8¢ | $2-ETOL | OST6 | TIL . nae TILT | $0-T || G¥-29 | 06-FZOLT | O8OT | TI-6 “- =. 96-0 | T8-0 || 10-19 | 00-TZOT | OLGT 6-4 19.0 | 9F-0 |) LF-0 | 99-0 || 6F-F9 | OL-9GOL | OTIL | 4-9 8¢-:0 | L¥-0 he at 64:8S | 0Z-IZOT | OOST | 9-% “| || S&T | 68-1 || 00-89 | 0S-F20T | OV6 | b% . ‘ 92-L | TPL || ZE-E9 | T€-SZOL | 096 G-GL rx ey 66-0 | 16:0 || 40-F¢ | 18-LZOT | OLPT | GI-IT G&.0 | PFO || T¥-0 | 6F-0 || SZ-6h | GZ-61OT | OST | TL-OL 68:0 | 02-0 || 61-0 | €2-0 || 2&-Zh | O-0Z0T | OL8 | OL-6 — a Z8-0 | 99-L || 09-8 | 00-4201 | OTS 6-8 oe 69-0 | OF-G d id OLG 8-4 moy || anoy moy sod Aep ete | Cass all"Spnos | S49C | semop out VARIATIONS IN THE ACIDITY OF THE URINE. 265 TABLE XVI. exhibits the Variations in the Acidity for the several days composing Table XV. Feb. 3, Feb. 7 Feb. 8, Feb. 9, First day Second day Third day Fourth day re | ——, eH - = Per Bers) Per 7 her Per 1000 jour ||Per 1000 aien | Per 1000) pour ||Per 1000} Rowe + 3°68) + 0°86]] + 2:13/ + 0-79 + 0°78) + 0°48)| + 1°45) + 0°58)! + 2°36) + 1:10|] + 2:04) + 1°12] Breakfast at 8. | — 0°20) — 0°39)! 0°00) + 0:00) + 0°46/+0°37)| 0-00) 0-00) Meat or eggs, 0:00} 0-00)| — 0-44] — 0-35) + 1-56 + 1:22|| + 0-40) +.0-43]| or both, milk, + 0°22) + 0°73}| + 0:80) + 0-60) + 1-28) + 1-18] +137) +117] water. | + 1:04) + 1°33)/ + 1°44) + 1°03)) + 1°20] + 1°34)| + 2-00) + 1°33 Diet + 1°25) + 0°46)| + 2°53) + 0°38 | + 0°46) + 0°51)| + 1°80) + 1°37|| + 1°50) + 1°61)| + 1°80) +1°51|| Dinner at 2. — 0°60} — 0°83}| — 0°76] — 0°81]| — 0:45] — 0°42]! — 0:07| — 0:27] Meat and milk, — 0°57| — 0:97) — 0'34| — 0°36)| + 0°52] + 0°52)| + 0°60] + 0°42/| cheese, water. || + 0°14] + 0°17)| + 1:14) + 1°21) + 0°96} + 1'39)|| + 1:00 +1-07 + 0°62) + 0°76|| + 1:24) + 1:24) + 1°32) + 1:27]| + 1:00) + 1°15]! A cup of water + 0°21} + 0°49)| + 1:06] + 1°18]| + 0°40} + 1:14]| + 0:37) + 0°85]| between 9 and + 1:09} + 0°58)| + 1°52) + 0°75]| + 1°04] + 0-61)| + 2°06 +0°64)| 11. || + 2°25] + 0°72/| + 3°68! + 0°86]| + 2°13] + 0°79]| + 2°30! + 0°72 The urine lost its acid reaction each day, both after breakfast and dinner, except on the third day, and even then it was reduced nearly to the neutral line. The diet used in these two sets of observations was va- riously composed. For breakfast, a mutton or pork chop, or beef steak, and water ; sometimes eggs and boiled milk, and once fried sole and boiled milk. For dinner, roast fowl, partridge or hare, broiled salmon, oysters, beef steak, mutton chop, cheese and milk. On the Effect of Food in general. Inasmuch as all our ordinary articles of diet, whether they be drawn from the vegetable or the animal kingdom, present the same elements of composition beneath a great diversity of outward condition, it might naturally be an- ticipated that their effects on the system would not be greatly dissimilar. In every article of diet, as offered by the hand of na- ture—in the flesh of beasts, birds, fish, and all other forms of animal life; in the seed of the various orders of cereals; in the succulent stems, roots and tubers of fresh 266 DR. ROBERTS ON THE DIURNAL vegetables; in the sweet and subacid fruits of our own and tropical climates — in all these, may be found repre- sentatives of the albuminous, oleaginous and saccharine groups of alimentary substances, together with certain saline ingredients — phosphates, sulphates, chlorides and carbonates, having for bases soda, potash, lime and mag- nesia — whose universal presence sufficiently attests the essential importance of their functions. Nevertheless the wide differences of proportion which are known to exist in the admixture of the organic and inorganic substances in various articles of diet, and espe- cially in the contrasted classes of animal and vegetable foods, prepare us to expect that in the final products of the vital operations there will be found certain peculiari- ties attributable to the nature of the aliment. One of these is the reaction of the urine, which is notoriously dissimilar in carnivorous and herbivorous creatures, being acid in the former and alkaline in the latter. And this difference has been universally laid to the account of the food of the two classes. The urine of the herbivora is alkaline, it is asserted, because they feed upon matter rich in alkaline carbonates, citrates and tartrates, all of which appear in the urine as carbonates. And it has been shown that when these creatures are made to fast, their urine becomes acid. Dr. Cl. Bernard* was able to trace still more decisively the connection between the reaction of the urine and the nature of the food. He found that when rabbits (whose urine is normally alkaline) were fed for some time on an exclusively animal diet, they passed an acid urine; and that its alkalinity was not restored until a vegetable diet was substituted. Dogs also, when restricted to a vege- table fare, secreted an alkaline urine, turbid from depo- sition of phosphates; but when restored to animal flesh * Comptes Rendus, 1846, tom, 22, p. 534. VARIATIONS IN THE ACIDITY OF THE URINE. 267 their urine resumed its natural clearness and acid re- action. It is more easy to reconcile the experiments detailed in the preceding pages with the first of these considerations than with the second. All food was found essentially to affect the reaction of the urine alike, but, contrary to what one would expect, animal food produced usually a stronger and more enduring impression than vegetable food. Yet it may be pointed out as worth notice, that of the three consecutive days of exclusively animal and exclusively vegetable diet, the greatest effect in the former was on the first day, and it fell progressively on the second and third days; whereas the reverse took place on the days of vegetable food. On the first day the urine did not be- come alkaline at all; on the second it was neutral after breakfast and alkaline for two hours after dinner; on the third day it was strongly alkaline for an hour after breakfast and for three hours after dinner. Ordinary food, whether it was purely animal, purely vegetable, or, as was more usual, an admixture of the two, was invariably found to cause a diminution in the amount of acid separated by the kidneys. In the tables there is record of thirty-two days on which the urine was examined both after breakfast and after dinner (with the exception of one day on which the observations did not commence until after dinner); and I have notes, in addition, of the state of the urine on five other days after breakfast and on four days after dinner. The following table shows how often the urine became neutral or alkaline, or sustained its acidity, after these two meals, with various kinds of food. VOL. XV. NWN 268 DR. ROBERTS ON THE DIURNAL TABLE XVII. shows the reaction of the urine at the time of greatest depression after breakfast and dinner on mixed food (twenty days breakfast and nineteen days dinner); vegetable food (eight days breakfast and nine days dinner); and animal food (eight days breakfast and dinner ). Mixed diet Vegetable diet Animal diet Alk. | Neut.| Acid || Alk. | Neut.) Acid || . | Neut-) Acid 2 3 OO Alk Breakfast} 12 | 2 6 2 1 5 3 Dinner...| 17 2 0 3 0 6 8 Out of seventy-two meals the urine became alkaline after forty-five, neutral after seven, and sustained its acid- ity after twenty. After thirty-six breakfasts the urine became alkaline seventeen times, neutral five times, and remained acid fourteen times. After thirty-six dinners the urine became alkaline twenty-eight times, neutral twice, and it continued acid six times. The effect of dinner is thus seen to be very considerably greater than that of breakfast. Indeed after a dinner of mixed or animal food the urine never failed to sink to the neutral line, its acidity being preserved only with vege- table food. The cause of the distinction lay, probably, simply in the fact, that breakfast was a much lighter meal than dinner, and its impression on the system conse- quently smaller. But although the urine preserved its acidity frequently after breakfast, and sometimes even after dinner, there was a notable falling off in the intensity of its reaction, whether regard be had to the degree of acidity per 1000 parts, or the quantity discharged per hour. In one set of experiments only, namely the first on vegetable food, does this appear at first sight somewhat doubtful, and seem to require some additional explanation. The decline in the hourly discharge of acid after breakfast, as seen in Table TX., seems so small that a doubt might be cast on its reality ; but if we compute the hourly separation of acid as it stands related to the hourly discharge of solids, we VARIATIONS IN THE ACIDITY OF THE URINE. 269 shall find that the fall after breakfast is brought out in its true prominence, as the following table shows. TABLE XVIII. The first column is a transcription of the hourly discharge of acid from Table IX. The second column shows the per-centage of acid on the solids at successive hours Srom seven a.m. to four p.m. (Breakfast at Eight). Pe Acid corres- Acid discharged ponding to 100 Time of day per hour grains of solids 7-8 1:20 3°87 8-9 1°35 2°87 9-10 114 2°11 10-11 1:00 1:72 11-12 119 2°12 12-2 1:32 2°32 2-4 1:30 3°82 By this table it is made evident that the solid constitu- ents of the urine, or what, for brevity, may be called the solid urine, became steadily less and less acid after break- fast until eleven o’clock ; from that time its acidity rose as steadily until dinner. At eleven o’clock the solid urine had less than one-half the acidity it possessed before breakfast, or just before dinner. The same result is brought out after dinner, and in about the same degree. Taking the two hours before dinner, and the sixth and seventh hours after dinner, every 100 grains of solid residue had an acidity of 3°65; whereas during the third and fourth hours after dinner (the period of the supposed alkaline tide) the solid residue of the urine had but 1°82 per cent. of acid. The apparently exceptional cases, where meals do not appear to lower the acidity at all, or where the hourly dis- charge rises even for a while after a meal, are thus made conformable to the general result. The taking of a meal greatly increases the excretion of solids by the kidneys; and even if these be of diminished acidity, the quantity passed per hour may overbalance this diminution, and, for a while, actually cause an increased hourly discharge 270 DR. ROBERTS ON THE DIURNAL of acid. For example, on the 25th of January (Table X.), the hourly separation of acid before breakfast was 1-08. During the first hour after breakfast it rose to 1:25; but if the solid matters be brought into consideration, it is found that there is here in reality a depression, instead of an elevation, of the acidity. At the former period the solids had a per-centage of acid of 3°81; whereas at the latter, the per-centage had declined to 3:17. I have been thus at pains to prove that the evidence of the first set of experiments on vegetable food is unequi- vocal as to the depressing effect of vegetable food on the reaction of the urine, not because other proof was want- ing —the second set abundantly supplies that — but in order to uphold the conclusion contended for, that all the observations on the individual under experiment were perfectly concurrent; and that the law, so far as he was concerned, came out absolute and without exception, that food lowered the acid reaction of the urine. There is considerable advantage in comparing the oscil- lations of the urinary free acid with the oscillations in the hourly discharge of solid urine. It is from this point of view that we can best see the relations of the former to the state of the blood. It is not necessary here to go into the proof that the degree of alkalinity of the blood regu- lates strictly the rising and falling acidity of the urine. By adding to the alkalescence of the blood through artifi- cial means, as by exhibiting caustic or carbonated alkalies internally, we are able to depress in corresponding pro- portion the acidity of the urine. On the other hand, also, by exhibiting acid (although this seems less readily ac- complished) we can similarly heighten the reaction. By taking the solid urine as a standard of comparison, we avoid two fallacies which respectively affect the deter- minations per 1000 parts and the determinations per hour. We escape, in the first place, oscillations arising from mere VARIATIONS IN THE ACIDITY OF THE URINE. 271 dilution, which sometimes sink the determinations per 1000 grains almost to zero, although the separation of acid at the time is high. In the second place, we avoid oscillations arising from the varying activity of the kid- neys. I shall presently have to call attention to the columns showing the hourly discharge of solid urine. It will be sufficient here to state, that, after meals, the hourly dis- charge of solids rises very considerably, that at periods remote from meals it sinks again, and that during sleep it falls to a minimum. So great is the oscillation from this sole cause that it gives an entirely false complexion to the columns indicating the acid separated per hour. It makes it appear as if the acidity, after having recovered from the first depression consequent on a meal, rose for a few hours to an unusual height, and then fell away again a second time — as if, in fact, an acid tide succeeded to the alkaline tide previously to the subsidence of the reaction to its normal level. If the eye be cast along the columns of hourly discharge of acid in the various tables, this will be seen at a glance. During the hours of continued absti- nence after dinner it comes into especial prominence. It is seen that when the alkaline tide has subsided, and the acid reaction become re-established, there is exhibited for about two hours an unusually high rate of discharge; in fact higher than at any period of the twenty-four hours ; and that after this again there appears a constant fall which is maintained and increased during the subsequent hours of abstinence and sleep. All the tables concur- rently demonstrate this (see Tables I., III., XI. and XV.), and without the correction here indicated it would lead to an erroneous interpretation of facts. Now if we take the solid urine, and calculate its acidity per 100 parts, we find that after recovering from the de- pression of the alkaline tide the acidity shows no sign of 272 DR. ROBERTS ON THE DIURNAL falling off again until after the next meal. The determi- nations in Table III. are especially worthy of attention in reference to this point. This table gives the mean num- bers for seven successive days, on which the utmost endea- your was made to avoid irregularity or inaccuracy; and the densities were all taken by the specific gravity bottle. If we now place, side by side, the numbers indicating the hourly discharge of acid and the numbers indicating the per-centage of acid in the solid urine, from the hour of seven p.m., when the alkaline tide subsided, to eight o’clock next morning, just before breakfast, the two series of numbers will be seen to be entirely different. In the former there is a rise and then a fall; in the latter there is a continued rise. For comparison, the acidity per 1000 grains of the liquid urine is also added. TABLE XIX. shows the varying results obtained by computing the acidity of the urine in three different ways, namely, per 1000 grains of the liquid urine; per hour; and per 100 grains of the solid urine. (The two first columns copied from Table L111.) Dinner at Two p.m. Per 1000 grains Per 100 grains Time of day | of liquid urine Renee of solid urine 7-9 pm. | 0°55 0:48 1:02 9-11 ,, 0-93 0°77 2°02 tia 1-07 0°62 ee 1-7 am. 1:30 0:38 2°37 3 YS See 0°45 2°50 From the third column, with which the first also is in agreement, we may conclude that when the alkalescence of the blood regains its level, after the influence of a meal has passed off, it continues nearly at that level until the next meal; or, rather, there appears a tendency to a gradual diminution in the alkalescence of the blood as fasting is prolonged; but I scarcely dare to rely with cer- tainty on the calculation for the determination of so nice a point. The other tables of means, Nos. I., XI. and VARIATIONS IN THE ACIDITY OF THE URINE. 