s^xy-^ MEMOIRS LITERABT AND PHEOSOPHICAL SOCIETY MANCHESTER. BRAMHAW AND BtACKLOCK, PRINTEHa, MANCHESTEB AND hOSDOS. ^. a //, A /A MEMOIRS LITERARY PHILOSOPHICAL SOCIETY MANCHESTER. Secottir %tvie»* VOLUME NINTH. LONDON: H. BAILLIEEE, PUBLISHER, 219, REGENT STREET, AND 290, BROADWAY, NEW YORK. PARIS : J. B. BAILLIERE, LIBRAIRE, RUE HAUTEFEUILLE. BBADSHAW & BLACIiLOCK, 47, BROWN STREET, MANCHESTER. MDCCCLI. 1 y\ ^ \ «_, rr *, 'r>,J NOTE. 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- ticulars the Society must not be considered as in any way responsible. i'^il, MANCHESTER. I. — Report o/PfeTER Clare, F.i2.^.>S.,ancf John Fredbrig Bateman, F.G.S., M. Inst C.E., being the Committee appointed for superintending the Measurement of Bain falling along the Lines of thi Rochdale^ Ashton-under- Lyne^ and Peak Forest CanaU. With Observations upon the Retutms, and other particulars. By John Frederic Bateman. Read April 11, 1848. ^HREE years have elapsed since the last Repott on this subject was presented to the Society. During this period the observations have been regularly continued, and the f-esults are as follow. It is almost unnecessary to remarkj that the old rain gauges, called the Canal Company's gauges, are placed on the roofs of dwellings, and the Society's gauges on the ground, the observations being taken with a view of ascertaining the difference in the quantity registered in the two situations. REPORT AND OBSERVATIONS ON ALONG THE LINE OF THE ROCHDALE CANAL. SLATTOCK3, MOSS LOCK, TOLL BAE, near near JBlackstoae BLACK HOUSE Middlcton, Rochdale, Edge, Edge, near 460 feet above about 500 feet about 1000 feet Ripponden. 1845. sea. above sea. above sea. above sea. Canal Isodety CCS Gauge. Gauge. 1 " Canal Cc's Gauge. Society Canal Co.'s Gauge. Society Cf°?l Society Canal Cc's Gauge, Society Canal SodetT Gaage. Gauge, Gauge. Gauge, Gauge. In. dec i in. dec. in. dec. 1 in. dec in. dec in, dec in, dec iB.dec in, dec! in. Ipc January 1 47 1 ... 180 2 05 i 3 8 2 09 4 0 1 80 2 5 1 26 I o FebrtiaiJy ... 127 1 ... 1 05 0 90 12 0 80 13 0 51 1 1 000 0 5 March 2 10 ... 1 85 ! ... 1 60 1 40 2 09 6 0 2.30 3 0 2 30 5 1 1 99 3 0 3 00 2 00 4 9 3 0 1 9! 1 17 2 1 5 q April May 130 1 08 1 90 1 3 8 1 86 4 0 1 80 2 7 1 81 2 1 2 75 3 25 2 55 2 85 3 80 5 0 3 80 4 3 3 36 3 70 5 0 4 5 3 60 3 91 3 7 4 7 2 01 •) 5 0 July 3 17i 4 August 4 75 6 00 5 00 8 3 4 60 8 a 4 80 8 0 4 93 5 5 September... 3 17 ... 2 55 3 60 4 7 3 30 4 6 3 40 6 3 2 91 2 3 October 3 85; ... 3 45 4 36 8 9 4 20 8 8 4 20 7 0 3 02 3 5 November ... 2 20 ... 2 15 2 90 4 3 2 12 4 2 2 70 5 1 1 81 2 2 December ... 4 25 1 ... 3 95 7 15 6 70 12 0 7 70 12 8 4 03 ! 5 4 32 21, ... 29 43 39 85 52 3 37 02 64 7 39 42 61 8 28 03 34 3 ill mo. i 1846. January 3 15 ... 4 60 4 5 5 00 5 7 4 85 fi 1 4 15 5 0 2 79 3 4 Februai-y ... 0 95 ... 0 65 1 0 1 80 ! 2 0 174 2 0 2 00 1 4 5 0 61 1 2 1 .50 ... 2 88 1 ... 1 00 ... 1 35 2 40 2 1 3 8 2 70 4 0 4 18 5 4 1 00 0 9 2 60l 3 1 2 08 4 55 1 15 2 72 3 9 5 1 1 0 3 5 2 60 3 5 4 20 6 2 1 20 18 2 68 4 0 1 95 •? 5 4 0 6 April 5 19 1 5 May..... 0 80 111 000 1 84 0 2 June 2 00 1 70 3 0 July 3 30 2 75 3 0 3 20 1 4 1 2 80 4 0 3 40 5 2 1 31 2 9 August 3 97 2 45 3 8 5 50 3 4 5 20 3 8 4 00 j 2 8 1 25 2 1 September... 1 20 0 85 1 4 1 06 2 6 1 00 2 5 0 90 ! 1 0 0 79 0 9 October 3 91 2 50 4 2 6 20 5 7 4 50 5 6 5 00 i 5 5 3 95 4 4 November ... 1 70 1 76 ! 2 1 3 70 4 3 3 50 j 4 4 3 60 ! 3 0 1 62 9 0 December ... 0 20 j ... 1 05 j 1 7 1 20 2 0 1 00 2 0 1 07 1 1 0 0 18 0 5 25 76 22 76 j 31 7 37 14 \ 43 2 35 09 43 9 34 80 42 5 21 48 27 9 1847. January 100 1 10 1 4 1 90 3 2 1 70 3 1 1 80 3 0 1 55 I 6 February ... 2 50 141 1 9 2 90 3 8 263 3 4 2 70 3 0 1 32 2 2 0 90 1 63 2 30 2 90 ) 80 1 65 0 .S4 'd'm'ged 1 29 ^d'mVivi 1 40 2 40 3 60 2 3 3 8 3 4 1 CO 2 30 3 72 2 47 0 75 1 80 2 6 4 0 3 5 2 5 1 3 2 3 1 20 2 20 3 90 2 57 0 60 1 90 no ret. 2 5 3 5 3 4 1 3 2 3 0 95 1 25 2 87 1 20 0 25 0 87 0 2 3 1 0 1 9 2 6 9 8 6 April May 3 25 2 55 0 80 1 35 3 5 3 1 1 6 2 0 2 60 2 6 1 July 0 65 1 70 1 1 2 3 August September... 3 80 3 41 4 5 360 4 4 390 4 5 3 40 3 a 1 10 3 0 October 4 00 4 00 4 3 4 50 5 5 4 30 5 6 4 30 5 5 2 77 3 8 November ... 3?0 3 15 3 8 4 30 4 5 3 80 4 5 3 90 4 5 1 55 2 4 December ... 4 20 380 5 2 6 30 11 0 6 20 1 11 0 6 50 13 8 4 63 3 0 29 38 26 95 31 3 35 75 47 -8 34 67 ! 48 1 34 97 i 46 3 20 3! 26 9 10 mo. 1 ' 1 |!1 mo. THE MEASUKEMENT OF KAIN. 3 ' On the Peak Forest and Ashton Canals, there are only two places where gauges remain on the tops of houses. These are at Marple and at Comb's Reservoir, both on the Peak Forest Canal. The comparative results between the old gauges placed on the tops of the houses, and the Society's gauges placed on the ground, are as follow : — 1845. January,.. February March ... April May June July August ... September October... November December Marpi.e, 531 feet. Old Society"! Gauge. Gauge. In. Dec 1 04 0 48 1 01 1 36 0 71 2 82 3 60 6 31 1 92 1 94 1 20 2 28 23 67 In. I'eo. 2 20 2 50 1 70 3 00 1 45 4 25 4 75 8 25 2 75 2 70 1 75 3 50 38 80 Comb's Reservoir, 850 feet. Old Gauge. In. Dec. 1 68 0 79 2 00 1 68 1 85 3 57 3 97 7 62 4 00 2 75 2 36 4 73 Society's Gauge. In. Dec. 3 70 1 80 3 60 3 00 2 70 5 00 6 10 9 10 5 00 4 10 4 00 8 00 36 90 66 10 Mr. Wood, the Canal Company's engineer, in whose charge the gauges are placed, is of opinion that the Society's gauge at Combo's Reservoir for this year (1845) indicates a greater quantity of rain than was received by the gauge, as he found it to be leaky on removing it to a more conve- nient position. The returns do not show, apparently, any greater discrepancy than is to be observed at other places where no leakage occurred. REPORT AUB QHSfiRVAXJONB ON 1846. 531 feet.' Old Gauge. Society's Gauge. Comb's Rese^toib, 850 feet. Old I Spcj^y's Gauge. Gauge. January.... February . March .... April May June July August .... September. October.... November . Decemljer . 1847 Ji^nuary ,... February March April May June July August September October November December In. D«c. 2 81 0 54 1 16 3 46 0 60 1 80 2 37 2 52 0 94 6 11 1 23 1 03 In. Deo. 4 15 0 80 1 60 5 00 0 85 2 45 In. Deo. 3 23 23 57 32 35 1 20 2 18 0 75 2 30 part 23 57 33 50 22 71 22 16 26 37 1 58 2 28 1 14 2 35 4 15 2 88 0 93 3 48 29 .93, 88 10 2 60 3 50 1 60 4 00 5 30 3 90 1 00 3 40 8 20 5 40 4 60 7 90 I 45 70 , 35 57 61 30 Accompanying the above returns from the Peak Forest and Ashton Canals, there have also been receiv^4 t^^ tct turns of the rain which has fallen at other places on or near the lines of the Canal, as indicated by gauges which ham always been placed n^ar the level of the ground. are as follow : — TI^^ MEAi^UB^ENT QF ^MJSI. « 184& January February March April May..,..^ Ji)ne ..,.^..._... J^ly August September.... October November .... Decenjber ..,. 1846. Jf^nuary Febinufj March April May June July ,,..,,,.,.... August September.... .,< October November December ....... Total Fairfield, 320 feet. Cld 6*uge. ^ few feet alvbve Ground. New Gauge, 1 foot above Ground. Water- houses I-ock, Ash ton Canal, seofeet above sea. Inclined Plane, Cbapel- le-Frith, 1121 feet. 20 60 60 90 30 20 60 30 00 70 S 00 7 10 42 60 3 60 1 30 1 70 3 00 0 90 2 90 3 90 3 60 0 80 5 30 8 70 2 10 32 80 90 00 20 40 60 90 80 6 40 2 70 4 00 3 00 8 00 as 90 3 30 I 30 1 70 3 00 0 90 2 40 3 90 3 60 0 80 4 70 3 10 1 50 3 90 1 40 1 80 3 60 0 80 2 80 4 20 3 80 0 70 4 90 2 40 1 60 30 20 31 80 6 00 42 80 4.3 2Q 38 80 For the year 1847, returns have been received of the rain which has fallen at various places within the district traversed by the Peak Forest and Macclesfield Canals, and the Manchester, Sheffield and Lincolnshire Railwa,y, in ad- dition to those already alluded to. Many of these are of agi cijperipi^ntal nature. Those which appear to bear upoij & REPORT AND OBSERVATIONS ON the subject of this paper are introduced in the following table : — 1847. Fairfield, 320 feet above Sea- h it. m ^1 ■c - n Comb's Mow, 1670 feet ITT feo« IP ml iSjCO si [i O OS * Old Gauge. New Gauge, January February 1 50 2 90 0 80 2 20 4 95 3 20 1 00 4 00 5 50 4 90 3 50 5 20 1 50 3 00 0 80 3 30 5 10 3 30 0 95 3 70 5 50 4 80 3 50 5 30 1 20 2 00 1 20 2 80 530 3 10 0 80 3 30 5 50 4 70 3 80 4 60 2 80 3 90 1 30 3 90 4 50 3 20 1 SO 2 90 6 40 4 30 3 60 5 70 \ 78 2 33 0 88 3 08 4 17 3 70 0 97 2 95 6 17 3 71 3 05 5 60 1 14 1 68 0 77 1 93 3 95 1 05 0 95 2 40 4 40 2 77 3 35 .5 05 2 60 1 78 0 79 2 45 4 50 3 44 1 S3 2 96 4 17 4 48 i 52 4 63 1 05 2 58 3 91 3 18 1 43 4 77 6 35 2 97 2 84 3 95 2 20 2 64 1 84 3 21 3 16 2 62 1 29 1 57 2 98 4 72 2 45 4 44 April May July September .... November .... December 39 65 40 7r< 38 30 44 00 38 39 29 44 35 85 33 03 lOmos. 33 12 The gauges employed at the five new places in the pre- ceding table, viz., Todd's Brook Reservoir, Brink's Edge, Comb's Moss, Bosley Reservoir, and Woodhead, are all similar, and of a new construction. The results indicated by them vary so greatly from those of other gauges in their immediate neighbourhood, as to occasion great suspicion of their accuracy. For instance, the gauge at Woodhead, which is situated at the head of the Longdendale Valley, hereafter alluded to, shows only 33 inches of rain at an elevation of 1000 feet above the sea. From this district it was ascertained by careful daily measurement, that 40^ inches of water had actually flowed off the ground, and the rain indicated by two neighbouring gauges in the same dis- trict, within a distance of about 2^ miles, was, respectively, THE MEASUKEMENT OF RAIN. 7 50^ inches and 62^'^ inches — the latter being at an elevation of about 1750 feet, and the former 700 feet above the sea. It seems exceedingly probable, from an examination of this new gauge, that it is liable to be affected in a serious degree by evaporation, and the difference may perhaps be assigned to this cause. The whole apparatus is placed above the surface of the ground — the water is received by a metal funnel, and conducted from thence by a short pipe at the bottom, through the top of a wooden box, to a glass bottle of large area placed within the box and open at the top. The water is not covered by a float, and the surface is there- fore in contact with the atmospheric air. It is not suggested that the evaporation would be as great as from the surface of an open pond; but that it does take place to a consider- able degree seems to be evident. As evaporation is greater in proportion to the altitude of the situation, the supposition that the discrepancy is to be assigned to this cause, will also account for the anomaly (as compared with the general result of all the other returns), at Todd's Brook and Comb's Moss, where it would appear, from the results furnished by these new gauges, that less rain falls on high land than on low. If the returns were properly corrected by a due allowance for evaporation according to the altitude, the true state of the case would probably be found to agree with the evidence from other places. The rain gauge at the top of the inclined plane at (^hapel- le-Frith is an exception to this rule, as it appears to show, pretty regularly, less rain than falls at Comb's Reservoir at a lower elevation. This may probably arise from local causes, as it is at variance with the general testimony within the range of the same elevation. All the Society's gauges, — those at Fairfield, and that at the top of the inclined plane at Chapel-le-Frith, put down by the Canal Company, — are cylindrical gauges, with an S REPORT AJriJ OfiS^if ATtONS ON Upright graduated rod attached to a flod.t covering the surface of the water, which itidicates the depth of rain cailght within the Cylinder. Objections have been made to this fbrtn of gauge from the effect alleged to be produced by thd rod as it rise^ above the level of the top of the cylinder, exposing addi- tional surface, and adding in that rnahner tO the collectitig surface of the gauge. £xperitrients have befen ttidde fdt sotfie years by Ml'. Wood, the engitiecft to the Ashton atid Peak Forest Canals, f o ascertain the effect produced hy tods standing sotrie height above the top of the gauge. That a Sensible effect is produced, incre^irlg the quantity of water caught, appeal^ to be cleariy established from thes^ experiments ; but they do liot afford any assistance in det^f- minidg the hfeigbt at which the rod begins to affect the accuracy of the register, not do they fbrnish atiy data frotil which to calculate the ptopoi*tionate increased quantity du^ fo the elevation of the rod. For instariCe, in one situation a rod Of 1 inch ifli did- iiietef, fetauditig 24 inches afeov'e tfce ib^ Ht the gaugd, collected in 12 months 3-18 Inches Of taiti ; While in atiothei* ptaCe a tod of the same diametef, but Only ISf iriche^ ^otig^ collected in the same time 27 -S^ inches, the quantity of raiii as indicated by the raia gauges beitig respectitely 4d'75 dtid 61-30. Again, a staff or rod of 2 inches diameter, and l8| itiches long, collected in 12 months 20-67 inches, the raiii being apparently at that place SS'SS'. At another plaCC^ a rod at fhe same diametef and length, where the rairi appears td have been 35 -85, collected in the same period 58 '99 inched. These differences are enormous, and ap^iarently anac^ countable. dii cohdderation, it appears probable that the rod wllf produce fad feasible effect until it rises tO a height greatW THE MEASUREMENT OF BAIN. 9 than half of the diameter of the gauge. This supposes that the rain will not often reach the ground at a more acute angle than 45°. If the rain descend at that angle, and the rod stand at a height equal to half the diameter of the gauge, any rain which would be intercepted by the rod would have fallen, had the rod not been there, within the area of the top of the gauge, as will be seen by the sketch in the margin. vs ^•■*\'^v^XV\v;> V If the rain come straight down, or v^'^^N^^^^S^^Vv^X^^ \ nearly so, as it often does in heavy „ „ 136-98 „ Sea Fell 3166 „ „ 12815 " An apparent exception to this law occurs at the gauge stationed at Burnt Rigg, about midway between the top of Stye-Head and the vale of Wast dale, and which in 1847 has received about one- eighth, or twelve and three quarters per cent., less rain than the valley. " This is the only one of the gauges situated on the slope of a mountain ; it is on the windward side, and I imagine that in such a position, eddies or counter-currents are produced in windy weather which cause a less quantity of water to be deposited in the instrument than is due to the elevation. We know that all sloping roofs, from the same cause, materially diminish the receipts of rain gauges. " It will be observed that the amount of water received by the Seatollar gauge, is invariably less than the deposit in the adjacent vale of Sea- thwaite, and the deficiency is pretty equable in every month of the year. •' I am -unable to give any satisfactory reason for this anomaly, or to * The height of Seatollar common has not been accurately asoertained. 