273 XV., support the main conclusion as uniformly as could be expected, if it be remembered that the densities were taken by a hydrometer, without correction for tempera- ture, and not with that care and precision which would have been used had it been foreseen that they would have been employed for calculating the solids. The rise in the hourly discharge of acid from nine to eleven and eleven to twelve (eighth, ninth and tenth hours after dinner) is, therefore, entirely due to the fact that the increased activity of the kidneys, called forth by the meal, persists for two or three hours after the blood, and by consequence the urine, has recovered its normal reaction. So that by taking the solid urine as a basis for calcula- tion, two distinct corrections have been shown to be ne- cessary in reading the determinations of acidity per hour. By the first correction, the apparently doubtful or contra- dictory cases, where there was but a slight or no fall in the numbers after breakfast, are made to agree satisfactorily with the general law; and by the second, the apparent ~ existence of an acid tide following in a few hours on the ebb of the alkaline tide is shown to depend on a fallacy. The duration and time-of-setting-in of the alkaline tide were both subject to considerable variations from day to day, and they differed too for breakfast and dinner. Under the term alkaline tide is embraced the whole period of depressed acidity, whether the urine at the time was alkaline, neutral or only of diminished acidity.* The effect of breakfast appeared earlier than that of dinner, and was always distinctly perceptible at nine * It might be objected to this term, alkaline tide, that it is applied to urines which do not become alkaline. But, although in such instances the urine does not altogether lose its acidity, there is only a difference of degree between such and those in which the urine becomes actually alkaline. There is the same movement in both cases; but in the former it does not crop out from beneath the surface of the neutral line, whereas in the latter it makes itself sensible by a change of reaction. 274 DR. ROBERTS ON THE DIURNAL o’clock, that is, within forty minutes after the conclusion of the meal. The urine, however, never became alkaline, nor even neutral, so soon. During the succeeding hour, from nine to ten, the alkaline tide usually culminated; but in about a third of the cases the point of least acidity was not reached until eleven o’clock. Then the tide turned ; and from eleven to twelve the urine was found fast re- covering its reaction, and about one the normal level was generally attained. But although the acidity after breakfast was depressed for a period of from four to five hours, it was not absolutely alkaline usually for more than one hour; generally from nine to ten. Not unfrequently, however, it continued alkaline, and even increased in alkalescence, until eleven ; and on one occasion it continued alkaline for three hours, that is, until noon. After dinner, which usually occupied half an hour, the acidity maintained itself through the first hour (that is, the hour at the beginning of which dinner was taken), but declined during the second in varying degrees on different days; and on five occasions the urine became alkaline. During the third, fourth and fifth hours the alkaline tide ran in its greatest strength. On the third and fourth hours the urine was always (with two exceptions) found alkaline when the meal had been of mixed food or animal diet. On the fifth hour it was also nearly always alkaline, but not so invariably so as on the two preceding ones. At the end of the next (sixth) hour the tide had generally turned and the acid reaction been restored. This change appeared often to take place with considerable suddenness, and the rise of the acidity went on with such celerity that in two hours (that is, at the end of the seventh hour) it had reached the ordinary standard. Three hours was the usual duration of the alkalescent state of the urine after dinner ; sometimes two hours, more rarely four hours, and — VARIATIONS IN THE ACIDITY OF THE URINE. 27 Or one occasion five hours. The amount of free alkali hourly discharged after dinner was, generally, not far from double the quantity observed after breakfast; so that in duration and intensity the effect of dinner proved about twice as great as that of breakfast. The Alkaline Urine of Food. The alkaline urine which followed the taking of food deserves special description. It is important, in the first place, to state that its alkalescence did not depend upon ammonia, but upon a fixed alkali. It had no ammoniacal smell; caustic potash failed to evolve any; and the phos- phates which were thrown down in it did not, when freshly passed, contain any crystals of the ammoniaco-magnesian phosphate. Under the microscope the fresh deposit was always found to be amorphous. No effervescence could ever be observed in the urine on the addition of hydro- chloric acid. As might have been anticipated, the occur- rence of an alkaline reaction determined the precipitation of the earthy phosphates, and the urine, when passed, was frequently turbid. But this was not always so. Not unfrequently, especially after breakfast, the urine, al- though alkaline, retained its transparency. Generally such a urine was of feeble alkalinity and dilute; but now and then it was observed to be tolerably concentrated, highly alkaline, and still clear. All transparent alkaline urines were rendered immediately turbid by caustic am- monia and by heating, so that the transparency did not depend on the absence of earthy phosphates. It was also found that, in the turbid urines after subsidence of their deposits, caustic ammonia caused an additional pre- cipitation. Out of twenty-eight specimens of alkaline urine after breakfast, nineteen were more or less turbid and nine clear. Of sixty-four specimens alkaline after dinner, fifty- one were turbid and thirteen clear. The proportion of VOL. XV. 00 276 DR. ROBERTS ON THE DIURNAL clear alkaline urines after breakfast was, therefore, con- siderably greater than after dinner. The degree of turbidity varied from a barely perceptible cloudiness to a thick muddy opacity. The deposit sub- sided quickly and left a clear, yellowish-amber supernatant liquor, often with a greenish tinge. The odour of this urine was peculiar, and so distinctive that its alkaline reaction could with certainty be predi- cated, without the aid of test paper, simply by the sense of smell. It was altogether devoid of the characteristic urinous odour, and exhaled a strong sweetish aroma re- sembling that of the fresh urine of the horse. The more strongly alkaline it was, the more powerful was this odour, and vice versd. The proportion of alkaline and earthy phosphates in the urine of the alkaline tide was found to be considerably increased. The quantity separated during the two hours preceding dinner and that separated during the third and fourth hours after dinner was subjected to comparison. The hourly separation of the earthy phosphates was found, on an average of six days, nearly doubled after dinner ; and the alkaline phosphates rose from 3°47 grains to 4°90 grains per hour. ‘This increase was not owing to the quickened activity of the secretory organs, for the propor- tion per 1000 of the liquid urine and the proportion per 100 grains of the solid urine exhibited an equally marked elevation. The quantity of wric acid was ascertained for three periods on each of the seven days composing Table III., namely, in the urine of the alkaline tide after dinner (from four to seven) ; in the acid urine passed between nine and eleven; and lastly, in the urine of sleep. The average for the seven days, at each of these periods, may be seen from the following table. VARIATIONS IN THE ACIDITY OF THE URINE. 277 TABLE XX. shows the mean amount of Uric Acid separated on the seven days composing Table [11—during the alkaline tide after dinner ; from nine to eleven p.m.; and during the night. Uric acid Uric acid Uric acid Time per 1000 grains per 100 grains of liquid urine perbour of solid urine 4-7 p.m., alkaline tide 0-40 0°36 0-83 9-11 p.m., acidity restored 0-18 0-13 0°34 1-7 a.m., urine of sleep 0°39 0-10 0-60 The urine of the alkaline tide was, therefore, rich in uric acid (as calculated per 1000), rather more so even than the night urine, which in nearly every instance de- posited urates copiously on cooling. The hourly quantity was almost three times greater than during the succeeding period when the acidity was restored (from nine to eleven), and more than three times greater than during sleep. The differences are not so great when the per-centage of uric acid in the solid residue is calculated; although here, also, the alkalme urime gives notably the highest figure. The urine of the alkaline tide may, therefore, in every sense be regarded as pre-eminently charged with uric acid. The remarkable poverty of the urine in uric acid, from nine to eleven, not only as contrasted with the urine of the alkaline tide but also with that of sleep, is probably closely connected with the increased excretion of uric acid during the preceding hours of the alkaline tide. If it be true, as these results lead us to believe, that a diminished acidity or, a fortiori, alkalescence of the urine is very fa- vourable to the separation of uric acid from the blood, and causes an increased quantity of it to pass through the kidneys, it follows that on the cessation of the alka- line tide the blood must be unusually poor in uric acid; and the separation of it by the kidneys therefore, on the 278 DR. ROBERTS ON THE DIURNAL re-establishment of the acid reaction, must be propor- tionally diminished. It is also extremely probable that there is, to some extent, an increased production of uric acid in the blood after food, and that a portion of the increased elimination is due to this. Remote Effects of a Meal. Although, as we have seen, the immediate effect of a meal was to depress the acidity of the urine, the more remote consequence was to uphold and even to increase the acidity. It has already been pointed out, that if we take the amount of acid separated per hour as our standard of comparison, the quantity discharged was greater during the period immediately following the re-establishment of the normal reaction than at any other period either pre- vious or subsequent. And this has been explained to have depended on the high activity of the kidneys at that time. But there is another and still more remote effect of a meal, which comes out under different relations, and which is seen most distinctly, when a comparison is made between the acidity of the urine on mornings suc- ceeding supperless nights and that of the urine on morn- ings following a hearty supper. In the former case, the mean hourly rate of acid discharged between seven and eight a.m. was only 0°51; im the latter it was 0°88, or nearly double. And not only was the hourly discharge thus increased, but even the degree of acidity per 1000 showed a slight rise—the mean numbers being for the mornings after supperless nights 1:83, and for mornings after a supper 2°15. This latter calculation, however, did not always exhibit results in accordance with this general mean, and exceptions occurred; but for the hourly calcu- lation, all the separate results were consistent, sometimes in a greater, sometimes in a less degree, with the general mean. It is important to bear these particulars in mind, for VARIATIONS IN THE ACIDITY OF THE URINE. 279 they enable us to explain certain irregularities and discre- pancies which appear when the effect of a meal on one day is compared with the effect of a similar meal on another day. For it frequently comes to pass that the remote effect of a previous meal interferes with, or completely masks the immediate effect of a succeeding one. Here, no doubt, lies the cause of the slight effect produced by vegetable food in the first set (Table IX). The remote effect of the highly animalised diet of the previous days masked the immediate effect of the meals on the days following. It is a marked feature in these experiments, that, al- though there was the greatest constancy and certainty in the successive changes of reaction, there were very consi- derable — indeed the widest — differences in the absolute amounts of free acid or free alkali separated by the kid- neys at corresponding hours after a meal on different days. For example, in Table IV., the acid passed between eight and nine a.m. on the first day was more than four times greater than on the third day at the same hour; and this, although the breakfast on the two days was as nearly as possible the same both in quantity and quality. Two considerations may be offered in explanation of this discrepancy. First, there was the supper on the night preceding the first day ; secondly, digestion and absorption of the meal was probably going on more rapidly on the third morning than on the first, from the comparatively empty state of the vessels, consequent on the long pre- vious abstinence. But these explanations by no means meet all the cases of irregularity, as is shown by a comparison of the state of the urine from ten to eleven a.m. on the fourth and fifth days of the same table. On the fourth the urine was neutral at this period, whereas on the fifth it was acid to the extent of 0°20 per hour; yet on both days the antece- dents, so far as food was concerned, were identical. Doubt- 280 DR. ROBERTS ON THE DIURNAL less there are secret causes of unequal action in the animal system of so subtle a nature as to be altogether beyond our present powers of appreciation. I would suggest, however, that the state of the body in respect to repose and exercise, and the external temperature, probably exert an important influence. Muscular juice is highly acid, and its quantity is, in all likelihood, greatly increased by exercise; and it seems not improbable, as Vogel has suggested, that the degree of acidity of the urine has some connection with the quantity of this acid of muscle thrown into the blood, and so through the kidneys out of the body. The external temperature, too, by quickening or re- tarding the cutaneous transpiration and the respiratory fuuction, may affect the amount of acid circulating and generated in the blood. Differences in the nature of the food, especially as re- gards the proportion of earthy and alkaline phosphates, may almost with certainty be named as operative in caus- ing differences in the effects of different meals on the reaction of the urine. This and other matters connected with this part of the subject will come under discussion again when the inquiry is entered on — Why should a meal depress the acidity of the urine ? The amount of free acid separated in the course of the twenty-four hours was fouud, on an average of nineteen days, to be sufficient to neutralise 14°10 grains of dried carbonate of soda, or an average of 0°58 grains per hour. The maximum quantity was 22°34, and occurred under a purely animal diet ; the minimum was 5-90 under a mixed diet. Some days were found exhibiting throughout a feeble acidity; others a high acidity, quite independently of the nature of the diet. The average amounts of acid per a oe — VARIATIONS IN THE ACIDITY OF THE URINE. 