12 REPORT AND OBSERVATIONS ON Chester with water from the hills beyond Stalybridge and Mottram— lying at a distance of from ten to twenty miles east of Manchester. In the highest part of this range of hills, known by the name of the Penine Chain, the river Etherow and its various mountain tributaries take their rise. Some of these uniting near Woodhead, form there a deep romantic valley account for the very great excess of rain in this valley over all others in the Lake districts. As the gauge on SeatoUar is two or three miles dis- tant in a direct line from the others, the near approach of its receipts to those of the Stye-Head gauge, both about the same elevation, is rather remarkable. lu 1846 the Seatollar exceeded the Stye gauge in quantity, which it should do if the assumed height be correct. " By referring to the table for the summer months, we find that be- tween the 1st of May and the 31st of October, the gauge at 1290 feet has obtained nearly twelve per cent, more rain than the valley ; at 1334 feet, nine and a half per cent, more ; at 1900 feet, twenty-nine per cent, more ; at 2928 feet, seven and a half per cent, less ; and at 3100 feet, thirteen and a half per cent, less than the valley. In the winter months (November to April inclusive) the gauge at 1290 feet has received four and a half per cent, more than the valley ; at 1334 feet, the same quantity as the valley ; at 1900 feet, eleven and a quarter per cent, more ; at 2928 feet, thirty-eight and a half per cent, less, and at 3100 feet, forty-two and a half per cent, less than the valley. " The difference in the proportion to the valley between the winter and summer half-year, as shown by the tables, is rather startling. " When much snow falls, no doubt a considerable portion is lost to the gauge, either by its being blown out of the funnel, or by tlie aperture getting choked up. But I do not think that this cause alone is at all adequate to account for the great comparative deficiency in the winter season ; for there was very little snow on the mountain tops during the winter 1846-7, less, I am told by one of the oldest inhabitants of the Fell dales, than he almost ever remembers. At "WTiitehaven, we had no snow worth naming, except on the night of the 23rd of December, where it lay to the depth of nearly an inch on the ground, but disappeared in course of the ensuing morning. " The late Mr. Crosthwaite of Keswick, by means of marks on the side of Skiddaw, and with the assistance of a telescope at his residence, made two or three daily observations on the heights of clouds for several years ; and it is clearly shown by his tables, that the clouds are lowest in the THE MEASUBEMENT OF RAIN. 13 called Longdendale, running for several miles nearly due west between hills which rise abruptly on each side to a considerable height, reaching in some cases nearly 2000 feet above the level of the sea. The valley is hemmed in to the west by the high land at Mottram, which, however, is not high enough to intercept the clouds driven before the westerly winds. three first and three last months of the year. Moreover, Dr. Dalton affirms that the clouds are seldom a mile high (or little more than one and a half times the height of Sea Fell), in our climate, in winter. Now the Doctor here probably alludes to, or at least includes, the most elevated clouds, such as the Cirri, and some variety of the Cirrostratus. But there can be no doubt, that between the months of November and March the under surface of the Nimbus or raiu cloud (the lowest except the Stratus) is far below the tops of our highest mountains, and, I have reason to believe, not imfrequently its upper surface also ; when this is the case, the gauges on Sea Fell, Gabel, &c., will receive no rain at all, when it is descending abimdantly in the valleys beneath. The lowness of the rain- cloud at this season, is, I apprehend, the principal cause of the small quantity of rain in proportion to the valley, during the winter as com- pared with the summer months." He also observes as to the value of the experiments, that " they have already shown us that at least sixty inches more rain is deposited in Eng- land than we were previously aware of: — that one hundred and fifty inches sometimes descends in the Lake districts in a year, — more than falls in most parts of the Tropics with which we are acquainted, and sufficient to drown standing two of the tallest men in Great Britain, one on the top of the other. They have further informed us, that six and a half per- pendicular inches of water is sometimes precipitated from the atmosphere in twenty-four hours, and ten inches in forty-eight hours ; a quantity which would be thought large for any two consecutive months in most parts of England. We have further ascertained that the almost incredible depth of thirty inches occasionally descends in a single month — a fall nearly equal to the calculated yearly average for all other parts of Eng- land. The experiments have, in short, enabled us to collect a number of new and curious facts, bearing on the quantity and very unequal distribu- tion of rain in this island. We have also ascertained with a high degree of probability, the law of the gradation in the amount of rain, at various intermediate points, between the valleys and the tops of the highest moun- tains. 14 KEPOBT AND OBSERVATIONS ON Li the upper part of the valley the tributary streama, falling frora 1000 to 1200 feet in a few miles, join the main stream in the valley of Longdendale, nearly at right angles, thus breaking the surface of the country into various cross valleys and deep ravines. The summit of this district is Holme Moss, nearly 2000 feet above the sea. It is the highest eminence in the whole chain, though it rises but A little consideration will greatly lessen our surprise at the enormous quantity of water deposited in the hilly districts of Cumberland and West- moreland, and at the consequent unequal distribution of the rain in the climate of Great Britain. To those unacquainted with these localities it mfay be briefly stated, that the Lake District valleys radiate from a series of mountains of slate and primitive rock, having the Gabel, 2928 feet in height, as a nucleus or central point ; and in the immediate vicinity of which are Sea Fell and Pillar, of the respective elevations of 3166 and 2893 feet, and Great End, Bowfell, and Glaramara, not much inferior in altitude. These mountains are distant only about thirteen or fifteen miles in a direct line from the Irish Channel, and, as no hills intervene, they are consequently fully exposed to our wet and prevailing winds, which are the south-west. The warm south-westerly current arrives at the coast loaded with moisture, obtained in its transit across the Atlantic : — Now, our experiments justify us in concluding, that this current has its maximum density at about 2000 feet above the sea level ; hence, it will travel on- ward until it is obstructed by land of sufficient elevation to precipitate its vapour; and, rttaining a portion of the velocity of the lower parallel of latitude whence it was originally set in motion, it rapidly traverses the short space of level country, and with little diminution of its weight or volume ; but, on reaching the mountains, it meets with a temperatmre many degrees lower than the point at which it can continue in a state of vapour ; — sudden condensation consequently ensues, in the form of vast torrents of rain, which in some instances' must descend almost in a con- tinuons sheet, as when nine or ten inches are precipitated in forty-eight hours. When we reflect that a warm moist current, perhaps only 3° or 4o above the point of saturation, in coming in contact with the mountain ridge, probably meets with a stratum of air 10° or 15° lower than its own inherent temperature, we shall cease to marvel that such quantities as four, five, or even six inches of water should be deposited in these localities in the course of a few hours. The mountains are, in fact, huge natural con- densers, destined to force from the atmosphere the mighty volumes of water requisite for the supply of our lakes and rivers." THE MEASUREMENT OF BAIN. 15 slightly above the surrounding table land, the elevation of which, from the crests of the valleys just described, varies from 1500 to 1900 feet. — The millstone grit caps the sum- mits in various parts, forming occasionally perpendicular precipices several hundred feet in height; and in other parts the sides of the valleys consist of beds of indurated shale. Some considerable portions of the tops of the hills are covered with peat and others with gravelly clay. The more easterly portion of the district consists of the lower coal measures. For hills of this elevation it is scarcely possible to find any which, from their position and character, would be more likely to induce a large fall of rain, or to allow a larger pro- portion of that which falls to flow down the streams. It is from this district that the town of Manchester is to be supplied with water. Measurements of the volume of all the various streams have been made daily since the end of 1846, and rain gauges have been placed at various elevations and at different parts of the district, from which the quantity of rain which has fallen has been ascertained. In an adjoining valley, down which flows the Swineshaw Brook, a tributary of the river Tame, similar observations have been made since the end of 1844. This valley lies nearly east and west, and is completely land-locked, turn- ing abruptly to the north through a narrow glen just before it joins the river Tame. The summit of the valley is at Windyate Edge, which is the summit also of two tributaries of the Elherow, the Hollingworth and Amfield brooks. Rain gauges were placed at the bottom of the Swineshaw valley, near the point at which the volume of the stream was measured, and on Windyate Edge near the summit. For some time also a gauge was kept about midway. The brook was measured regularly twice a day. During the years 1846 and 1847, the index rod of the rain gauges 16 REPORT AND OBSERTATIONS ON was constantly tied down so as to prevent its rising above the top of the cylinder — and partially so during 1845. These observations therefore are free from objection on account of the additional surface exposed by the rod, though there may have been some loss from evaporation. In taking the rain-gauge observations in the Longden- dale district, the index rod was tied down from their com- mencement in November 1846 to Midsummer 1847, since which time the rod has been detached, and inserted only at the time an observation is being taken. The streams were measured, one (the river Etherow) three times a day, some twice a day, and others once a day. But in the results, which are given in monthly amounts, it has been found necessary to omit many months, in conse- quence of the gauges being frequently injured and rendered unfit for use for some days,' by the effects of violent and destructive floods. THE MEASUBEMENT OF BAIN. 19 SWINESHAW BROOK. RAIK, AND DEPTH OP WATER FtOWINO 0»F aBOUND. The extent of ground draining to the point at which the Tolume of tho stream is measured is 1250 statute acres. Baix. Flow. 1845. 5^^ ?; ♦» January... February . March April May Jiuoe July August .... September October... November December 2 6 2 8 4 3 3 4 3 10 In Dec. 3 1 1 8 5 5 4 7 14 8 4 1 5 9 4 0 11 0 6 0 9 6 3 2 In. Dec. 3 0 21 2 4 7 10 0 3 7 11 0 38 6 Deo. 668 764 892 424 436 364 13 648 3 048 7 236 2 736 4 608 2 904 6 624 27 166 Depth of rain in first six months 21 '2 inches, of which there flowed off the ground 13*548, or nearly two-thirds. Depth of rain in last six months 38*6 ins., of which there flowed off the ground 27*156 ins., or nearly three-fourths. Rain tor the whole year 59*8 ins., of which there passed down the brook 40*704, or upwards of two-thirds. It is possible that in this year the fall of rain has been registered too high, in consequence of comparative inatten- tion to the index rod.— The fall of rain in other places was just an average. 18 EEPOBT AND OBSERVATIONS ON SWINESHAW BROOK.— 1846. In this year the index rod of the rain gauge was tied down. The measurement of the stream was suspended during July, August, and September; but it is probable that the 7 inches supposed to have flowed off the ground during this period is not far from the truth. This year was con- siderably below the average fall of rain : nearly one of the driest on record ; and the above results may be taken as the fall and produce in such an extreme period. THE MEASUREMENT OF EAIN. SWINE SHAW BROOK.— 1847. 19 Rain. 1847. January February,... March April May June July August September.. October November... December... WhcJ.e Year 1 7 4 4 1 7 4 0 6 8 3 5 22 1 24 6 46 6 1 8 3 9 1 4 5 1 8 0 3 7 23 9 28 2 62 1 1 75 4 15 1 55 4 55 7 40 3 60 23 00 1 55 3 25 6 25 4 20 4 40 6 70 26 35 49 35 Flow. ■3>S P. »; bo ►-" o o 18 2 1 3 1 0 3 3 3 2 4 0 6 1 18 9 37 1 The fall of rain during this year was about an average, in some places rather more. It fell, however, very unequally, the last three or four months making up for previous deficiency. The proportion of the water flowing off the ground to that which fell, was about 3 to 4. The quantity of the rain- fall which was lost to the river, was about 12 ins. : that being apparently the annual amount required for evaporation, the supply of vegetation, and absorption by the ground, in a year of averse rain. The springs in this valley are very copious; and though neither the fall of rain, nor the average volume of the stream, are equal to the Longdendale district, yet the supply 20 REPORT AN1> OBSBBVATIONS ON of spring water in dry weather is greyer in proportion to the extent of drainage ground. In December, 1844, no rain fell, and yet the springs yielded a quantity of water equal to a depth of 1 inch over the whole drainage ground, the mean volume of the stream being the same as in August, 1847, in which month the fall of rain was 3^ inches. The following table shows the fall of rain for the year 1847, and for the two last months of 1846, at all the places in the district at which rain gauges have been put down. ' 1846. Brushes, 480 feet. Wind- yate Kdge, 1700 feet Crow- den Hall, 700 feet Rakes Moss, 1620 feet. Butterly Moss, 1750 feet Mean of all the Obser- Tations. Mean, omitting Brushes. November In. Deo. 2 4 2 1 In. I)«c. 2 4 2 7 In. Dec. 2 0 In. Dec. In. Dec. 9 4 S 0 In. Dec 2 4 2 9 In. Dec. 2 4 3 1 December 2 8 3 1 4 0 Two months 4 6 5 1 4 8 5 5 7 0 5 3 6 5 1847. 1 7 4 4 1 7 4 0 6 8 3 5 1 8 3 9 1 4 5 1 8 0 3 7 2 2 4 3 1 7 4 7 2 4 4 6 2 3 3 3 3 7 4 3 1 4 9 0 4 8 3 4 2 4 4 3 1 7 5 2 6 6 3 4 2 6 4 3 1 7 6 6 6 4 3 4 February March May 4 9 3 1 7 9 3 3 June Six months 22 1 23 9 26 9 23 8 26 6 23 6 23 8 July 1 6 3 2 6 5 3 5 3 8 5 9 1 5 3 3 6 0 4 9 6 0 7 6 1 4 a 6 7 6 6 1 6 0 6 1 2 0 5 1 8 7 4 4 4 3 8 2 3 1 6 5 8 2 6 4 8 3 6 0 1 5 4 3 7 4 4 7 6 5 6 7 1 5 4 6 7 6 4 9 6 9 6 9 September October November Six months 24 5 28 2 29 6 32 7 35 5 30 1 31 4 Whole year 46 6 52 1 50 6 56 6 62 1 53 6 55 2 THU MEASUBEMENT OF EAIIf. 