281 twenty-four hours for the different sets were: mixed diet (Tables I. and III.), 14-21 and 10°30; purely vegetable diet (Table XI.), 15°36; and purely animal food (Table XV.), 18:03. These numbers show that the daily amount of acid eliminated by the kidneys was not much or uniformly affected by the nature of the food. The smallest numbers occurred under mixed food; but I am not disposed to at- tribute this to the nature of the food so much as to other circumstances, inasmuch as on some of the days of mixed food the acidity ruled unusually high, but the averages (especially of those of Table III.) were greatly reduced by one or two days of very low acidity. The degree of acidity per 1000 grains is of more impor- tance, practically, than the amount per hour, inasmuch as the occurrence of urinary deposits depends on it. If no liquids were taken, the degree of acidity after the pass- ing away of the alkaline tide, gradually increased until food was again taken. The highest acidity, therefore, was always found after the longest fasting, or just before break- fast and dinner. Between seven and eight in the morning the urine was uniformly found excessively acid, invariably depositing abundance of urates on cooling. The night urine likewise, except when liquids were taken on going to bed, was highly acid and sedimentary. It may be easily conceived, therefore, that this is the period most favour- able to the formation of renal and vesical concretions. The urine flows slowly aud rests for a lengthened period in the bladder, while its excessive acidity and concentra- tion diminish its solvent powers over oxalate of lime, uric acid and the urates — the three substances most liable to unuatural precipitation. When no liquids were taken before going to bed the urine of sleep had an acidity varying from 1°50 to 2°16 per 1000. In the morning, before breakfast the numbers ran from 1°50 to 2:80 and 3:00, rising on one occasion to 3°68 282 DR. ROBERTS ON THE DIURNAL per 1000. The mean acidity, taking all the hours during which the urine flowed acid, was 1:13 per 1000; if the hours of sleep, the two hours before dinner and the hour before breakfast be excluded, the mean acidity for the remaining hours of the acid flow falls to about 0°80 per 1000. The amount of free alkali passed in the twenty-four hours varied according to the duration and intensity of the alka- line tide. On the nineteen days represented in Tables I., IIL., XI. and XV. the urine was alkaline, on an aver- age, for more than three and half hours each day, and contained a quantity of free alkali equal in saturating power to 3°32 grains of dried carbonate of soda. The mean quantity discharged per twenty-four hours, with mixed food, was 4°14 grains for the first set and 4°72 grains for the second (Tables I. and III.) ; for vegetable food, 1:71 grains (Table XI.) ; and for animal food, 1-68 grains (Table XV.) The numbers for the separate days ranged from zero to 9°35 grains, and great irregularities prevailed throughout, even when the days were consecutive and the diet the same. It is to be remarked that the days of mixed diet showed a greater discharge of free alkali than those of purely animal or purely vegetable diet. This corresponds with the lower figure representing the free acid on those days. The hourly discharge of free alkali oscillated from 0-00 to 2°60; and it usually ranged between 0°50 and 1:00. The degree of alkalinity per 1000 grains varied also in the same manner. The highest grade observed was 4:12; but the usual numbers were from 0°40 to 1:60. In comparing the effects of the three different kinds of diet it must be owned that the results are very contradic- tory. In the first set of experiments on vegetable food the urime did not once become alkaline (Table IX.) nor VARIATIONS IN THE ACIDITY OF THE URINE. 283 even neutral. This must be attributed to the remote effect of the flesh meat diet used on the alternate days. In the second set (Table XI.), however, vegetable food was found to possess a great and apparently increasing power of depressing the acidity of the urine when persevered in for successive days. So that the daily average discharge of alkali for the four successive days of vegetable food was raised even above that for successive days of animal food — the mean total amount of free alkali per day in the former case being 1:71 grains and in the latter 1-68 grains. In the first set of observations on animal food, however, the mean daily discharge of free alkali was 2°17 grains. But the effect of mixed food was found on an average considerably greater than that of purely vegetable or purely animal food, both in duration and intensity— the mean daily separation of free alkali being 4°14 grains and 4-72 grains respectively for the first and second sets. This contradictory result appears at present quite inexplicable, for there did not seem to be any difference in the rate of absorption nor in the quantity of the meals. The degree of acidity before meals (in other words, the remote effect) was found greatest after animal food, while the difference between purely vegetable food and mixed food was not very considerable ; the numbers on the whole, however, being favourable to vegetable food. In the sub- joined table may be seen the degree of the acid reaction during the hours of fasting, before breakfast and dinner, with the three kinds of food. TABLE XX, exhibits the Acidity of the Urine before meals, with the different kinds of food. om 1 to8a 12 noon to 2 pym duane breakfast) (Before dinner) Mixed food | 1-47 per 1000 | 0: 3 per hour 4 85 per 1000 | 0°74 per hour Vegetable food|/1-52 —s,, 0°4: a "OL, Oe 1:20 Fe Animalfood |2-01 ,, 0’ 70 | - AB iis 1°27 iy VOL. XV. PP 284 DR. ROBERTS ON THE DIURNAL Before proceeding to discuss the explanation to be offered of the power of a meal to depress the acidity of the urine, attention must be called to the variation in the amount of solid matter—%in other words, solid urine — secreted at different hours of the day. The numbers were obtained, as already explained, by calculation from the density. With the exception of the seven days included in Table III., the specific gravity was taken by a hydro- meter, and not with that scrupulous care, nor under that correction which is required for calculating the solids. While therefore able to state that all the tables show re- sults remarkably concurrent, I am inclined to place most reliance on the determinations in Table III., which were made with the necessary precautions. On passing the eye down the columus of solids in the six tables of means, it is seen that the quantity per hour began to increase within an hour after breakfast, that is, between eight and nine, and went on increasing rapidly until eleven. It re- mained stationary until about noon, and then began to fall, and continued to diminish until dinner (see especially Tables I., IX. and XIII., when dinner was at four), and even for an hour after. At the end of the second hour after dinner there was usually, but not always, a slight rise. During the third hour there was a very decided rise, which went on increasing until the fifth or sixth hour, about which time it reached the culminating point. From that time the discharge of solids fell gradually until bed- time, and sank to the lowest point during the hours of sleep. All the tables of means exhibit the same process- sion of numbers, whatever the nature of the food might be. Exceptional instances occurred, it is true, here and there; but with so great a multiplicity of details it was impossible altogether to avoid errors, from faulty obser- vation, calenlation or record. The contradictory facts, however, were quite insignificant beside the overwhelming majority of concurrent observations. VARIATIONS IN THE ACIDITY OF THE URINE. 285 Now, what does this increase in the solids of the urine after a meal indicate? It evidently marks the passage of digested food into the blood. We see reflected in the renal secretion, as in a mirror, the flow of aliment into the blood. First the small beginning, then the increase, then the full tide, followed by a gradual ebb as the last portions of the food continue to be taken up from the lower divi- sions of the alimentary canal; until at length, after all has been absorked, the blood returns to its condition as before a meal. No doubt the passage of food into the blood is not the only circumstance that causes an increased separation of solid urine. Active functional exercise, whether of muscle or brain, probably quickens the activity of the kidneys by throwing into the blood an increased quautity of effete materials — the products of the destructive metamorphosis of the muscular and nervous elements. The invariably great and sudden rise in the solid urine from eight to nine in the morning is certainly not entirely the effect of breakfast, as it is incredible that digestion and absorp- tion should have made such progress in forty minutes. The awakening of the whole system from the torpor of sleep— the aroused mental and muscular activity, and the consentaneous acceleration of the circulation and respira- zion — contributed, unquestionably, in producing the re- sult. No such sudden rise is seen at any time in the subsequent hours, nor after dinner.* The question now arises: How far is the depressing action of food on the acidity of the human urine a univer- sal phenomenon? It is impossible to solve this question * It may be noted that the increase in the hourly secretion of solid urine after dinner was not so prominently marked as after breakfast. I believe this arose from the almost complete inactivity usually observed by the subject of experiment for two or three hours after the former meal, whereas after breakfast his condition was one of activity. 286 DR. ROBERTS ON THE DIURNAL except by greatly multiplied observations conducted on the principle of hourly determinations. But the facts at pre- sent known warrant the statement, that at least some healthy persons, whose urine presents no appreciable peculiarity, suffer constantly a diminished acidity of urine after meals, and that the secretion may flow alkaline and turbid for four and six hours daily without the least evidence of impaired function. I am disposed to believe that the depression of the acidity of the urine after a meal is of universal occur- rence ; but, at the same time, observation leads me to con- clude that there are very great differences in the degree or intensity of the phenomenon in different individuals — these differences arising, apparently, from fixed peculiari- ties of constitution. Without pretending to be able to point out all of these, or to estimate their influence, the following may be indicated as probably tending to pro- duce that effect. 1. The cutaneous transpiration varies greatly in differ- ent persons. Those with an active skin eliminate a large amount of acid by the cutaneous surface, and consequently have less for removal by the kidneys. In such persons the alkaline tide might be expected to flow with greater intensity than where the skin is habitually dry and torpid. The sweat seems incapable of ever changing from acid to alkaline, for I have invariably found it sour, even when the urine has been rendered strongly and uninterruptedly alkaline by bicarbonate of potash for four and five weeks continuously. 2. The respiratory capacity of individuals varies. This bears on the present question in this way. Recent obser- vations, especially those of Dr. E. Smith,* show that shortly after a meal the respiratory function is materially * See his Papers read before the Royal and Medico-Chirurgical Societies during the present year, 1859. Neate be Dae VARIATIONS IN THE ACIDITY OF THE URINE. 287 quickened ; more carbonic acid is exhaled, and conse- quently more oxygen absorbed into the blood. Those persons in whom this increase takes place promptly after a meal, and is unusually great, would probably show but a feeble alkaline tide, because the oxygen, thus thrown in excessive quantity into the blood, would cause increased formation of acid, and in this way mask the contrary effect of the food. 3. The quicker the digestion and the absorption of a meal, the greater cwteris paribus would be its depressing effect on the acidity of the urine. It seems not unlikely that the two last mentioned causes determine by their mutual relation in an especial manner the degree of intensity assumed by the alkaline tide. A rapid absorption of a meal and a small amount of respi- ratory acceleration would present the combination most favourable to an intense depression of the urinary acid. On the contrary, a slow absorption of a meal and a prompt exaltation in the respiratory functions would so balance the opposing tendencies that the reaction of the urine would suffer only a minimum of depression. These modifying circumstances apply not only to differ- ences between individuals, but, in a minor degree also, to differences in the same individual at different times. This seems the proper place to offer some considerations which may explain why alkaline urine after meals is not oftener met with in actual experience; and whence arise the grave discrepancies in the experience of different observers in this country and in Germany. It is essential, in order to trace the effect of a meal, fo examine the urine at short intervals. For if this be neg- lected the acid product secreted before and after the period of depression becomes mixed in the bladder with _the urine of the alkaline tide, and when the whole is ejected by micturition it is found acid, even although a 288 DR. ROBERTS ON THE DIURNAL portion of it was highly alkaline as it left the pelvis of the kidney. And micturition seems to occur, in the usual course, at such intervals as most effectually to prevent the alkaline urine of food from being observed unless by an unusual accident. The first micturition of the day usually takes place on leaving bed, and the urine is highly acid. The second does not occur, unless the bowels be emptied after breakfast, for some five or six hours; and it includes some very acid urine secreted before and immediately after breakfast, together with the urine of the morning alkaline tide, as well as the secretion with recovered acidity which is produced for two or three hours after the subsidence of the alkaline tide. Such a urine is sure to be acid, not- withstanding an alkaline flow through the kidneys for one or two hours. The same thing occurs after dinner. The bladder is usually emptied before the meal, and then for a couple of hours the urine flows acid. If there be excessive potation the urine may require discharging at this period, and it will always be found acid. If, on the contrary, potation has been restricted, or the system drained of water at the time of the meal, the next micturition may be delayed an hour or two, so as to cut the alkaline tide in half. Even then the urine will be acid, unless the depression was unusually great. The next micturition may occur after tea, and the recovered acidity of the urine would then conceal the alkalinity of the portion secreted at the beginning of the period. From these remarks it may be gathered that unless the product of the alkaline tide be isolated, by emptying the bladder before and after its flow, no reliance can he placed on observations concerning it. This is, I apprehend, the reason why the observations of Beneke and Vogel, and probably also those of Dr. Sellers, have failed to support the conclusions of Dr. Bence Jones. : In Dr. Beneke’s experiments the bladder was not emptied VARIATIONS IN THE ACIDITY OF THE URINE. 