21 In the preceding table, the last column, which is the mean of all the rain observations, omitting Bruslies on account of its being in another valley, may be taken as the mean fall of rain for the year 1847 in the Longdendale district, so fer at least as the gauges may be supposed to indicate the real quantity. The measurements of the streams, hovrever, lead to the belief that more rain has fallen than the rain gauges show. They are placed in the valleys, and at mean heights, none being quite on the tops of the hills; and it is probable that there heavy rain has fallen, and contributed to swell the streams, which has been beyond the range of the rain gauges. The next table exhibits the depth of water flowing off tl)e ground, as measured in various streams in the Longden- dale valley. It shows also the drainage or collecting ground to the point of measurement on each stream, and the mean^^ flow as deduced from all the observations. 2S REPORT AND OBSERVATIONS ON LONGDENDALE VALLEY. DEPTH OP WATER FT.OWINe OFF THE GROCND. 1847. Jan Feb March .. April ... May.... June.... July August Sept October Nov. ,.., Dec Rain duringl period of > observations^ In. Dec. 1 20 1 60 1 90 1 06 In. Dec. 1 70 1 80 1 08 1 41 10 40 9 24 5 76 9 95 4Mos. 25 63 22 42 '6 Mos, M In. Dee. 0 60 4 80 0 13 0 24 0 78 4 90 8 60 Si In. Dec. 3 6 2 2 0 7 1 8 6 2 6 9 5 6 20 05 26 9 25 80 7 Mos. 32 66 7 Mos. 49 3 64 7 11 Ms. In. Dec. 5 89 4 80 6 90 8 40 25 99 26 10 4 Mos Dec. 25 40 17 65 10 59 0 92 1 29 5 73 6 50 6 70 9 16 49 46 In. Dec 2 6 12 Ms. 56 2 12 Ms By comparing this Table with that of the Swineshaw brook in the same year, it will be seen that in the Long- dendale district the quantity of rain falling and that flowing off the ground are considerably greater. In the Swineshaw valley the mean rain was 49*35 inches, and the water flowing off* the ground 37'1 inches: In Longdendale the rain was 55-^is inches: the produce of which was 49^ inches — i. e., the rain in the two districts is as 49 to 55, the produce as 37 to 49. THE MEASUREMENT OF EAm. ^ By uniting the Swineshaw observations with those in Longdendale, the mean rain and flow would be as follows :— 1847. January ... February.. March April May June July August September October... November. December . Raik. 111. Dec. 2 36 30 70 22 48 40 62 32 38 66 48 74 53 56 Flow. In. Dec. 85 10 30 12 75 65 0 99 24 12 67 25 8 55 46 59 On examining the last Tables, it will be found that in many months, particularly during October, November, and December, the quantity of water flowing off the ground appears to be larger than the rain which fell during the same period. During months in which little rain fell, this would be ac- counted for by the produce of the springs ; but in periods of excessive rain, such as the last four months of 1847, in which the rain was 24^ inches, and that which flowed off the ground 25^ inches, although it is reasonable to suppose that the ground would be so saturated that very nearly all the rain would flow down the streams in torrents, yet we could scarcely calculate upon more. The produce of springs from water previously stored up would no doubt add something to the quantity, but not enough to account for the whole. It is most likely, therefore, that either the streams have been over-estimated, or the rain under-measured. On a careful examination of all the returns from which the tables have been constructed, it seems probable that the latter supposition is the correct one. Every stream bears the Wl REPORT AND CmSBBVATIOlTS ON same sort of evidence, although the measurements were necessarily taken at different periods of the day. It is true that at all times, and in swollen states of brooks particularly, the measurement of streams by daily gaugings, although they are repeated several times a day, can only be considered as a tolerable approximation to the truth. Ac- cording to the height of the flood at the time the measure- ment is taken, they may indicate rather more, or rather less, than the average quantity. Still the observations, regularly continued, will in the course of the year pretty well correct each other, and the result, obtained by taking the mean of upwards of seven hundred measurements of the same stream, at equal intervals of time, cannot be far from the truth. It is probable that had rain gauges been placed on the summits of the highest hills, as well as in the valleys, and on elevated parts about midway of the whole rise, the returns would have shown a greater fall. The result of observations made by Mr. Hawksley during the past year in the district surrounding Bivington Pike, near Chorley, from whence the town of Liverpool is pro- posed to be supplied with water, confirm the accuracy of the observations which have been made in the Longdendale district, the nature of the ground and the character of the hills being very similar in both cases. Mr. Hawksley found the mean rain, from gauges placed in various parts of the district, varying from 430 feet to 1800 feet above the level of the sea, to^ be 56^ inches, and the quantity of water which flowed ofi" the ground to be about 44 inches, leaving 12^ inches to supply evaporation, vege- tation, and absorption. ^he greatest rain-fall was found to be at an elevation of about 1000 feet. On this point some valuable information may be obtained from observations made at the Edinburgh Waterworks. THE MEASUREMENT OF EAIN. Sff* That city is supplied with witer from the Pentland hiHs. It is mainly collected in a large reservoir called the Glen- corse reservoir, about 730 feet above the level of the sea, which affords the means of ascertaining with the greatest accuracy the quantity of water which flows into it from the elevated tract of country above. The mean height of this; collecting ground, which consists of 3820 statute acresy is about 1100 feet above the sea, the summits varying from 1300 to 1500 feet. A rain gauge is kept at the Glencorse reservoir, the mean rain of 1844, 1845, and 1846 being 37*403 inches — the average for sixteen years is 37*067 ; the maximum (1836) 49*080 ; the minimum (1842) 25*675 inches. From the 1st December 1846 to the 3 1st March 1847, there came into the reservoir a quantity of water equal to a depth over the whole drainage ground of 4*53 inches : the rain during the same period as registered by the rain gauge being 4*35 inches. This period was remarkably dry in Scotland, the mean of sixteen previous years being 11*442 inches, and the mean of the four lowest G*150 inches. Here is an instance, from the most accurate measure- ment, of the flow of water exceeding for four months the fall of rain. The gauge is certainly placed at the lowest point; but the fall of rain was so small, that a larger propor- tion than usual would be lost by absorption and evaporation. The feet must be accounted for by supposing that the fall of rain was much heavier in the highest parts of the district, as the springs alone, unswollen by rain, would not have" yielded the quantity. In another more elevated part of the Pentland hills, the Bonally district, the fall of rain from the 1st of December 1846, to the 28th February 1847, was 4*71 inches; the pro- duce 4*55 inches: this was ascertained by repeated gaugings. 26 REPORT AND OBSEEYATIONS ON Observations are now being carried on by the Bolton Waterworks Company which will throw much light on this subject ; but they are not yet in a condition to be laid before the public, though it is hoped, at a future time, when the observations are completed, permission may be given to use them. The rain at Belmont, in the Bolton Waterworks district, for the last five years, is as follows, — the elevation 850 feet. January February ... March 1843. 1844. 1845. 1846. 1847. Mean In. Dec. 3 0 1 9 2 7 12 0 4 4 5 0 8 0 4 6 1 0 11 1 7 4 2 0 In. Dec, 6 3 5 9 4 5 1 8 1 9 3 3 3 7 10 0 6 4 3 5 2 4 0 3 In. Dec 4 1 1 9 4 0 2 9 1 7 6 2 3 9 10 2 4 1 6 0 4 0 8 0 In. Dec 7 0 2 1 3 3 4 4 1 9 2 4 6 6 6 0 2 4 7 1 3 7 3 9 In. Dec. 1 9 3 3 1 8 2 8 6 3 6 3 1 6 3 7 9 7 7 5 8 5 8 0 In. Deo. 4 4 3 0 3 2 4 8 3 2 4 4 4 7 6 7 4 7. 6 8 6 2 4 5 April May June .,.., July August September ... October November ... December ... 63 0 50 0 65 0 49 8 61 4 55 6 The returns for 1843 and 1844 have been given in former papers, but they are introduced again for the purpose of bringing all together. Here the gauge has been emptied every month, the index rod being allowed to rise during that period, and the returns are therefore liable to the ob- jection before alluded to. The quantities registered may be somewhat too high. By the kindness of Mr. John Ecroyd of Rochdale, the following table of rain which has fallen at that place for the last sixteen years is introduced. THE MEASUREMENT OF RAIN. 27 » "~ ^^■" s o * •* CO l» « ^ 1 5 I. ^ o ■* o eo o d ^ „ « •♦ M m eo o» w ^ ^ « „ A ^ « to i -2 ^ lO 00 •^ -" -' ^ ,a ra •^ eo « '* (0 ■ in 5i 1 ^. ^ ^ ^ o eq M ■^ IM o> n ■* o •w i « C4 t* - O - ■* •0 eo 04 CT o ^ 1 ^ ■» _ ^ 5 to O eo o d ■* - « o to en * - o« eo U5 eo 1 ■w ^ •^ c>< to to - 1C o KS Oi eo o> o> ■* 00 o g t- '-' m « o> r- t- 00 •-^ m e^ o o ■• US •o •a >o c« ■* o 5 ^ 1- ^ ^ o « >1» o ■* t- o» ■« eo 'w ■"* S o « «o IN e» •o "5 to " » eo e« !5 p ^ f. ^ ^ -* CO ■* 'N CO t~ to eo t-* m ■n — "" "■ «n eo « eo ■o o to « ■^ 6 IM t- 00 r-. (-1 ■O lO >o "H » (N M CO ' " o o *" »-* o ts - « - «~ ■* •a CO ^ 00 - to o 1 es _ CO « o t- «• m eo 00 P-* ^ " '" ■* CN « eo « w -• ICI ^ w 5 g o E I 1 ^ 4 ^ 1 < ■• 1 1 1- 1 '1 t- I 1 1 28 EEPORT AND OBSEEVATIONS ON EAIN, ETC Mr, Ecroyd's gauge is exactly similar to that used eo long by the late Dr. Dalton, and placed at Mayfield, in Manches- ter. J^he rain is received from a large funnel into q gradu- ated vial or cylindrical receiver of smaller area than the funnel, and graduated accordingly. When the quantity is measured the receiver is emptied. Mr. Ecroyd's gauge stands about four feet from the ground; it is placed in an exposed situation in his garden, near Castle-hill, about half a miJo to the south-west of Rochdale. The garden is on the top of the hill forming the southerly summit of the valley of the river Roch, probably about 600 feet above the level of the sea, and is little more than half a mile from Moss Lock on the Rochdale Canal. The correctness of the observations are said <^o be corroborated by those of Mr. Haworth, who keeps. a similar gauge nearer Moss Lock: the variation be- tween the two is stated to be very slight. As bearing upon the accuracy not only of the observar* tions of the Rochdale Canal Company, but of those instituted by the Society, these observations are very important. They sJiow a much larger fjall of rain than has been regis- tered at Moss Lock, and go far to prove that the index rods, in rising above the tops of the gauges put down by the Society, have not materially, if at all, influenced the returns. The following table will afford a comparison : — Canal Co, Society. Mr. Ecroyd. 1844 In. Dec. 20 50 29 43 22 76 2$ 82) 10 Mos. i In. Dec. 30 3 31 7 31 3 \ 10 Mos. ) In. Dec. 34 41 51 •» 42 04 •m 58| 10 Mos. ) 1845 , 1846 1847 l^ //•^' It 16 not easy to account for the great discrepancy. It may be that Mr. Eeroyd and Mr. Haworth are pniore accu- rate observers than the lock-keeper of the Canal Company. 29 II,— On Mnemonic Aids in the Study of Analym. By Rev. Thojias P. Ktrkman, M.A., Rector of Croft-with- Southworthf Lancashire. Read February 8, 1848, (I.) That mnemonical a,\dla in the study of the mathemadcs are considered valuable, a«d to be worth the notice even of adepts in analysis, appears from the following article, at page 291, vol. iii., of the Cambridge Mathematical Journal, O. S. : — '^ Mnemonic JRule. — The following mnemonic rule fw the * Cotangent* formula in spherical farigonometry may be found uaeful. " If in any spherical triangle four parts be taken in succession, as for example A, b, C, a, consisting of two means b, C, and two extremes A) a ; then, ' iha product of cosines of the two means is equal to the sine of the mean side X cotangent of the extreme side - — sine of the mean anffle X cotangent of extreme angled That is, Cos b. cos C rr sin b. cot a — sin C. cot A." Granting fully the usefulness of this consideration, particularly to a mind familiar with mathematical analogies and symmetries, I may be allowed to entertain a doubt as to how far ordinary students of spherical trigonometry will find their memory's bur- den hereby lightened. But I speak the result of some experi- ence when I say, that few youths who have capacity to compre- hend the proof of this formula, will fail to retain it firmly in their memory, if it is at first presented to them thus : — Cot 'Ang. si Gang and c6 b. co Caug are cot a. si b; with the explanation and directions following. 30 EEV. T. P. KIEKMAN ON MNEMONIC AIDS The angles are to be distinguished from their opposite sides, a, b, c, not only to the eye by the letters A, B, C, but to the ear by the sounds Aug, Bang, Gang. Read now, or rather chaunt, the above mnemonic slowly, and very often, with as strong an emphasis as possible on the accented syllables ; and you will thus teach it to your ear, your tongue, and your Itps^ which have all their own powers of memory. The meaning of the abbreviations si. and co. is obvious: they are put for sin. and COS., for smoothness merely. The proposition cot A. sin C -|- cos b. cos C zz cot a. sin b, is difficult to remember, chiefly because, when you attempt to pronounce it unambiguously, it is long and inharmonious. Con- tract it and smooth it as above, giving it a little sing-song cadence, and the ear and other organs cheerfully undertake the task of remembering its twelve monosyllables, which never can fail to suggest the expanded formula with accuracy j a task which, in some men who have more talent for language than for science, these organs will continue faithfully to discharge, after the reasoning faculty has forgotten the proof and much of the application of the theorem. But I do not see, although some persons may have sagacity enough to perceive this, that a student is the less likely to be able to prove a proposition be- cause he can easily recall the enunciation : this may suggest, but can hardly conceal, the argument. The mnemonical rule quoted above is a hint offered to the judgment ; that which I have proposed is a short lesson to be taught by rote to the unreasoning sensuous organs. What these have once engraved on their tablets becomes a ready instrument of rapid thought ; the judgment reads the formula without effijrt, and performs at leisure its proper function of interpreting and applying it. (2.) The 12th and 13th propositions of the second book of Euclid's Elements are as follows : — " In an obtuse-angled triangle, the square of the side (6) subtending the obtuse angle, exceeds the sum of the squares of the sides (a and «) which contain that angle, by double the rectangle under either of these IN THE STUDY OF ANALYSIS. 