289 oftener than about five times in the twenty-four hours ; and by comparing the times of the meals with the times of micturition, it becomes evident that it was impossible for him to obtain results other than nugatory. The urines he examined were mixed urines, and he did not in any wise isolate the secretion at the critical periods. The same objection applies to the observations con- ducted under the supervision of Vogel. The urines were collected during three periods— namely between breakfast and dinner (morning urine), between dinner and evening (afternoon urine), and during the hours of night. All such urines would be acid mixtures ; but it by no means follows that they did not pass through an alkaline state, possibly even of some hours duration. So true is it that the existence of the alkaline tide may be concealed for an indefinite time, even from those who are in the constant habit of observing the state of the urine, unless the urinary product be, as it were, analysed by fre- quent micturitions, that the urine of the subject of these experiments, though under close observation for some years, was not once known to have departed from its usual acid reaction; and it took me almost by surprise to find the phenomena of the alkaline tide so strongly and so remarkably pronounced after such long and effectual concealment. A second circumstance, which must be borne in mind, is the remote influence of meals. As already fully ex- plained, this remote effect of a previous meal frequently altogether masks the immediate effect of a recent one; and this statement seems especially to apply to the effect of supper on the succeeding breakfast. For this reason it is well, in order to obtain distinct results, to fast for eight or ten hours, or more if night intervene, before taking the meal whose effect it is wished to observe. Finally, we come to the inquiry: Why should a meal 290 DR. ROBERTS ON THE DIURNAL depress the acidity of the urine? Dr. Bence Jones im- putes it to the diversion of the acid present in the blood to the stomach for the purposes of digestion. When the stomach is empty its ling membrane is neutral, or nearly so, and the acid generated in, or thrown into, the blood passes off by the skin and kidneys, rendering the urine acid; but when food is taken, acid gastric juice is poured from the blood vessels into the stomach, and the alkali- nity of the blood is consequently raised. This causes the kidneys to separate a less acid or even an alkaline pro- duct; but when digestion is completed, the gastric acid returns to the blood with the chyle, and the urine regains its normal reaction. So that the stomach and kidneys are antagonistic in their reaction, the former being least acid when the latter are most so, and vice versd. So plausible an explanation was at once adopted; and it is difficult to imagine that it may not have some operation in the way supposed, though, as I believe, it must be re- jected as the main cause of the alkaline tide. From Dr. Beaumont’s twenty-fifth and twenty-sixth ex- periments, second series, we are led to conclude that suffi- cient gastric juice for the digestion of a meal is poured into the stomach within fifteen or twenty minutes after its ingestion ; and that in half an hour, or an hour at most, the flow of acid into the stomach has ceased and absorp- tion of the digested meal commenced. If this be so, the reaction of the urine ought to be most depressed within an hour or an hour and a half after a meal, instead of from two to five hours after. Moreover, how can it be explained, on this supposition, that the alkaline tide after dinner sets-in an hour or an hour and a half later than after breakfast ? But, whatever may be thought of the validity of these objections, is there not another solution more consonant with the facts observed? Is the alkaline tide not the VARIATIONS IN THE ACIDITY OF THE URINE. 291 effect of the absorption of a meal into the blood, rather than of digestion? If it be true, as Liebig maintains, that the alkalescence of the blood —and in all animals that possess blood its reaction is alkaline — depends simply on the chemical composition of the alimentary sub- stances, is there not here a solution for our question ? In his twenty-eighth Letter,* Liebig points out that phosphoric acid and the alkalies are present in such pro- portion in bread, meat, and our ordinary food, that if we suppose them dissolved the alkalies invariably prepon- derate. Hence arises, he says, the alkalinity of the blood. If this be so, every meal that is dissolved and absorbed into the blood must increase the alkaline reaction of that fluid and raise it for a time above the natural level. But it is well known that when salts of the fixed alka- lies which have an alkaline reaction — such as carbonates, basic phosphates and borates, or vegetable salts, which become carbonates in the system — are artificially exhi- bited, they change the reaction of the urine from acid to alkaline; evidently from inducing an excessive alkales- cence of the blood, which it is the function of the kid- neys to diminish by allowing the excess to escape in the urine. Conformably to this hypothesis, the earthy and alkaline phosphates were found greatly increased in the urine after meals. A meal, therefore, viewed in this light, is a dose of alkali, which, when digested and absorbed, necessarily adds to the alkalinity of the blood; and, as a more remote but equally inevitable consequence, lowers the acidity of the urine or, if in sufficient quantity, renders it actually alkaline. It has been already pointed out that the set- ting-in of the alkaline tide coincides, in point of time, with the passage of the digested food into the blood, as indicated by the increased amount of solid urine secreted * Familiar Letters, Letter 28. VOL. XV. QQ 292 DR. ROBERTS ON THE DIURNAL by the kidneys; and as absorption goes forward and in- creases, the acidity of the urine diminishes more and more. It is nevertheless true that the subsidence of the alkaline tide is not synchronous with the cessation of absorption, for we have seen that the passage of food into the blood appears at its highest activity when the alkaline tide is beginning to ebb. This want of coincidence appears, prima facie, to militate against the solution here offered ; but it may be explained in two ways. Either it arises from the phosphatic salts being absorbed with more cele- rity than the rest of the food, and producing their effect before the other materials are all taken up; or, more pro- bably, it depeuds on the increased absorption of oxygen by respiration, already noticed as occurring after a meal, which, after the lapse of five or six hours, by generating acid, counteracts the contrary effect of the food—in other words, from the remote effect of a meal overlapping the immediate effect. If this solution be admitted, it brings all ordinary food into the same category with sub-acid fruits, which have long been acknowledged to possess the power, in virtue of their saline constituents, of rendering the urine alkaline ; the only difference being that in the latter the effect is produced by salts, which become carbonates in the blood, and in the former by basic phosphates, which pass as such into the urine.* * It must not for a moment be supposed that the urine is never alkaline (from fixed alkali) except after food. The urine not unfrequently loses its acid reaction in disease, independently of food, and presents to all appear- ance the characters of the urine of the alkaline tide after a meal. Ihave observed this character of the secretion repeatedly in the debilitated and — ansemic condition which sometimes follows chronic subacute gout and obstinate subacute articular rheumatism; also in the course of some other protracted and exhaustive diseases that induce a chlorotic or spanemic state; of which perhaps the most common is a certain form of atonic dys- pepsia. In such cases the urine may be alkaline all the day through, though . '. a, tk ee VARIATIONS IN THE ACIDITY OF THE URINE. 293 The results of the foregoing observations may be sum- med up in the following propositions : — 1. The immediate and primary effect of a meal, whe- ther of purely vegetable, purely animal, or mixed food, was, in from one to three hours to diminish - the acidity of the urine; and very frequently to render it alkaline. The term “alkaline tide” is suggested to designate the period of depressed acidity. 2. The remote or secondary effect of a meal was to uphold and increase the acidity of the urine. This effect of a meal was especially observed over-night after supper. 3. The remote effect of animal diet appeared consi- derably greater than that of vegetable food. So that a highly animalised diet tends in the long run to heighten the acidity of the urine. 4. After breakfast, the greatest depression occurred at the second hour; and the period of depression continued from two to four hours. 5. After dinner, the greatest depression occurred at this is very rare. In my experience I have not found such a condition con- tinuing from day to day for any length of time, but rapidly passing away, not lasting more than some hours or a day or two; but perhaps returning again and again. Such urines must be carefully distinguished from ammo- niacal urines, which invariably indicate some disorder in the urinary pas- sages, generally in the bladder. Another cause of alkaline urine is the immersion of the body in water. Homolle and Duriau found that after a bath the urine always lost its acid reaction, even when nitric acid had been added to the bath.— Archiv. Génér., T. ii. 1856. I am also convinced that an overworked and depressed state of the system promotes a diminished acidity of the urine; and that a high state of health and yigour tends to heightened acidity. 4 The explanation offered in the text, therefore, applies only to one particu- lar case of alkaline urine, and must not by any means be taken as a general explanation. 294 DR. ROBERTS ON THE DIURNAL the third, fourth and fifth hours, and lasted from four to six hours. The effect of dinner was greater, as well as more prolonged, than that of break- fast. . The effect of mixed and purely animal diet seemed almost identical. Vegetable diet, when used on alternate days with mixed or animal food, had a decidedly feebler effect ; but when used contiuu- ously for several days successively its effect was equally powerful. . Alkaline urine after a meal owed its reaction to a fixed alkali. It was generally, but not always, turbid, when passed, from precipitated phosphates. Its odour resembled that of the fresh urine of the horse. It was richer in uric acid and in earthy and alkaline phosphates than the urine of fasting. . The depression of the acidity after a meal coin- cided in point of time with absorption rather than with digestion. The solids of the urine began to increase simultaneously with the declension of its acidity. So that the passage of food into the blood and the diminished acidity of the urine seemed to be connected together as cause and effect. The following deductions appear also to be warranted : — 1. That the power of a meal to depress the acidity of the urine depends on its mineral constituents. These contain phosphoric acid and the alkalies in such proportion, that if we suppose them dissolved the alkalies invariably preponderate. Hence arises the alkalinity of the blood. If this be so, every meal that is dissolved and absorbed into the blood must for the time raise the alkalescence of that fluid above the natural level. VARIATIONS IN THE ACIDITY OF THE URINE. 295 2. But the kidneys have the special function of regu- 5. lating the degree of alkalescence of the blood. When it is too high they separate alkali, and the urine becomes alkaline ; when it tends to become too low, on the other hand, they separate acid, and this gives to the urine its common acid re- action. A meal then, in so far as its mineral ingredients are concerned, is but a dose of alkali, and its ab- sorption causes, like any other dose of alkali, a depression of the acidity of the urine. . The emission of urine turbid with phosphates is, within certain limits, a natural phenomenon ; and ; earthy phosphates constitute the only urinary de- posit which can appear in the healthy urine on passing. Urines may be divided into two chief classes : — First, urines of fasting (urine sanguinis) ; secondly, urines of food (urine cibi). Fasting urines are scanty and of high acidity; they present only one variety, namely, that of sleep, which differs from other fasting urines in possessiug more colouring matter. The urines of food fall naturally into two divisions — those with a diminished, and those with a restored acidity; they are abundant in quantity. The urines of either class may be concentrated or dilute, according to the relation between potation and the requirements of the system in regard to water. So that urine potis do not merit to be regarded as a distinct class. Most commonly the urine of micturition belongs exclusively to no divi- sion, but is a mixture of several kinds. 296 DR. ROBERTS ON THE DIURNAL APPENDIX. On two days the effects of cane sugar and honey were tried. Neither seemed to produce any depression of the hourly quantity of urinary acid discharged. On the first day a quarter of a pound of loaf sugar was dissolved in water and taken at eight am. At two p.m. half a pound was taken in the same way, instead of dinner. No solid of any sort, or other liquid than water, was taken during the experiment. On the second day half a pound of honey was taken for breakfast at eight in the morning, with water; and no _ solid food was again taken until the conclusion of the experiment. The annexed tables exhibit the results obtained. 