31 sides, and the external segment between the obtuse angle and the perpen- dicular let fall from the opposite angle. " In any triangle the square of the side (6) subtending an acute angle, is less than the sum of the squares of the sides (a and c) containing that angle, by twice the rectangle under either of them, and the segment between the acute angle and the perpendicular let fall from the opposite angle." These enunciations, to say nothing of the proofs, are something for a beginner to remember ; nor would it be easy to express them in fewer words. Any teacher who will put a youth, even of moderate capacity, in possession of the following mnemonic, for both the proof and the properties, will be able to judge for himself of its value: — Read ba, as well as ac, as one syllable, marking well the accent and the rhythm. Pronounce Sq.'b squib : SUD and DUQ are syllables, as are perc and Bang. (3.) Draw perc : SUDba is SUDsegs of 'c, and Sq.'b is DUQac mol seg. op. two 'c, (A.) as 'tiise or 'cute is Bang op. 'b or Sq.'b is DUQac le coBang two ac. (B.) (a.) Here perc signifies the perpendicular Cp on c from C : draw this ; then if * be the segment Bp of c, not adjacent to 6, 62 _ a^ — (c 4: 5)2 — **, or 62 =r o2 ^ c^ + *. 2c ; (A), the upper or lower sign being taken as B is obttise or acute, (b.) SUD is >S'Mm X -Difference of the two quantities in- dicated. SUDba is {b -f- a) (6 — a) or b^ — a^ i SUDab would be a2 — 62. SUDsegs of c is Sum X I^iff* o£ the segments Ap and Bp, or (c + sf — «2. (c.) DUQ means duo ^uadrata of the indicated pair of quantities ; DUQac is the two squares a^ -|" ^^' Sqb is squared 6: pronounce squib ; squab might mislead. {d.) mol is a contraction of wore or fess, + 2Sf REV. T, P. KIKKMAi» OK MNEMONIC AIDS («i^ le is an abbreviation of less, or — seff. op. in (A) means tke segment of e opposite, noC adjacent to, 5, whose square is the subject of the proposition. Pronounce two long, to avoid confu- sion of it with to. Bang is the angle opposite to 6. By putting in A for s its value IfT a cos B. we obtain ¥ — a^-\-c^ — cos B. 2ac. (B.) (4), The following: two groups of mnemonics give the solu- tions of ail plane and spherical triangles. SubyD is a dis- syllable; sleb' ajid slec are syllables ;. s. sla is a dissyllable, as is b'c also. Side to Sin op. as Side to Sin op. (a.) SubyD Sines or sides is taf Sum by taf Diff. (6.) T^quaf 'A is sleb slec by s. sla, (c.) the num, i&b'cof sisquaf 'A; (d.)- In sphere put sines of facs. (e.) le cotasquaf in p6. has co's 'bove and low. (/I) side 'c is coy. sum (ab) {ff.) where Siy is chaCang two mean kb by S (ab.) In these two lines ab is a syllable. These are abbreviations explained thus :^- (a.) Side b is to the sine of its opposite angle, as side c to the sine of its opposite angle, as side a, &c. (5.) SubyD means Sum by Difference; to/" stands for ten- gent of hal/I sine A -|- sine B a -\- b tan. ^ ( A -j- B) ^ (5.) sine A — sine B a — b tan. -^ (A — B) ' sines or sides means sines of angles, or' sides opposite those angles. (jS.y Tasquaf is tan squared of haif: there is no risk of error here in pronouncing A and a alike ; sfeb is (s -— i), * being semiperimeter; vide Ik (3) ffi- tan.HA-^^-/>^^7"> (c.) ■* s (s <0J X^y The wwrnerator on the right (c) is be X 8^-^ i A = (* — &)(« — c) id.) ^equaf is «»n sg-wared of bal/"; ^ means times ot X • (fi,} fn ^^erica, the two last formulae (c) (c?) become, putAj)^ tjlie 5»»6^ ©/"the^ctow tan.» i A = jaC^---6)«n(..~c) ^^^ ^0 * sin (^ — .«) *• sua o sm c (Jl) In the supplemental polar formula made jfrom (ec), we have le cotasquaf (/ess cotasguared of hal/} on the left, and cosine* above and helow on the right : or, c,,.^ _Cos(S~B). Cos(S-~C) — leb a sySable, as also tan <{>, cot t.) (A.) Sing is «ne of an^le ; sin A : sin a rz sin B : sin h zn sin C : sin c. (i.) Neper's circular parts are a, b, and Mjpplements of the rest, or of c AB ; and sine middlQ part is joroduct of Cosines of opposite*, or product of Tangents of acQacent*. u REV. T. P. KIRKMAN ON MNEMONIC AIDS (k.) taf, vid (b). SorD is sum or diff. Cotk is coisxxg of Aalf. CorSha is cos or sin of ha\S. Dab is difference of a an^ b. 3Vl stands here, and in many other mnemonics, conveniently for BuM ; and refers to the quantities a and h in Dab. In CorS in the second, and SorD in the first line, the cos goes with sum, and sine with difference : no ambiguity can arise from the two mean- ings of S in CorS and SorD, "We have here the two formulae : cos , ,s t^n i (A ± B) sin ^ <>^ ~ ^> cot^C cos^^^_^j^ or, read at sm length, {k) is the tan, of half sum (or diff.) of angles AB, divided by the cotan. of half C, is the cosine (or sine) of half the difference of the sides aJ, divided by the cosine (or sine) of half their sum, il.) The cfenominator on the left of (k) has taf instead of cothf in the supplemental po/ar formula; or, tan ^ (g + 6) __ sin i (-^ — ^) " ' ^ im.) cos (p. cos c zr cos a. cos (f — h), where tan

A Cos (n — 1) 6, &c., writ- ten repin : and Sin to odd o, Sin'^+ * 0 when w + 1 is odd, has the series Sin (n -f 1) ^, A Sin (n — 1) ^, B Sin (n — 3) d, &c., written repin. (c.) CorS 0619 (Cos or Sin) (n + 1) 6, as the index « -f- 1 is even or odd, the ors in (b) and (c) corresponding as to antece- dents and consequents. ^ Ji REV. T. P. KIEKMAN ON MKEMftJTC AIDS pin is ^n abbreviation of jjlua iniiwuS, + — ^ H ••..•• rqnn is plus -and minus alternately, -reversed j or -|- — -j ^ read backwards froln the la^ttenn. Examples are 2* Cos ^6 = Cos 5^ + 5 €os 3^ + 10 Cos S 2» Cos ^'d =: Cos bd 4- 6 Cos 4^ -f l5 Cos 2^ + 10 2* Sin '6 =. Sin 5^—6 Sin 3^ -f 10 Sin 6 2* Sin ^d =— Cos 6^+6 Cos 4^ — 15 Cos 2^ -f 10 t8.) I shall now proceed to Select a few mnemonics, from the multitude which I have been accustomed to employ, so as to exhibit instances of various mnemonical devices, and shall pay no regard to the order of the mathematical topics to which they belong. None of these devices have been invented for one emergency only, being all of them repeatedly applicable in the concise expression of formulae. (a.) per ab is (a' — b^)K per ba is (b^ — a^)*^ (6.) poth ab is (a^ -j- J^)*. (a2 — b^^ is the ^perpendicular in the right angle triangle whose base is b, and whose hypothenuse is a : and (a* -|- 6^* is hjpo^Aenuse to the sides a and b about the right angle. Thus, to remember the equation to the conchoid, say,'-*- (xy a dissyUabie,) , In conch, a pole ord, and^od, xy is moi per^ Sum yci. In the concJtoi^i a being the oreKnate of thfe pole, and jfi^ modaluB, xyz=±if' —y)*. 0/ + a). Again, Witch x^p is two a per &I)ax. In the witch, xi/ =z 2a (a^ — (a — - a:)-)* =r 2a (.2ax — a:*)* Dax being a — x, per a Dax is (a? — (a — xY)* i D means Diff. here. (c.) If TO rz ax and n zz by, (aV — W^f ZZL (m? — ^^)* is a perpendicular of the same kind : call this perprod {ax by), or smoother, peprod (ax by), or shorter still, pep. (ax by): i. e-, the /mpdtidictila^ in tiie triangle tviiose base is the product i^ awl hypothenuse the product auk The equation of tbe ienMtecate i8> fifA ][»ie ifijriUable,) DUQjib; itt Lemni is pepr^jd {ax bf), and pep (co^ a ^bsi^) is r; i. e.^ y" + a:* =z (a V — ^V)*, or, i^coa.'^jft' '— 6*sin.