297 VARIATIONS IN THE ACIDITY OF THE URINE. *g 4@ 104A qt Aouoy jo punod v jv PIC punod wv jyepy *g 48 1909eM pue ‘avsns jeoy jo punod ® Jo aojaenty "Z ye TO¥eM pue “1esns jo PI ‘soqnuror £4107 Jo yonpoad oy ATpeor svat SIT, » “Auepunge soqzeyqy ty Suyrsodop £ aoqure yor ‘awozQ || ¢.0 | ZzT || 94-4z0T “Suryisodep you ‘aoquae oped ‘aeaqQ || PE-0 | 04-0 || PP-910L | OSb | 9-F “Surptsodop you f raquae oped “aea{D || 9F-0 | 92-0 || 99-F00T | O9ZL | Fz ‘Surptsodap you ‘aoquie ‘aeetO || 4F-0 | 12-0 || #Z-9T0T | 099 | Z-IT “Surjtsodep you faoqure ‘reetQ || O¢.0 | 99.0 || ze-etot| og4 | TI-OT “SuITOO TO sogeyy] Surpisodop ‘ coqure ‘awoIQ || 09.0 | O&-T || 8z-2ZZz01 | Ser | OT-6 “Sul[ooo uo saquqy] Suyytsodop ‘ zoqure ‘avojQ |} 49.0 | ZhT || P8-Z20T | Z8e | 6-8 “SUI[OOD UO soyVIAIT Suryisodep doqure ‘aeatD |} 4-0 | SFL || F9-FZOT | 898 S-L e z ~ |\anoy seq! 000T 10d Moy Ap aourireddy Ayrproy A809 | gmoyy go ‘at0fag qysiu ayy aadgns ony “Mauogy ‘TT ATIVLI ‘SUIpUTys wo saqeyyt] surpisodep £ zoqure-ystppot doap “rwoqp “‘SUIPURYS TO sozeyyT] surqIsodop { zoquae-ystppod doop “rea ‘SUIpULys Uo soqvyyt] surgisodep £ coquie-Yystppot doop “xeeTO “avers oped “reato *SSOTINOTOD *MUIqS “TBAT “Toque Yard “award ‘meays oped ‘avoto “mays oped ‘eatQ “roquie “avay— “doquie “realo “‘SUI[OOD WO soyRYyyT, SurpIsodop { zoqure “eo[D aourivaddy 96:0 | 9L-6 || Z8-080T 9¢-0 | 9§-2 || 9§-6Z20T Gc-0 | 00-L || Z8-920T ¥S-0 | ZE-0 || P8-900T €S-:0 | 40:0 || 09-000T 6S-0 | 68-0 || FS-GOOT GS-0 | GL-T || ¥2-LZOT TP-0 | SLO || PZ-900T GS-0 | 1TE-0 || 98-L00T SP-0 | 89-0 || 00-GZ0T 9F-0 | G6-0 || 9T-SZ0T 66-0 | 00-T || 08-420T moy sq} 0001 19d. Ayprey yeaa. ‘at0fog qysiu ays uay sogddns qysys py ‘avsns uvy 'T ATIVL IL-6 298 DR, ROBERTS ON THE DIURNAL VARIATIONS, ETC. Cane sugar and honey do not seem to possess any power of diminishing the excretion of acid by the kidneys. The evidence is rather the other way; for although the ten- deucy to a rise after breakfast on the two days is too slight to insist much upon, yet it is very notable that the honey and sugar prevented the fall that would otherwise have taken place from mere prolongation of the fasting. The hourly flow kept up without appreciable change until about the seventh hour after the saccharine meal; then there occurred a marked declension. The absorption of the sugar and honey was far from rapid; for sweet eructations took place for four and five hours after their ingestion, showing that the stomach was still hampered with their presence. Some considerable difficulty was experienced in swallowing and keeping down these large doses. It is worthy of note, that the second dose of cane sugar produced a most copious, though eva- nescent, diuresis. Not less than 7780 grain-measures of a watery urine were produced in forty minutes, or at the rate of 264 fluid ounces per hour, presenting a remarkable contrast to the flow five hours later, when only 166 grain- measures were secreted per hour, or at a rate seventy times slower than during the first period. Appendty. PROCEEDINGS OF THE MICROSCOPICAL SECTION OF THE LITERARY AND PHILOSOPHICAL SOCIETY. ESTABLISHED 1859. VOL, XV. RR INAUGURAL ADDRESS. By Proressor W. C. Wiiuiamson, F.R.S., President of the Section. Printed from a Short-hand Report by John Slagg, Esq., the Secretary of the Section. GentLemMeN, —I will not waste your time in listening to apologies for the position I occupy to night; sufficient that the request of your Council to inaugurate your pro- ceedings by an Introductory Address, was one that I did not feel justified in declining; the more so since your object is one in which I naturally feel a deep interest. I have hailed with much pleasure the formation of a Microscopic Society ; and I think we have wisely deter- mined that it shall be a section of a still older institution, which I hope it will aid im antagonizing those materialistic tendencies that abound in this mercantile community. Hitherto the devotion of all our energies to the accumula- tion of wealth and material comforts has been too promi- nent a characteristic of Manchester society, to an undue exclusion of more elevating and soul-inspiring objects. Where such tendencies increase, and continue dominant, either in nations or individuals, we cannot be at a loss to foretel the issue. All former examples recorded in the world’s history have shewn that as soon as a people make wealth and luxury their sole object of pursuit, the result is rottenness, and their ultimate fall merely a question of time. Any project, therefore, calculated to antagonize such tendencies cannot be received without extreme satis- 302 PROF. WILLIAMSON’S ADDRESS TO THE faction by thoughtful men. Let me not be misunderstood here: far be it from me to underrate the value of God’s substantial blessings; they are amongst his choicest gifts to man, aud properly used are designed to make him happy whilst they are employed for his Creator’s glory. The pursuits in which you intend to engage have no such material purpose; on the contrary, by fillimg the mind with less selfish aims and loftier conceptions, they are cal- culated to bring man nearer to his Maker, and help to fit him for the position he is finally destined to occupy. I trust that the members of this Society, shunning high sounding titles and airy assumptions, will continually keep the highest objects in view; so that whilst they are lowly in profession they may be lofty in their aspirations. There appear to be three definite purposes which may have influenced us in the establishment of this section: 1st, Amusement; 2nd, Self-instruction; 3rd, Instruction of others and the enlargement of the common stock of knowledge. The first of these objects is the lowest, but still a legitimate one. No thinking man can witness without pain the frivolity of the amusements popular even in fashionable society at the present day; many of them are unworthy of dignified manhood ; consequently an im- portant advantage is gained if we can substitute some- thing which, whilst amusing, tends also to elevate and instruct. That the second object is still a higher one needs no demonstration; but the third purpose to which I have referred merits a moment’s attention. It behoves every thoughtful man to feel himself responsible for helping to leave the world wiser and better than when he entered it. On questions of religion and morality God has made to man a direct revelation of His will; but on questions of science He has made no such revelation: the solution of these He has intrusted wholly to human instrumentality. MEMBERS OF THE MICROSCOPIC SECTION. 303 But He has indicated His will, on this subject, by planting in man’s bosom an intuitive craving for knowledge, which impels him to investigation and research, whilst it enables him to receive pleasure from those lofty generalizations that science finally aims to establish. If each man who, from the beginning, ever entered upon scientific inquiries, had but left behind him the trustworthy record of one single and previously unknown fact, how much larger a mass of information should we have possessed than at present ? I trust that our members may record many such facts ; for the field of operation is so boundless, that steady perseverance is certain to be rewarded with success. In one respect we labour under a disadvantage compared with earlier observers. In the days of Leeuwenhoek and Hooke, when the microscope was a new instrument, and its revelations an unknown world, the observations made even by means of their imperfect instruments were won- drous, and their results great. They could not possibly go wrong in the search for subjects. They lay strewed around them on every hand. Consequently, we find the Memoirs of Leeuwenhoek abounding in notices of the structure of flies and spiders, sections of wood and such like matters, now familiar to every tyro. Not only was new material abundant, but his whole time could be given to its examination. Ploughing in a virgin soil, he was under no bondage to past and contemporaneous journal- ism — not troubled with the study of other men’s re- searches, or in danger of being charged with appro- priating other men’s discoveries. With us the case is different. All common objects have been examined again and again. It is true a new field has been opened to us by the discovery of the achromatic lens; but the obser- vations to which this improvement has led have assumed a more minute and abstruse character than was formerly 304 PROF. WILLIAMSON’S ADDRESS TO THE necessary, whilst our time for original inquiry is more limited by the necessity of reading and searching out what has already been done by other men. Microscopic inquiry involves two processes which are essentially distinct, namely, the observation of facts, and their interpretation, I would earnestly recommend you to begin by disciplining yourselves in the observation of facts, ever honestly recording what you see. This is not by any means so easy a task as may at first sight appear ; for we frequently find ourselves at a loss to distinguish facts from mere appearances. An illustration of my meaning presents itself in the case of the Navicule: How long were microscopists disagreed as to whether the dotted lines sculpturing their siliceous cases were due to eleva- tions or depressions? Could the microscope but be made stereoscopic, much of this difficulty would be removed. In estimating the forms and relative positions of objects in a surrounding landscape, man derives his conclusions from two sources — from the indications afforded by linear and aerial perspective, and from the adjustment of the visual angle to objects at various distances. In working with the microscope, all the aids derivable from perspective are unavailing, from the nearness of the object and the almost uniform plane of all that can be brought within the focus of the microscope ; and as the instrument is essentially mono- cular, we are equally deprived of the aid which the visual angle would afford us; hence, being deprived of both our ordinary means of measuring distances, we are compelled to fall back on more doubtful methods of estimating them. This circumstance demands that the microscopist should be strictly conscientious ; that he should never assert any- thing to be a fact, whilst the slightest doubt remains in his mind respecting it. By so doing he would save much needless labour to others, who now have to go over the same ground, and furnish their confirmation, before the SESS 825. dk. MEMBERS OF THE MICROSCOPIC SECTION. 305 supposed fact can take its place amongst the indisputable records of scientific progress. But however difficult may be the observation of histo- logical facts, their interpretation is still more so. As the former inquiry regards the structure of objects, the latter has mainly reference to their functions. Now observers have rarely, if ever, the means of watching the objects they are studying in all the phases of their vital opera- tions ; hence, to a logical mind, much collateral informa- tion becemes essential to the comprehension of what is seen. It was here that Ehrenberg, the most justly dis- tinguished of microscopic investigators, so signally failed. He observed facts of structure accurately and well, but his mind was not adapted to their interpretation; hence the conclusions he arrived at, respecting the functions of organs, were rarely such as can now be accepted. The essential characteristic of a truly philosophical ob- server lies in his power of inducing from an aggregation of facts the relations which subsist between them. No fact either stands alone, or is the result of accident; and when these relationships are ascertained and expressed, in some brief but comprehensive formula, we obtain what is called a law. ‘To discover laws is the legitimate ambition of the highest intellects, as the undying distinction which such discoveries afford is one of its loftiest rewards; but so great is the labour and difficulty involved in the dis- covery of one such law, that it is only given to the loftiest intellects to wear the laurel-wreath. But let it be the ambition of each one of you to aim at the achievement of some such result. I am anxious that this address should not deal exclu- sively with general topics, but that it should assume a practical character, calculated to aid you in your future proceedings. The course that I should most strongly re- commend to any one just entering upon these pursuits, is 306 PROF. WILLIAMSON’S ADDRESS TO THE to select some standard work, such as those of Schwann or Schleiden, on cell-growths, or Balfour’s Manual of Botany; in which the illustrations are mainly taken from common objects, easily obtained. I would recommend you to study these works with the microscope by your side, and, as you read, examine the objects which the authors describe. By thus repeating the experiments of others, on any subject, you gradually learn what to see and how to see, and are thus made familiar with the modes of conducting original researches. Whilst carrying on this process, you either confirm your author’s observations, or note the points in which they do not harmonise with your own. When the latter is the case, the discrepancy suggests new observa- tions, correcting or confirming your independent opinion ; and you thus take your first steps as original observers. Above all, I would guard you against desultory obser- vations. Let whatever you do with the microscope be part of a fixed plan. The reward which, sooner or later, such a system brings will amply repay you for your labour. Another course which I would strongly recommend to our younger members is, that each should undertake to lay before the Society a report on the existing state of knowledge on some fixed subject, accompanied by illustra- tive specimens. The twofold result would be the produc- tion of good and interesting Papers, and the acquirement by the reporter of that practical power of observation that would best fit him for original investigations. But, whilst adopting this plan, I cannot too strongly warn you against an uninquiring acceptance of all published results, and undue submission to the authority of great names. Aware beforehand of what we may expect to see, there is danger of our imagining that we do see all that has been described. I have myself been led astray in this manner, so that I speak feelingly. When I began my researches on the A MEMBERS OF THE MICROSCOPIC SECTION. 307 Foraminifera, I was strongly influenced by Ehrenberg’s opinion that they were Bryozoa, and gave countenance to that opinion in a Paper on the “ Levant Mud,” published in the Transactions of the Manchester Literary and Philoso- phical Society. Thus I was, for some time at least, the means of diffusing an erroneous idea. On the other hand, a youthful, overweening confidence, and a deficiency in modesty and merited respect for truly great men, cannot be too strongly reprehended. Try to hit the golden mean. One of the most important studies that can claim your attention is that of the mysterious principle of life. Far be it from me unduly to check the aspirations of any phi- losophic student; but it may save us from uselessly ex- pending our strength, to remember that there are some subjects which must be studied under limitations; sub- jects which, whilst the phenomena they present demand the devotion of our best energies, we may never hope, at least in this world, fully to comprehend their cause. Life is one of these subjects. We cannot apprehend it; we can only study its functions. Its source lies beyond the field of our mortal experience, and, probably, we shall never know more of its primary nature than we do at present. But the microscopist has already done much to remove misapprehension respecting it, and will do more. When Leeuwenhoek commenced his researches on this subject, he found it loaded with absurdities, most of those who preceded him, from Aristotle to Philippo Bonani, believed in spontaneous generation; even in his day this idea was not only applied to the obscure lower animals, but to those of comparatively higher organization, such as shell-fish and corn-weevils ; and Leeuwenhoek found himself under the necessity of refuting such absurdities, by demonstra- ting that these objects were produced, like other animals, by the ordinary modes of generation. Driven from these positions, men were still unwilling to abandon the idea, VOL. XV. Ss 308 PROF. WILLIAMSON’S ADDRESS TO THE and took refuge amongst the less perfectly understood creatures, such as the Entozoa and infusorial Monads; but even here we have seen the error dispelled by the increas- ing light of science. Siebold has exploded it in the case of the Tape-worms, and other writers have done the same for the Infusoria; so that the fallacy is now left without a refuge. Greatly is the world indebted to science for this work, remembering what is the question involved. Let it be admitted that one of these lowest living creatures comes into existence independently of a Creator, and wherefore should a Creator be needful to man himself! Though scarcely credible, there still exists a class of men who doubt the interpretations of the microscrope and distrust its revelations. In his Socratic dialogues, Plato makes Meno compare Socrates, who was constantly in- fusing doubts into others, to a torpedo, which benumbs whoever touches it. We occasionally meet with anti- microscopic torpedos. I fear some of these find an avowal of doubt the readiest means of excusing their ignorance ; but where such doubts have been honestly entertained, I have never found them capable of withstanding the proof afforded by ruled-glass micrometers when placed under the microscope. Here, we know beforehand the angles at which the ruled lines intersect one another, and the geo- metric forms of the spaces they enclose; we can at once ascertain whether the microscope disturbs such angles, or whether it faithfully transmits them to the eye. If the latter, it is equally to be trusted in all its other demon- strations; and such we know to be the case. Other de- murrers occasionally ask the question, Has not all been done in the way of discovery that can be done? is not the field exhausted? The absurdity of this objection to fur- ther inquiry is best shown by the fact that, as yet, we do not know the entire correct history of one single object! Many magnificent monographs have been written, and MEMBERS OF THE MICROSCOPIC SECTION. 309 none more remarkable than the classic one of Trembley “On the Fresh-water Polype,”’ which, though penned a century ago, seemed to approach as nearly as any other to an exhaustion of the subject; but recent inquiries have shown us in how many points this history was incomplete. What a vast field do these and similar facts open to you. How many great subjects remain to be developed in Phy- siology — especially as bearing on the healing art, in Che- mistry, in Geology, and, most of all, in the endless themes which Natural History brings before us ! You cannot go wrong in your search for subjects of study. The rocky crags of the everlasting hills abound with minute memorials of former life. The desert sands of Egypt and the rice-swamps of Carolina tell alike of ocean waves that once rolled over their spreading plains. The dusty ash, cast from the volecano’s mouth and float- ing away upon the passing breeze, reveals organic atoms drawn from ocean depths. The minutest of the innumer- able insects that dance in the laughing sunbeam display beauteous forms and elaborate structures. Earth, air, fire and water cast their wondrous treasures into the lap of the microscopist, and invite him to an exercise of his observa- tive skill. I must apologize for the rambling nature of this dis- course; but I am anxious in every way to confirm and encourage you in studies of the fruits of which you cannot be robbed. They will abide with you in life and, I be- lieve, will survive death. Of how few sources of earthly pleasure can this be said! ‘“Tedet me vite” is an old cry: the feelings that prompted it were experienced long before the exiled Roman orator thus gave utterance to disappointed hopes and ungratified ambition; and it is still the cry of thousands wearied with ennui and wanting interest in the world around. But let our hearts first be right, whilst our heads and hands are occupied in cultiva- 310 PROF. WILLIAMSON’S ADDRESS, ETC. ting those fields of research that we know to be so bound- less, and the cry of ‘‘ Tedet me vite” will never be heard from any of us. That great intellectual results may crown your labours is my earnest wish. Though I would not teach that intel- lect affords any title to eternal life, I cannot believe that, other things being equal, the giant mind of a Newton and that of a clown will occupy the same level in the great day of account. That a man who, besides being a good and faithful follower of his Lord, has fostered his intellect, benefiting his fellow men and glorifying his God, will not find such culture influencing his condition in a future world, constitutes no part of my religious creed. 311 The first Paper of this Session, being the first read before the Section, was one by Mr. Go. Mostey, on “ Daphnia Pulex,’ March 21st, 1859. —_—_— Mr. Mostey commenced his account of the animal by enumerating some of its principal historians and describers from the year 1669 to the recent account of it by Baird, published in the Ray Society’s volume for 1850. After giving descriptions of the animal, as it appears under 50 and 200 linear magnifying power, Mr. Mosley proceeded specially to examine a doubtful organ on the head of this crustacean. “ Proceeding,” he says, “from the upper frontal extre- mity, a nerve about 335 of an inch in length is produced to a small oval black spot soo of an inch in its long dia- meter, situated between the eye and the beak. Baird does not name this organ in the Daphnia. In the Microthrix it is so much developed that he marks it as characteristic : ‘Eye accompanied with a black spot.’ This black spot was noticed by Miiller, who considered it to be another eye. Jurine differs from this opinion, though unable to dis- cover its real utility ; so does Strauss, who found it in the embryo before birth, exactly as in the adult. Baird agrees with these last authors, that it is not an organ of vision. Dr. Zenker, in the Microscopical Magazine for 1850, vol. ii. second series, says it has been looked upon as an auditory organ, but compares it with the tripartite azygos eye, which occurs extensively in the crustaceans ; but he leaves the question much as he found it. He says it is the first- deyeloped organ of sense, and, reasoning from analogy 312 MR. GEO. MOSLEY ON DAPHNIA PULEX. with the laws of development of other Branchiopoda, that it may be an eye in the embryonic state. It is evidently of great importance to the animal; and with all due deference to these distinguished authors (the suggestion is made with diffidence) may it not possibly be the organ of smell? Schédler’s supposition that it is the ear, and Miiller’s that it is another eye, do not bear the impress of probability ; whereas in most animals the sense of smell is situated be- tween the eye and the mouth. Zenker confesses that an eye in the embryo could be of but little use; whereas a highly developed olfactory might be so, Again: immedi- ately behind the eye may be observed several round spots each about ;,)5 of an inch in diameter, from which a branched nerve extends across the nerves and muscles of the eye to the front of the larger lobe of the brain. These do not appear to have been noticed by any of the said writers; their use is unknown; but from their position in the head, and their connection with the brain, it is not unreasonable to suppose that they may form the true auditory organs. If these suggestions be correct, the organs of hearing and smelling will be placed behind and in front of the eye, in accordance with the laws which prevail in comparative anatomy.” After describing the male Daphnia, which Mr. Mosley says is not often to be met with, he concluded this paper by a lengthy description of the singular method of the reproduction of the animalcule. 313 Al the meeting of the Section, April 18th, 1859, Mr. H. A. Horst read a Paper on the “ Structure of Starch Granules.” ——. Mr. Horst gives a long series of quotations from the works of the most celebrated writers on this subject, illus- trative of the various opinions entertained by high au- thorities on the Structure of these Granules, showing the prominent opinion on the subject to be, that they owe the peculiar striated appearance of their surfaces to their being formed of a number of concentric layers converging to a nucleus. This theory, however, Mr. Hurst disputes ; and after giving a detailed account of the manner in which the granules behave under the influence of various chemical reagents, proceeds to support a theory of his own on the subject as follows : “ Payen, in his Mémoires sur le développement des Végé- taux, in 1842, gives plates of the appearances of Starch Granules, which on first inspection appear to prove satis- factorily the concentric layer theory; but on further ex- amination it will be found that they require to be roasted, iodized, acidulated, and otherwise acted on by chemical agency to such an extent that the results cannot be ac- cepted as evidence of structure existing before all these influences were brought to bear upon them; and in con- nection with this I would remark that immersing an object for three weeks in iodine, as done by Dr. Allman, is hardly a fair test, and may very possibly cause the concentric layers, to prove the existence of which it is used. Mr. Tuffen West remarks, and I consider very justly, that 314 MR. H. A. HURST ON THE appearances presented by, more especially, vegetable struc- ture, after having undergone prolonged torture by chemical agency, are no more to be depended upon than those pre- sented by a human being under similar circumstances. “The striz, if they were the optical expression of lines of separation of concentric layers, would, I submit, show strongest when the plane of the focus of the object glass coincided with the central plane of the granule; but this appears to me not to be the case. On the contrary, the strie are only shown when the surface of the granule is in focus. Some starch grains do not show striz at all, whilst those of the Canna Indica show them only on three sides. “But a strong argument against the concentric layer theory is that the nucleus, where all the supposed layers are superimposed one over the other, is actually the most transparent part of the granule, instead of being the most opaque, which would be the natural consequence of the presence of a larger number of layers than those existing at the opposite end of the granule. «This extreme end, where there can be only one layer, is, if there be any difference, the darkest part of the granule, as can be verified by inspection. “In examining the action of any mineral acid, or of an alkali, on starch granules — say the largest —I find the following phenomena: “ Should the agent be much concentrated, the granule bursts at once the moment it is touched by it, and lies on the glass slide a flat ovate single membrane, no trace being discernible of strize or internal layers. If more dilute, a gradual increase of size takes place, one or more of the striz becoming strongly marked. The hilum or nucleus seems to be an aperture through which the agent obtains access to the interior, and from it stellate fissures arise, caused apparently by the absorption of the agent STRUCTURE OF STARCH GRANULES. 315 and consequent expansion of the outer cell-wall or mem- brane; these gradually increase in length, exposing the interior to view, and allowing the granule to expand later- ally till they almost reach the edge of the greatly enlarged granule, when, of course, the band of the cell, which has still retained its cellular or enclosing character, gives way, and the granule assumes the aspect previously described as immediately resulting from the application of a strong alkali or mineral acid. It is worthy of notice that, while these fissures are extending, the strip of membrane be- tween any two, still retains the striz, which are clearly visible. Now these fissures are, undoubtedly, not in the body of the grain, but are merely fissures of the outer layer or investing membrane; at least they appear to me to be so; and if such should be the case, I would suggest, it is a positive proof that these strize do not arise from the existence of concentric layers, nor even from an unfolding of the outer layer, since they appear even if it has been rent by distention. “That this exterior membrane or investing layer is of a different nature from the contents it encloses, I think no one can doubt who has observed the above described phenomena; but from what Dr. Gregory says, it would appear to be merely a difference of structure or texture, ' not of chemical composition; and after a careful consi- deration of my experiments, I am inclined to attribute the striz to the circular deposition of rings of starch on the inner surface in precisely a similar manner to the deposi- tion of circular or spiral rings of woody fibre on the sides of the so-called annular and spiral ducts in the tissue of many plants, more particularly those belonging to the natural family of Cucurbitaciz, Orchidiz and Filices. Drawings of the cells found in the leaves of species of Oncidium might easily be taken for representations of starch granules, so exact is the resemblance. VOL, XV, rT 316 MR. H. A. HURST ON STARCH GRANULES. “This view derives confirmation from the fact, that the deeper marking of some striz, when the granule is under the influence of acids or alkalies, is evidently caused, not by its own increase in depth or thickness, but by the ex- pansion of the membrane in which it is deposited, each side clearly separating it further from the adjacent striz ; while the moment the extension of stellate fissures allows lateral expansion of the granule no further deepening of striz is observable. “I therefore consider the starch granule to be consti- tuted of an enveloping membrane, or shell, of very firmly consolidated starch, enclosing a certain amount of the same substance in an amorphous condition and a more lax state, while the hilum, or nucleus, would, according to my view, be merely an aperture.” At the conclusion of this Paper some discussion took place as to the effect of various chemical reagents on the starch granule, and also as to the possibility of producing sections of them. 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Adams, John Couch, F.R.S., F.R.A.S., F.C.P.S., Mem. Amer. Acad. Arts and Sc. Boston. Pembroke College, Cambridge. Agassiz, Louis, F.R.S., Mem. Imper. Instit. France. Cambridge, Massachusets. ae George Biddell, M.A, D.C.L., F.R.S., Astronomer Royal, V.P.R.A.S., Hon. Mem. RSE, R.I.A., M.C.P.S., Chev. of the Prussian Order “ Pour - Meérito,” Corr. Mem. Nat. Inst. Washington U.S., Imper. Instit. France, Imper. Acad. Sc. Petersburg and Roy. Acad. Sc. Berlin, Mem. Acad. Se. and Lit. Palermo, Roy. Acadd. Se. Stockholm and Munich, Roy. Soc. Sc. Copenhagen, and Amer. Acad. Arts and Sc. Boston. The Royal Observatory, Greenwich, London, S.L. Barlow, Peter, F.R.S., F.R.A.S., Hon. M.C.P.S., Mem. Imper. Acad. Sc. Petersburg, and Amer. Acad. Arts and Sc. Boston, Corr. Mem. Imper. Instit. France and Roy. Acad. Se. Brussels. Woolwich, London, S.L. Bosworth, Rev. Joseph, LL.D., F.RS., F.S.A., M.R.LA., Corr. Mem. Roy. Soc. Northern Antiqu. Copenhagen, Mem. Roy. Soc. Sc. Drontheim and Soc. Se. Gothenburg. Islip House near Oxford, and Etwell, Uttoxeter. Brewster, Sir David, K.H., LL.D., F.R.S. L. & E., Hon. M.R.1A., F.G.S., F.R.A.S., Chev. of the Prussian Order “ Pour le Meérite,’ Mem. Imper. Instit. France and Roy. Soc. Se. Got- tingen, Principal of St. Leonard’s College. St. Andrews. Brisbane, General Sir Thomas Makdougall, Bart., K.C.B., G.C.H., D.C.L., F.R.S., Pres.R.S.E., Hon. M.R.LA., F.R.A.S., F.LS., Corr. Mem. Imper. Instit. France. Makerstown, Kelso. Dumas, Jean Baptiste, F.R.S., Mem. Imp. Instit. France, &e. Paris. Faraday, Michael, D.C.L., F.R.S., Fullerian Prof. of Chemistry in the Roy. Instit. of Great Britain, Hon. Mem. R.S.E., ie C.P.S., and Med. Chir. Soc., F.G.S., Chev. of the Prussian Order “ Pour le Mérite,” Comm. Legion of Honour, Mem. Imper. Instit. France, Imper. Acadd. Se. Vienna and Peters- burg, Roy. Acadd. Sc. Berlin, Turin, Stockholm, Munich, Naples, Amsterdam, Brussels, Bologna and Modena, Roy. Socc. Sc. Gottingen, Copenhagen, Upsal and Harlem, Amer. Acad. Arts and Sc. Boston, and Amer. Phil. Soc. Philadelphia, Corr. Mem. Acad. Sc. and Liter. Palermo, Acad. Georg. Florence, Philom. Soc. Paris, Nat. Instit. Washington U.S., and Imper. Acad. of Med. Paris, &c. 21, Albemarle Street, London, W. Hamilton, Sir William Rowan, Knt., LL.D., M.R.LA., F.R.A.S., Astronomer Royal, Ireland, Andrews Professor of Astronomy, T.C.D. Observatory, Dunsink, Dublin. Harcourt, Rey. William Venables Vernon, M.A., F.R.S., Hon. M.R.LA., F.G.8. Bolton Percy, Tadcaster. Hartnup John, F.R.A.S. Observatory, Liverpool. Herschel, Sir John Frederick William, Bart., D.C.L., F.R.S. L.& E., Hon. M.R.1.A., F.G.S., M.C.P.S., Chev. of the Prussian Order “ Pour le Mérite,” Corr. Mem. Imper. Instit. France, Mem. Imper. Acad. Se. Petersburg, Roy. Acadd. Se. Berlin, Turin, Naples and Brussels, Roy. Soce. Sc. Gottingen, Copenhagen and Harlem, Acad. dei Nuovi Lincei, Rome, Acadd. Padua, Bologna, Palermo, Modena, Acad. Gioen. Catania, Imper. Acad. Se. Dijon, Philom. Soc. Paris, and Soc. Nat. Sc. Switzer- land. Collingwood, near Hawkhurst, Kent. Hind, John Russell, F.R.A.S., Regents Park, London, N.W. Hopkins, William, M.A., LL.D., F.RS., F.G.S. Fitzwilliam Street, Cambridge. Humboldt, Baron Alexander von, F.R.S., &e. &c. Berlin. Kirkman, Rev. Thomas Penynton, M.A., F.R.S. Croft Rectory, near Warrington. Le Verrier, Urbain Jean Joseph, F.R.S., Mem. Imper. Instit. France, &c. Paris. ~ 5) Liebig, Baron Justus von, F.R.S., &c. Munich. Mitscherlich, Eilert, Professor, F.R.S., &e. Berlin. Morin, General, A., Mem. Imper. Instit. France, &e. Paris. Moseley, Rev. Henry, M.A., F.R.S., Corr. Mem. Imper. Instit. France. Olveston, near Bristol. Mosley, Sir Oswald, Bart. olleston Hall. Murchison, Sir Roderick Impey, G.C.St.S., D.C.L., M.A., F.R.S., F.G.S., F.L.S. &c., Director Gen. of the Geol. Survey, Pr. R.G.S., Hon. Mem. R.S.E. and R.I.A., Mem. C.P.S. and Imper. Acad. Se. Petersburg, Corr. Mem. Imper. Instit. France, Roy. Acadd. Se. Stockholm, Turin, Berlin and Brus- sels, Roy. Soc. Sc. Copenhagen, Amer. Acad. Arts and Sc. Boston, and Imper. Geogr. Soc. Petersburg, Hon. Mem. Imper. Soc. of Naturalists, Moscow, &c. &c. 16, Belgrave Square, London, S.W. Owen, Richard, M.D., LL.D., V.P.R.S, F.L.S., F.G.S., V.P.Z.S., Director of the Natural History Department, British Museum, Hon.F.R.C.S. Ireland, Hon.M.R.S.E., Corr. Mem. Imper- Instit. France, Mem. Imper. Acadd. Sc. Vienna and Petersburg, Imper. Soc. of Naturalists Moscow, Royal Acadd. Sc. Berlin, Turin, Madrid, Stockhoim, Munich, Amsterdam, Naples, Brussels and Bologna, Roy. Soce. Se. Copenhagen and Upsal, and Amer. Acad. Arts and Se. Boston, Corr. Mem. Philom. Soc. Paris, Mem. Acad. Georg. Florence, Soc. Se. Harlem and Utrecht, Soc. of Phys. and Nat. Hist. Geneva, Acad. dei Nuovi Lincei, Rome, Roy. Acadd. Sc. Padua, Palermo, Acad. Gioen. Catania, Phys. Soc. Berlin, Chev. of the Prus- sian Order “ Pour le Mérite,” For. Assoc. Instit. Wetter., Philadelphia, New York, Boston, Imper. Acad. Med. Paris, and Imper. and Roy. Med. Soc. Vienna. British Museum, London, W.C. Peacock, Very Rev. George, D.D., Dean of Ely, F.R.S., F.GS., F.R.A.S. Deanery, Ely. 6 Playfair, Lyon, C.B., Ph.D., F.R.S., F.G.5., F.C.S., Professor of Chemistry, Univ. Ed. Zdinburgh. Poncelet, General Jean Victor, F.R.S., Mem. Imper. Instit. France, &e. Paris. “Rawson, Robert. Royal Dockyard, Portsmouth. Sabine, Major-General Edward, R.A., D.C.L., Treas. and V.P.R.S., F.R.A.S., Hon. Mem. C.P.S., Chev. of the Prussian Order * Pour le Mérite,’ Mem. Imper. Acad. Se. Petersburg, Roy. Acadd. Se. Berlin, Brussels and Gottingen, Roy. Soc. Se. Dron- theim, Acad. Sc. Philadelphia, Econ. Soc. Silesia, Nat. Hist. Soc. Lausanne and Roy. Batavian Soc., Corr. Mem. Roy. Acad. Se. Turin, Nat. Instit. Washington U.S., Geogr. Soce. Paris, Berlin and Petersburg. 13, Ashley Place, Westminster, S.W. Sedgwick, Rev. Adam, M.A., F.R.S., Hon. M.R.LA., F.G.S., F.R.A.S., Woodwardian Lecturer. Cambridge. Stokes, George Gabriel, M.A., D.C.L., Seer. R.S., Lucasian Pro- fessor of Mathem., F.C.P.S., Mem. Batavian Soc. Rotterdam. Pembroke College, Cambridge. Tayler, Rev. John James, B.A., Principal of Manchester New College. London. Thomson, William, M.A., LL.D., F.R.S.L,&E., Professor of Nat. Philos. University. Glasgow. Turner, Rey. William. Manchester. Whewell, Rev. William, D.D., F.R.S., Hon. M.R.LA., F.S.A., F.G.S., F.R.A.S., Master of Trinity College, Cambridge. Lodge, Cambridge. Woodcroft, Bennet, Professor, Superint. of Regist. of Patents. Southampton Buildings, London. DATE OF ELECTION. 1857 1856 1839 1854 1849 1837 1846 1824 1852 1842 1849 1840 1858 1847 1847 1857 1858 1844 1854 1842 1834 1821 1855 1839 1857 1855 1824 1844 1855 1846 1810 “I ORDINARY MEMBERS. January 27. April 29. April 30. January 24. April 17. August 11. January 27. January 23. January 27. April 19. Aprile Uy, January 21. January 26. January 26. January 26. April 21. January 26. January 23. January 24. January 25. April 18. January 26. January 23. October 29. Apsl 21; April 17. April 30. January 23. April, ©L7. January 27. Novem. 1. Acton, Henry Morell, B.A. Adshead, Joseph. Ainsworth, Ralph F., M.D. Allan, James, Ph.D., M.A, Ash, W. H. Ashton, Thomas. Atkinson, John. Barbour, Robert. Barlow, Henry Bernoulli. Barratt, Joseph. Bassnett, Rev. R., A.M. Bateman, John F., M. Inst. C.E., F.G.S. Baxendell, Joseph, F.R.A.S. Bazley, Thomas. Bell, William. Bellhouse, Edward T. Benson, Davis. Bevan, James. Beyer, Charles. Binney, Edward William, F.R.S., F.G.S. Birley, Richard. Blackwall, John, F.L.S. Bowman, Eddowes, M.A. Bowman, Henry. Brock, John. Brockbank, William. Brooke, Edward. Brooks, W. C., M.A. Broughton, Frederick. Browne, Henry, M.B. Buchan, Laurence, DATE OF ELECTION. 1846 1853 1854 1847 1858 1841 1855 1832 1852 1857 1854 1841 1851 1853 1853 1853 1851 1839 1848 1854 1843 1854 1842 1853 1854 1842 1855 1850 1818 1856 1854 1850 January 27. April 19. January 24. January 26. January 26. April, 15. January 23. January 24. April 20. April 21. April 18. April 15. April 29. April 19. January 25. January 25. April 29. January 22. January 25. January 24. April 18. February 7. April 19. April 19. January 24. Novem. 15. January 23. April 30. April 24. April 29. January 24. April 30. Burd, John. Bury, Henry. Callender, William, jun. Calvert, Frederick Crace, M.R.A.T., F.C.S. Casartelli, Joseph. Caw, John Young, F.S.A., F.R.S.Lit. Cawley, Charles E., M. Inst. C.E. Charlewood, Henry. Chadwick, David, F.S.S., Assoc. Inst. C.E. Churchill, George Cheetham. Christie, Richard Copley, M.A., Prof. Hist. Owens College. Clay, Charles, M.D. Cleminshaw, Charles. Clift, Samuel, F.C.S. Corbett, Edward. Cottam, Samuel. Crompton, Samuel. Crossley, James, F.S.A., Pres. Chet. Soc. Crowther, Joseph 8. Culley, Richard §., C.E. Curtis, Matthew. Dale, John. Dancer, John Benjamin, F.R.A.S. Darbishire, Robert Dukinfield. Davies, David Reynold. Dean, James Joseph. Dickinson, William L. Dyer, Frederick Nathaniel. Dyer, Joseph Chesborough. Ekman, Charles Frederick. Ellis, Charles. Fairbairn, Thomas. Date oF ELEcTION. October 29. 1824 1849 1852 1818 1856 1857 1855 1854 1840 1836 1847 1837 1817 1849 1848 1844 1839 1828 1854 1815 1833 1851 1845 1848 1839 1820 1854 1855 1846 1823 Vol. October 30. January 25. October 31. 29. 21. January 23. April April January 24. January 21. 20. 20. August 11. January 24. October 30. January 25. April April January 23. January 22. October 31. January 24. January 27. 26. 29. 29. October 31. January 22. April April April January 21. January 24. January 23. January 27. January 18. 15, 5 ##* Fairbairn, William, F.R.S., Corr. Mem. Imp. Inst. France and Roy. Acad, Se., Turin. Fairbairn, W. A. Fleming, David Gibson. Flint, Richard. Forrest, H. R. Foster, Thomas Barham. Fothergill, Benjamin. Fryer, Alfred. Gaskell, Rev. William, M.A. Giles, Samuel. Gould, John, Graham, John. Greg, Robert Hyde, F.G.S. Greg, Robert Philips, F.G.S. Grundy, John Clowes. Hampson, Richard. Hawkshaw, John, F.R.S., F.G.S., M.Inst.C.E. Henry, Wiliam Charles, M D., F.R.S. Hetherington, John. Heywood, Sir Benjamin, Bart., F.R.S. Heywood, James, F.R.S., F.G.S., F.S.A. Higgin, James. Higgins, James. Higson, Peter. Hobson, John. Hodgkinson, E., F.R.S., F.G.S.,° M.R.LA., Hon, Mem.R.1.B,A., Inst.C.E., Roy. Scot. Soc. Arts, and Soc. Civ, Eng. Paris, Prof. of the Mech, Prine. of Engineering Univ. Coll. London. Holeroft, George. Holden, Isaac. Holden, James Platt. Hopkins, Thomas, M. Brit, Met. Soe. DATE OF ExzcrTion. 1824 1854 1857 1857 1823 1850 1856 1836 1821 1848 1842 1846 1843 1852 1854 1822 1830 1850 1856 1825 1839 1852 1857 1842 1854 1850 1855 1855 1829 1838 1844 January 23. February 7. January 27. January 27. January 24. April 30. January 22. January 22. October 19. April 18. January 25. January 27. January 24. January 27. January 24. April 26. April 30. April 30. January 22. January 27. October 29. January 27. January 27. April 19. January 24. April 30. January 23. October 30. October 30. April 17. April 30. 10 Houldsworth, Henry. Hoyle, William Jennings. Hunt, Edward, B.A., F.C.S. Jackson, George. Jesse, John, F.R.S., F.R.A.S., F.L.S. Johnson, Richard, F.C.S. Johnson, William. Jones, Rev. Henry Halford, F.R.A.S. Jordan, Joseph. Joule, B. St. J. B. Joule, James Prescott, LL.D., F.R.S., Hon. M.C.P.S., Corr. Mem. Roy. Acad. Se. Turin. Joynson, William. Kay, Samuel. Kennedy, John Lawson. Kershaw, James, jun. Lane, Richard. Langton, William. Leese, Joseph, jun. Lingard, Thomas Taylor. Littler, Thomas. Lockett, Joseph. Long, Isaac Waithman, F.R.A.S. Longridge, R. B. Love, Benjamin. Lowe, George Cliffe. Lund, Edward. Lund, George T. Mabley, William Tudor. M‘Connel, James. M‘Connel, William. M‘Dougall, Alexander. Date oF ELECTION. January 24. April iis 1823 1849 1858 1842 1837 1849 1848 1822 1852 1854 1845 1827 1844 1844 1833 1841 1857 1854 1857 1854 1836 1847 1823 1822 1858 1842 1848 1851 April 20. January 25. January 27. January 23. January 25. April 26. January 27. February 7. January 21. January 26. April 30. April 30. April 26, April 15. April 21. January 24. January 27. February 7. April 29. January 26. January 18. January 25. January 26. January 25. April 18. April 29. 11 Macfarlane, John. Manchester, The Right Rev. the Lord Bishop of, D.D., F.R.S., F.G.S., F.C.P.S., Corr: Mem. Arch. Inst. Rome. Mather, Colin. Mellor, Thomas. Mellor, William. Morris, David. Neild, Alfred. Neild, William. Nelson, James Emanuel. Nevill, Thomas Henry. Nicholls, John Ashton, F.R.A.S. Nicholson, William. Ormerod, Henry Mere. Parr, George. Parry, John. Peel, George, M. Inst. C.E. Platt, William Wilkinson. Pochin, Henry Davis. Poynting, Rev. T. E. Ramsbottom, John. Ransome, Joseph Atkinson, F.R.C.S. Ransome, Thomas. Roberts, Richard, M. Inst. C.E. Robinson, Samuel. Roscoe, Henry Enfield, B.A., Ph.D., F.C.S,. Professor of Chemistry, Owens College. Royle, Alan. Salt, Samuel. Sandeman, Archibald, M.A.,* Professor of Mathematics, Owens College. DATE OF ELECTION. 1847 1842 1854 1855 1835 1852 1838 1845 1851 1852 1834 1847 1858 1814 1856 1852 1836 1821 1857 1823 1857 1858 1832 1853 1851 1851 1839 1855 1846 1839 1840 January 26. January 25. February 7. January 23. October 30. April 20. January 26. April, 29. April ) 29. January 27. January 24, April 20. January 26. January 21. January 22. April 20. April 29. April 19. January 27. January 24. January 27. January 26. January 22. April 19. April 29. January 21. January 22. October 30. April 21. April 30. April 28. 12 Satterthwaite, Michael, M.D. Schunck, Edward, Ph.D., F.R.S., F.C.S. Scott, Alexander J., M.A., Prof. of Logic and English Language, Owens College. Sharp, Edmund Hamilton. Shuttleworth, John. Sidebotham, Joseph. Smith, George S. Fereday, M.A., F.G.S. Smith, Robert Angus, Ph.D., F.R.S, F.C.S. Spence, Peter. Standring, Thomas. Stephens, Edward, M.D. Stephens, James. Stewart, Charles P. Stuart, Robert. Taylor, John Edward. Thom, David. Turner, James Aspinall, M.P. Turner, Thomas, F.R.C.S. Walker, Robert, M.D. Watkin, Absalom. Webb, T. G. Whitehead, James, M.D. Whitworth, Joseph, F.R.S. Williamson, Samuel Walker. Williamson, William Crawford, F.R.S., Pro- fessor of Natural History, Owens College. Withington, George Bancroft. Wood, William Rayner. Woodcock, Alonzo Buonaparte. Woodhead, George. Woods, Edward. Worthington, Robert, F.R.A.S. 13 CORRESPONDING MEMBERS. Dockray, Benjamin, Lancaster. Girardin, M. J., Rowen. Harley, Rev. Robert, F.R.A.S., Brighouse, Yorkshire. Kenrick, Rev. John, M.A., York. Koechlin-Schouch, Daniel, Mulhouse, France. Lowe, E. J., F.R.A.S., Mottingham. Mercer, John, F.R.S., Oakenshaw, Accrington, Lancashire. Pincoffs, Peter, M.D., Dresden. Watson, Henry Hough, Bolton, Lancashire. Wilkinson, T. T., F.R.A.S., Burnley, Lancashire. MANCHESTER : PRINTED BY CHARLES SIMMS AND CO, ee _ = ‘7 - 7 ¢ = * ~ ‘ ~ - , { \ } é - - £ THE Literary and BHilosophical Society of Manchester. APRIL 19, 1859. THE COUNCIL. PBrestvent. WittaM Farrparrn, CZ, RS. F.G.S., Corr. Mem. Imper. Lnstit. France and Roy, Acad. Sc. Turin. Pice-Yrestdents. James Prescott Joutz, ZZ.D., F.R.S., FCS. Hon. M. CoP Si Corr. Mem. Roy. Acad. Sc. Turin. JosEpH CHESBoROUGH Dyer. Tuomas Hopkins, JZ B.JIZ.S. Rosert Ancus Suit, PA. D., ERS, £-.C.S., Corr. Mem. Imper. Roy. Geol. Instit. Vienna. Secretaries. Epwarp Scuunck, Ph. D., ERS, FCS. Ricnarp CopLry Curistiz, I0.A., Profess. Hist. Owens Coll. ‘ Treasurer. Henry Mere Ormerop. VOL. XV. YY 16 Librarian. CHARLES FREDERICK EKMAN. OF the Council. Rev. WiLi1aM GaskELL, JZA. Ricuarp Roserts, JZ, Just. CL. JosrPpH BAxENDELL, 7.7.A4.S. Epwarp WitiiaM Binney, “2S, 7G-.S. PETER SPENCE. JosEpH ATKINSON Ransome, JZ.R#.C.S. 17 HONORARY MEMBERS. Date oF ELECTION. 1847, Apr.20. Adams, John Couch, F.R.S., F.R.A.S., F.C.P.S., Mem. Amer. Acad. Arts and Sc. Boston. Pem- broke College, Cambridge. 1843, Apr.18. Agassiz, Louis, F.R.S., Mem. Imper. Instit. France, &e. Oambridge, Massachusets. 1843, Feb. 7. Airy, George Biddell, M.A., D.C.L., F.R.S., As- tronomer Royal, V.P.R.A.S., Hon. Mem. R.S.E., R.LA., M.C.P.S., Chev. of the Prussian Order “ Pour le Mérite,’ Corr. Mem. Nat. Inst. Wash- ington, U.S., Imper. Instit. France, Imper. Acad. Se. Petersburg and Roy. Acad. Se. Berlin, Mem. Acad. Se. and Lit. Palermo, Roy. Acadd. Se. Stockholm and Munich, Roy. Soc. Se. Copen- hagen, and Amer. Acad. Arts and Sc. Boston. The Royal Observatory, Greenwich, London, S.L. 1843, Apr.18. Barlow, Peter, F.R.S., F.R.A.S., Hon. M.C.B.S., fh Mem. Imper. Acad. Sc. Petersburg, and Amer. Acad. Arts and Se. Boston, Corr. Mem. Imper. Instit. France and Roy. Acad. Sc. Brussels. Woolwich, London, S.E. 1849, Jan. 23. Bosworth, Rev. Joseph, LL.D., F.R.S., F.S.A., M.R.1.A., Corr. Mem, Roy. Soc. Northern Antiq. Copenhagen, Mem. Roy. Soc. Sc. Dron- theim and Soc. Se. Gothenburg. Islip House, near Oxford, and Etwell, Uttoxeter. 1843, Apr.18. Brewster, Sir David, K.H., LL.D., F.R.SS. L.& E., Hon. M.R.ILA. F.GS., F.RAS., Chev. of the Prussian Order “ Pour le Mérite,” Mem. Imper. Instit. France and Roy. Soc. Sc. Gottingen, Principal of St. Leonard’s College. St. Andrews. rm : - 18 Date or Exxcrion. 1843, Feb. 7. Brisbane, General Sir Thomas Makdougall, Bart., K.C.B., G.C.H., D.C.L., F.R.S., Pres. R.S.E., Hon. M.R.1.A., F.R.A.S., F.L.S., Corr. Mem. Imper. Instit. France. Makerstoun, Kelso. 1859, Jan. 25. Cayley, Arthur, M.A., F.R.S., F.R.A.S. 2, Stone Buildings, Lincoln's Inn, London, W.C. 1859, Jan. 25. De Morgan, Augustus, V.P.R.A.S., F.C.P.S., Professor of Mathematics in Univ. Coll. London. 41, Chalcot Villa, Adelaide Roads London. 1844, Apr.30. Dumas, Jean Baptiste, F.R.S., Mem. Imper. Instit. France, &c. Paris. 1843, Apr.18. Faraday, Michael, D.C.L., F.R.S., Fullerian Prof. of Chemistry in the Roy. Instit. of Great Bri- tain, Hon. Mem. R.S.E., C.P.S., and Med. Chir. Soc., F.G.S., Chev. of the Prussian Order * Pour le Mérite,’ Comm. Legion of Honour, Mem. Imper. Instit. France, Imper. Acadd. Se. Vienna and Petersburg, Roy. Acadd. Se. Berlin, Turin, Stockholm, Munich, Naples, Amster- dam, Brussels, Bologna and Modena, Roy. Soce. Se. Gottingen, Copenhagen, Upsal and Harlem, Amer. Acad. Arts and Sc. Boston, and Amer. Phil. Soc. Philadelphia, Corr. Mem. Acad. Se. and Liter. Palermo, Acad. Georg. Florence, Philom. Soc. Paris, Nat. Instit. Washington U.S., and Imper. Acad. of Med. Paris, &c. 21, Albemarle Street, London, W. 1843, Feb. 7. Frisiani, nobile Paolo, Prof., 2ud Astron. at the Observ. of Brera, Milan, Mem. Imper. Roy. Instit. of Lombardy, Milan, and Ital. Soe. Se. Milan. 1843, Feb. 7. Hamilton, Sir William Rowan, Knt., LL.D., " ees oe 19 Dare oF ELECTION. M.R.LA., F.R.A.S., Astronomer Royal, Ire- land, Andrews Professor of Astronomy, T.C.D. Observatory, Dunsink, Dublin. 1843, Feb. 7. Harcourt, Rev. William Venables Vernon, M.A., 1853, Apr. 19. 1843, Apr. 18. 1848, Jan. 25. 1853, Jan. 25. 1852, Oct. 19. 1848, Oct. 31. 1847, Apr. 20. 1843, Feb. 7. F.R.S., Hon. M.R.LA., F.G.S. Bolton Percy, Tadcaster. Hartnup, John, F.R.A.S. Observatory, Liverpool. Herschel, Sir John Frederick William, Bart., D.C.L., F.RS.S. L. & BE. Hon. M.R.IA., F.G.8., M.C.P.S., Chev. of the Prussian Order “ Pour le Mérite,’ Corr. Mem. Imper. Instit. France, Mem. Imper. Acad. Sc. Petersburg, Roy. Acadd. Sc. Berlin, Turin, Naples and~ Brussels, Roy. Soce. Se. Gottingen, Copen- hagen and Harlem, Acad. dei Nuovi Lincei Rome, Acadd. Padua, Bologna, Palermo, Mo- dena, Acad. Gioen. Catania, Imper. Acad. Sc. Dijon, Philom. Soc. Paris, and Soc. Nat. Se. Switzerland. Collingwood, near Hawkhurst, Kent. Hind, John Russell, F.R.A.S. egent’s Park, London, N.W. Hopkins, William, M.A., LL.D., F.R.S., F.G.S. Fitzwilliam Street, Cambridge. Kirkman, Rev. Thomas Penynton, M.A., F.R.S. Croft Rectory, near Warrington. Lassell, William, F.R.S., F.R.A.S., Hon. F.R.S.E., Hon. M. Philomath Soc. Paris. Bradstones, Sandfield Park, near Liverpool. Le Verrier, Urbain Jean Joseph, F.R.S., Mem. Imper. Instit. France, &e. Paris. Liebig, Justus Baron von, M.D., Ph.D., Prof. of Chem. Univ. Munich, Conservator of Chem. Labor. Munich, Chey. of the Bav. Order * Pour le Mérite,” &c., F.R.S.S. L. & E., Hon. 20 Dave oF ELECTION. 1849, Apr. 17. 1843, Feb. 7. 1854, Jan. 24. 1843, Apr. 18. 1821, Jan. 26. 1844, Apr. 30. M.R.1.A., Hon. Mem. Univ. Dorpat and Med. Phis. Facult. Univ. Prague, Hon. Mem. and For. Assoc. Roy. Acad. Sc. Vienna, Roy. Acadd. Se. Stockholm, Brussels, Amsterdam, Turin, Acad. Se. Bologna, Roy. Socc. Se. Gothen- burg, Gottingen, Copenhagen, Liege, Imper. Roy. Instit. of Lombardy, Milan, Corr. Mem. Imper. Instit. France, Imper. Acad. Sc. Peters- burg, Roy. Acad. Se. Madrid, Mem. Roy. Med. Chir. Soc. London and Perth, Roy. Scot. Soc. Arts, Botan. Soce. Edinburgh and Regensburg, Soce. Nat. Sc. Berlin, Dresden, Halle, Moscow, Lille, Ph. Soc. Glasgow, Agric. Socce. Munich, Giessen, &c. &c. Munich. Mercer, John, F.R.S. Oakenshaw, Accrington, Lancashire. Mitscherlich, Eilert, Professor, F.R.S., &c. Berlin. Morin, Arthur, General of Artillery, Mem.Imper. Instit. France, formerly éléve Polytechn. School, Dir. Conserv. of Arts, Paris, Corr. Mem. Roy. Acadd. Se. Berlin, Madrid and Turin, Acad. Georg. Florence, Imper. Acad. Metz, and In- dustr. Soc. Mulhouse. Paris. Moseley, Rev. Henry, M.A., F.R.S., Corr. Mem. Imper. Instit. France. Olveston, near Bristol. Mosley, Sir Oswald, Bart. Rolleston Hall. Murchison, Sir Roderick Impey, G.C.St.S., D.C.L., M.A., F.RBS., F.G.S., F.L.8., &c., Director Gen. of the Geol. Survey, Pr. R.G.S., Hon. Mem. R.S.E., and R.I.A., Mem. C.P.S. and Imper. Acad. Se. Petersburg, Corr. Mem. Imper. Instit. France, Roy. Acadd. Se. Stock- holm, Turin, Berlin and Brussels, Roy. Soc. Se. Copenhagen, Amer. Acad. Arts and Sc. Boston, and Imper. Geogr. Soc. Petersburg, Hon. Mem. Imper. Soc. of Naturalists, Moscow, &e. &c. 16, Belgrave Square, London, S.W. 21 Date oF ELECTION. 1844, Apr. 30. 1851, Apr. 29. 1856, Jan. 22. 1859, Apr. 19. 1849, Jan. 23. 1859, Apr. 19. 1844, Apr. 30. Owen, Richard, M.D., LL.D., V-P.R.S., F.LS, F.G.S., V.P.Z.S., Director of the Nat. Hist. Department, British Museum, Hon. F.R.C.S. Treland, Hon. M.R.S.E., For. Assoc. Imper. Instit. France, Mem. Imper. Acadd. Sc. Vienna and Petersburg, Imper. Soc. of Naturalists Moscow, Royal Acadd. Sc. Berlin, Turin, Ma- drid, Stockholm, Munich, Amsterdam, Naples, Brussels and Bologna, Roy. Socc. Se. Copen- hagen and Upsal, and Amer. Acad. Arts and Se. Boston, Corr. Mem. Philom. Soe. Paris. Mem. Acad. Georg. Florence, Soc. Sc. Harlem and Utrecht, Soc. of Phys. and Nat. Hist. Geneva, Acad. dei Nuovi Lincei Rome, Roy. Acadd. Sc. Padua, Palermo, Acad. Gioen. Catania, Phys. Soc. Berlin, Chev. of the Prus- sian Order “ Pour le Mérite,” For. Assoc. Instit. Wetter., Philadelphia, New York, Boston, Imper. Acad. Med. Paris, and Imper. and Roy. Med. Soc. Vienna. British Museum, London, W.C. Playfair, Lyon, C.B., Ph.D., F.RS., F.GS., F.C.S., Professor of Chemistry Univ. Ed. Zd- inburgh. Poncelet, General Jean Victor, F.R.S., Mem. Imper. Instit. France, &e. Paris. Rankine, William John Macquorn, LL.D., F.R.S.S. L. & E., Pres. Inst. Eng. Scot., Regius Profes- sor of Civil Engineering and Mechanics, Univ. Glasgow. 59, St. Vincent Street, Glasgow. Rawson, Robert. Royal Dockyard, Portsmouth. Reichenbach, Carl Baron von. Vienna. Sabine, Major-General Edward, R.A., D.C.L., Treas. and V.P.R.S., F.R.A.S., Hon. Mem. 22 Date oF ELEcrion. 1843, 1851, 1854, 1851, 1843, Feb. 7. Apr. 29. Apr. 30. C.P.S., Chev. of the Prussian Order “ Pour le Meérite,” Mem. Imper. Acad. Sc. Petersburg, Roy. Acadd. Sc. Berlin, Brussels and Gottingen, Roy. Soe. Se. Drontheim, Acad. Se. Philadel- phia, Econ. Soc. Silesia, Nat. Hist. Soc. Laus- anne and Roy. Batavian Soc., Corr. Mem. Roy. Acad. Sc. Turin, Nat. Instit. Washington U.S., Geogr. Soce. Paris, Berlin and Petersburg. 13, Ashley Place, Westminster, London, S.W. Sedgwick, Rev. Adam, M.A., F.R.S., Hon. M.R.I.A., F.G.S., F.R.A.S., Woodwardian Lecturer. Cambridge. Stokes, George Gabriel, M.A., D.C.L., Secr. B.S., Lucasian Professor of Mathem., F.C.P.S., Mem. Batay. Soc. Rotterdam. Pembroke Col- lege, Cambridge. Tayler, Rev. John James, B.A., Principal of Man- chester New College. London. Thomson, William, M.A., LL.D., F.R.S.S. L. & E., Professor of Nat. Philos. Univ. Glas- gow. 2, College, Glasgow. Whewell, Rev. William, D.D., F.R.S., Hon. M.R.LA., F.S.A., F.GS., F.R.A.S., Master of Trinity College, Cambridge. Lodge, Cam- bridge. Woodcroft, Bennet, Professor, Superint. of Regist. of Patents. Southampton Buildings, London, W.C. 23 ORDINARY MEMBERS. Date oF ELECTION. 1857, Jan. 27. Acton, Henry Morell, B.A. 1856, Apr. 29. Adshead, Joseph. 1839, Apr. 30. Ainsworth, Ralph Fawsett, M.D. 1854, Jan. 24. Allan, James, Ph.D., M.A. 1837, Aug.11. Ashton, Thomas. 1846, Jan. 27. Atkinson, John. 1824, Jan. 23. Barbour, Robert. 1852, Jan. 27. Barlow, Henry Bernoulli. 1842, Apr.19. Barratt, Joseph. 1849, Apr.17. Bassnett, Rev. Richard, M.A. 1840, Jan. 21. Bateman, John Frederick, M.Inst. C.E., F.G.S. 1858, Jan. 26. Baxendell, Joseph, F.R.A.S. 1847, Jan. 26. Bazley, Thomas, M.P. 1847, Jan. 26. Bell, William. 1857, Apr.21. Bellhouse, Edward Taylor. 1858, Jan. 26. Benson, Davis. 1844, Jan. 23. Bevan, James. 1854, Jan. 24. Beyer, Charles. 1842, Jan. 25. Binney, Edward William, F.R.S., F.G.S. 1821, Jan. 26. Blackwall, John, F.L.S. 1855, Jan. 23. Bowman, Eddowes, M.A. 1839, Oct. 29. Bowman, Henry. 1859, Jan. 25. Brittain, Thomas. 1857, Apr.21. Brock, John. 1855, Apr. 17. Brockbank, William. 1824, Apr. 30. Brooke, Edward. 1844, Jan. 23. Brooks, William Cunliffe, M.A. 1846, Jan. 27. Browne, Henry, M.B. 1846, Jan. 27. Burd, John. 1854, Jan. 24. Callender, William, jun. VOL. XV. J ZZ 24 DATE OF ELECTION. 1847, Jan. 26. Calvert, Frederick Crace, Ph.D., F.R.S., F.C.S., Corr. Mem. Roy. Acad. Sc. Turin, Acad. Se. Rouen, Pharmac. Soc. Paris and Industr. Soc. Mulhouse. 1859, Jan. 25. Carrick, Thomas. 1858, Jan. 26. Casartelli, Joseph. 1855, Jan. 28. Cawley, Charles Edward, M. Inst. C.E. 1852, Apr. 20. Chadwick, David, F.S.S., Assoc. Inst. C.E. 1842, Jan. 25. Charlewood, Henry. 1854, Apr. 18. Christie, Richard Copley, M.A., Prof. Hist. Owens College. 1857, Apr. 21. Churchill, George Cheetham. 1841, Apr. 20. Clay, Charles, M.D. 1853, Apr.19. Clift, Samuel, F.C.S. 1853, Jan. 25. Corbett, Edward. 1858, Jan. 25. Cottam, Samuel. 1859, Jan. 25. Coward, Edward. 1851, Apr. 29. Crompton, Samuel. 1839, Jan. 22. Crossley, James, F.S.A., Pres. Chet. Soc. 1848, Jan. 25. Crowther, Joseph Stretch. 1843, Apr. 18. Curtis, Matthew. ‘ ‘ 1854, Feb. 7. Dale, John. 1842, Apr.19. Dancer, John Benjamin, F.R.A.S. 1853, Apr. 19. Darbishire, Robert Dukinfield. 1854, Jan. 24. Davies, David Reynold. 1842, Nov.15. Dean, James Joseph. 1855, Jan. 28. Dickinson, William Leeson. 1859, Jan. 25. Dorrington, James. 1850, Apr. 30. Dyer, Frederick Nathaniel. 1818, Apr. 24. Dyer, Joseph Chesborough. 1859, Jan. 25. Eadson, Richard. 1856, Apr. 29. Ekman, Charles Frederick. 1854, Jan. 24. Ellis, Charles. 1850, Apr. 30. Fairbairn, Thomas. 25 DatE oF ELECTION. 1824, Oct. 29. 1849, 1842, 1817, 1856, 1857, 1855, 1854, 1840, 1836, 1847, 1837, 1817, 1849, 1848, 1844, 1858, 1839, 1859, 1828, 1854, 1815, 1833, 1851, 1845, 1848, 1839, 1820, 1854, Oct. Jan. Jan. Jan. . 29. atte . 20. . 20. 24. Fairbairn, William, C.E., F.R.S., F.G.S., Corr. Mem. Imper. Inst. France and Roy. Acad. Se. Turin. Fairbairn, William Andrew. Fleming, David Gibson. Flint, Richard. Forrest, Henry Robert. Foster, Thomas Barham. Fothergill, Benjamin. Fryer, Alfred. Gaskell, Rev. William, M.A. Giles, Samuel. Gould, John. Graham, John. Greg, Robert Hyde, F.G.S. Greg, Robert Philips, F.G.S. Grundy, John Clowes. Hampson, Richard. Harrison, William Philip, M.D. Hawkshaw, John, F.R.S., F.G.S., M. Inst. C.E. Heelis, Thomas. Henry, William Charles, M.D., F.R.S. Hetherington, John. Heywood, Sir Benjamin, Bart., F.R.S. Heywood, James, F.R.S., F.G.S., F.S.A. Higgin, James. Higgins, James. Higson, Peter. Hobson, John. Hodgkinson, Eaton, F.R.S., F.G.S., M.R.LA., Hon. Mem. R.I.B.A. Inst. C.E., Roy. Scot. Soc. Arts, and Soc. Civ. Eng. Paris, Prof. of the Mech. Prine. of Engineering Univ. Coll. London. Holcroft, George. 26 Date or ELECTION. : 1855, Jan. 23. Holden, Isaac. 1846, Jan. 27. Holden, James Platt. | 1823, Apr.18. Hopkins, Thomas, M. Brit. Met. Soc. | 1824, Jan. 23.. Houldsworth, Henry. 1857, Jan. 27. Hunt, Edward, B.A., F.C.S. 1859, Jan. 25. Hurst, Henry Alexander. : 1857, Jan. 27. Jackson, George. . 1823, Jan. 24. Jesse, John, F.R.S., F.R.A.S., F.L.S. 1850, Apr. 30. Johnson, Richard, F.C.S. 1856, Jan. 22. Johnson, William. 1821, Oct. 19. Jordan, Joseph. 1848, Apr.18. Joule, Benjamin St. John Baptist. 1842, Jan. 25. Joule, James Prescott, LL.D., F.R.S., F.C.S., Hon. M.C.P.S., Corr. Mem. Roy. Acad. Se. Turin. 1846, Jan. 27. Joynson, William. 1843, Jan. 24. Kay, Samuel. 1852, Jan. 27. Kennedy, John Lawson. 1822, Apr.26. Lane, Richard. 1830, Apr. 30. Langton, William. 1850, Apr. 30. Leese, Joseph, jun. 1856, Jan. 22. Lingard, Thomas Taylor. 1839, Oct. 29. Lockett, Joseph. 1852, Jan. 27. Long, Isaac Waithman, F.R.A.S. 1857, Jan. 27. Longridge, Robert Bentink. 1842, Apr.19. Love, Benjamin. 1854, Jan. 24. Lowe, George Cliffe. 1850, Apr. 30. Lund, Edward. 1855, Jan. 28. Lund, George Taylor. 1859, Jan. 25. Lynde, James George, M. Inst. C.E., F.G.S. 1855, Oct. 30. Mabley, William Tudor. 1829, Oct. 80. McConnel, James. 1838, Apr.17. McConnel, William. 27 Date oF ELECTION. 1844, 1823, 1859, 1849, 1858, 1842, 1837, 1859, 1849, 1859, 1848, 1822, 1852, 1854, 1827, 1844, 1844, 1833, 1841, 1857, 1854, 1857, 1854, 1859, 1836, 1847, 1859, 1823, 1822, 1858, Apr. Jan. Jan. Apr. Apr. Jan. Jan. Jan. Jan. Jan. Jan. Apr. Jan. Feb. Jan. Apr. Apr. Apr. Apr. Apr. Jan. Jan. Feb. Apr. Apr. Jan. Jan. Apr. Jan. Jan. 30. 24. 25. Wife McDougall, Alexander. Macfarlane, John. Maclure, John William. Manchester, The Right Rev. the Lord Bishop of, D.D., F.R.S., F.G.S., F.C.P.S., Corr. Mem. Arch. Inst. Rome. Mather, Colin. Mellor, Thomas. Mellor, William. Molesworth, Rev. William Nassau, M.A. Morris, David. Mosley, George. Neild, Alfred. Neild, William. Nelson, James Emanuel. Nevill, Thomas Henry. Nicholson, William. Ormerod, Henry Mere. Parr, George. Parry, John. Peel, George, M. Inst. C.E. Platt, William Wilkinson. Pochin, Henry Davis. Poynting, Rev. Thomas Elford. Ramsbottom, John. Ransome, Arthur, M.A., M.B. Cantab., M.R.C.S. Ransome, Joseph Atkinson, F.R.C.S. Ransome, Thomas. Rideout, William Jackson. Roberts, Richard, M. Inst. C.E. Robinson, Samuel. Roscoe, Henry Enfield, B.A., Ph.D., F.CS., Professor of Chemistry, Owens College. 28 DATE oF ELECTION. 1842, Jan. 25. Royle, Alan. 1851, Apr.29. Sandeman, Archibald, M.A., Professor of Mathe- matics, Owens College. 1847, Jan. 26. Satterthwaite, Michael, M.D. 1842, Jan. 25. Schunck, Edward, Ph.D., F.R.S., F.C.S. 1858, Oct. 19. Sever, Charles. 1855, Jan. 23. Sharp, Edmund Hamilton. 1835, Oct. 30. Shuttleworth, John. 1852, Apr. 20. Sidebotham, Joseph. 1859, Jan. 25. Slagg, John, jun. 1838, Jan.‘26. Smith, George S. Fereday, M.A., F.G.S. 1845, Apr.29. Smith, Robert Angus, Ph.D., F.RS., F.CS., Corr. Mem. I. R. Geol. Instit. Vienna. 1859, Jan. 25. Sowler, Thomas. 1851, Apr. 29. Spence, Peter. 1852, Jan. 27. Standring, Thomas. 1834, Jan. 24. Stephens, Edward, M.D. 1847, Apr. 20. Stephens, James. 1858, Jan. 26. Stewart, Charles Patrick. 1814, Jan. 21. Stuart, Robert. 1859, Jan. 25. Tait, Mortimer Lavater. 1856, Jan. 22. Taylor, John Edward. 1859, Jan. 25. Thompson, James. 1836, Apr. 29. Turner, James Aspinall, M.P. 1821, Apr.19. Turner, Thomas, F.R.C.S. 1857, Jan. 27. Walker, Robert, M.D. 1823, Jan. 24. Watkin, Absalom. 1859, Jan. 25. Watson, John. 1857, Jan. 27. Webb, Thomas George. 1858, Jan. 26. Whitehead, James, M.D. 1839, Jan. 22. Whitworth, Joseph, F.R.S. 1859, Jan. 25. Wilde, Henry. 1859, Apr.19. Wilkinson, Thomas Read. 1853, Apr.19. Williamson, Samuel Walker. i Date or ExxEction. 1851, Apr. 29. Williamson, William Crawford, F.R.S., Professor of Natural History, Owens College. 1851, Jan. 21. Withington, George Bancroft. 1836, Jan. 22. Wood, William Rayner. 1855, Oct. 30. Woodcock, Alonzo Buonaparte. 1846, Apr.21. Woodhead, George. 1839, Apr..30. Woods, Edward. 1840, Apr. 28. Worthington, Robert, F.R.A.S. CORRESPONDING MEMBERS. 1824, Jan. 23. Dockray, Benjamin, Lancaster. 1849, Apr. 17. Girardin, M. J., Corr. Mem. Imper. Instit. France, Rouen. 1850, Apr. 30. Harley, Rev. Robert, F.R.A.S., Brighouse, York- shire. 1816, Apr. 26. Kenrick, Rev. John, M.A., York. 1838, Apr. 17. Koechlin-Schouch, Daniel, Mulhouse. 1859, Jan. 25. Le Jolis, Augustus, Hon. Perpet. Archiviste Imper. Soc. Nat. Sc. Cherbourg, Mem. Imper. Leopold- Carol. Acad. Nat. Sc., Cherbourg. 1857, Jan 27. Lowe, Edward Joseph, F.R.A.S., F.G.S., M.B.M.S., Hon. M. Dublin Nat. Hist. Soc., M, Geol. Soc. Edinb., &c., Nottingham. 1851, Apr. 29. Pincoffs, Peter, M.D., Chev. of the Turkish Order of the “Medjidié” 4th Cl., Mem. Coll. Phys. London, Brussels and Dresden, Hon. and Corr. Mem. Med. and Phil. Soce. Antwerp, Athens, Brussels, Constantinople, Dresden, Rotterdam, Vienna, &c., &c., Con- stantinople. 1853, Apr. 19. Wilkinson, Tigi Tuner, F.R.A. a Bur re Lancashire. MANCHESTER: PRINTED BY CHARLES SIMMS AND CO, P a