^)* r^ r. for Dl5^ vide 3^* (rf.) If ax — A* and by r= /», (ax — iy)* is (;&« — /'')* ; or the perpendicular to base iarmonic mean, of the legs a and ^'4 Harm oA is two 06 fey Sab. (ff.) t;e in ^r is a contraction iox qusie of (di*Me)^ the-qootto of a and b is the quote of the segments of the base c. For an example of the use Qipoth^ take one from Alg. Geom. ^(^,) lorJs the con. by poth f ics-; with which may be given the corresponding (i.) plor^s the -con. by dig fics. In the line ax •■\- by -z c, referred to right «xes, if we Call lor the^ortest distance p of the /ine from the orighi, ^ EEV. T. P. KIKKMAN ON MNEMONIC AIDS lor ia the constant c divided iy the hypothenusQ to the Jics or coe^ient*, a and b, of the variables. In (i), plor is the shortest distance P from the origin, of the jp/ane Ax -|- By -}- C;? zn: H, referred to right axes. diff means diaffonskl, sometimes diaff or dta. Dig bac is dig Jics is (A^ -f" ^' 4" C^)*> diagonal of the right solid whose sides are A, B, and C. Then we have if) thus explained, p , H — (A^ + B^ -f C^)4 (9.) Useful formulae are the following on lines and planes re- ferred to right axes. plan's vagiU, if lor's con in h \ (a) plap is vapltf, if plor's con in 6. (b) (a.) plin is the distance of a point p from a /me ^. ay -f- Jo; -|- c rz: 0 ; gil means ^^tven line ; gil « is the f ^«» or zero just written j vagil** is the value of that expression, when a:,yi of the point p are put for xy, or ayi -|" ^^i ~f" ^' This is the length pliriy if the cowstant c in that a *«*' is the length lor^ (8, K). (b.) plap is the distance of a point p from a plane Ax + B^ + C2 -f- H =z 0. vapla is the value of this jo/a, or plane's ^^e", at jd (XiPiZt) : and this value, Axj -|- Byi -|~ QiZ^ -j- H, is the length plap. provided that the constant H in that a or zero be the length plor, (8,i,). Generally, let the co-ordinate axes contain an angle V : Plin by sine V^ is vagil « at '/> by *bas (fics V). (c) Plin and vagil u as above. (c.) bos jies Vis the base of the triangle, whose sides are the JicSy or coe^cientj of the variables x and y, and contained angle V. That is, Plin (axx -f- bxi + «) (c) SmTV — (a* — 2a^ Cos.V + 5')* and this is true whatever c may be, whether lor or not. If my space were not limited, I could easily give mnemonics, equally brief and simple, for all the formulae required about the IN THE STUDY OF ANALYSIS. 39 inclinations and intersections of lines and planes, for both right and oblique axes. (10.) The properties of the conic sections are easily remem- bered, when the student has learned to talk to himself about them in rapid and unambiguous syllables like the following. The equation to the central conies being— a:' I y^ — 1 the following mnemonics are a few examples out of many. [Pron. (1 d), un d, a dissyllable.] Sq}- is SUD (1 e) SUP (d x) ;* 2 squib's larec h, ; per (ab) is ek ; at or* foe the x* is ea mol x\ 6rfoc, r's larec by ('2 mo 2 eco^) : h mol ex or ex mol a are fops in 'li'pse or hyT)ola : diet is ba by mean fops : dift is b' of ro6q fops, and b' is mean of difts : ba by diet is cbnja : mean of fdps is cbnja : rho. ba is cu conjk j rho ba is cuper (a ex) : per (poc 'b) is xb. (a.) y* = (1 — c«) («' — a:^) (J.) 26^ rr /a, ? being larec, or /atus rectum : (a^ — J')* zr ea. (c.) The origin is at the^cus, when the equation is — a^ * b^ T(3*) («) ▼ (3e){8a) (*) V(3,a) («) (rf) T(3,<») (e) y{i.9) 00 v(8,/) (^) (0 T(8,o) (m) X e a dlisyl. r (8, a) («) T SUD, (3 8) mo ia + under a lincnlnm. '^ 2+2ecos. ^ (c.) (a + ex) are the distances from j^ci of a point p, or the fops, in the ellipse, and (jex + a) the^ps in the hyper- bola. if.) diet is rftstance of centre from fengent, and rr ha'. (a* — e* a:^)*, or 5a by mean of fops, zz ba : (^)*, if /and/' are fops. _ - 4CV EEV. T. P. KIEJKMA» OH MNEMONIC AIDS (ff.) d^ is a' — ba by diet j «' = (#0*. (A.) (./.) rAo is g, the radius of Qurvsture ; ^ 6a ~ a'' ; cwbed a*. (m.) g 6a zz (a^ — «'^)^i a cwbed ^perpendicular, (8, a.) (».) jpoc is distance of (arjr) or/? from centre, n R : ^ ; This last property (compare e) \b expressed by Ijeslie in his " Geometrioai Analysis," in the following luminous and encou- raging language j Prop. yii. p. 206 :— *' If the transverse axis of an ellipse or hyperbola, be divided into segments equal to lines dr^Tsvo fyom the foci to 8.ny point in tbe curve, the square of its distance from the centre will be equivalent to the sum or difference of the squares of the semi- conjugate axis, and the distance of intermediaite section from the centre." Alas for the student who is doomed tp pick yp his notions of advanced geometry from such authors as Leslie ! It is understood, of course, in all that precedes .in this article, tlaat 6* is, of either sign, positive for the ellipse ^nd negative for the hyperbola. (11.) Many persons find it difficult tp remerpbej: the principal theorems in combinations. I found it so, for on?, until I ta,ught them to my ear and tongue in the fashion following— If *p ele'ms. are 'm, a'sj ei, b's ; ^ c's ; The pe'rms in' p are 'p fagp (a) by 'm fags, e fags, i fags. The comb, repe's of n in ^p's are p-n-dps by p faga. (6) Com. non-repe's of ^n in 'p's fire *p-n-backs by 'p fags. ^^c) Jfou repe. vars. of 'n i» p's ^re 'p-n-ba'*,) say of 7 in fours, such as 7654, &c., are in number , four-7-backs by four fags. Two three backs are 3 '2. ( : a)} " :rz (a -j- b)™. "We say multiply with in (A) ; bt/ should always denote divi- sion. (C.) golt is fr" tong. This means .* zi t" + y + i^ + p^ + &c. I call g*, or log"^ t, golt j g" is gol(ax) ; gol is log read back- wards : fru t on vid (A) above, t ong differs from t on, in that every power is divided in the former by ^^fags of that power : the zero power is unity in both of course. There is no ambiguity in pronouncing t ong here as one syllable tong. (D.) a'"^ le one 's vi ong {y mod) ^^ ^'^^ ^^''^ ^^*-> and mod is ci loga: i.e. y :^ , (y • »»)' , (y-mf ay-^lz=:-^ + -Y^ +172^ + "* '' this is vi ong^ or a quote ong ; the quote of y and m, m being modulus of the system: and the mod. m is (log.a)-i ; the re«procal of hyp. log.a. (E.) If ii be loga vi (A>i niod)'s pin inca.** on by — dex. i. e., If M ir log a, and h be the increment of a ; and Wj zi log (a + A) ; mod — mod a 2a* •" 3a* 4a* pin vid. (7 c). incaths is the fraction — zz - *^ a a vi (Au mod) (8, g\ is this fraction pin on by index of the power. (F.) N"* val is n-'bino fun obp. (G.) Ator u *s r-bin fun up repm ; (pron. Deitteoo.) 44 EEV. T, P. KIEKMAI? ON MNEMONIC AIDS (F.) If «o Ui «a «8 ^ a^y function and its successive values, fim up is t or reversed dtf, I call^ LIGHTNING CONDUCTOKP. m lightning on the clock tower of Osborne Palace, to the forest of pointed conductors in its vicinity. Those conductors, as well as the fine tall trees that are about the building, will always be the means of giving a tendency to their locality for the reception of discharges of lightning ; and that ten- dency will necessarily be increased in some proportion with the number and altitude of pointed conductors on or about the building. Such at least appears to be the natural in- ferences derivable from an association of the well-established influence of pointed conducting bodies, with many observed circumstances attending discharges of lightning. 18. Damage by lightning in the vicinity of pointed con- ductors at sea, is no less remarkable nor less frequent than on shore. There are many striking instances of this kind on record, some of which afford lessons of no ordinary import. 19. Case E.— In January 1824, H. M. ship Milford, 74 guns, was struck by lightning within the distance of 80 fathoms from the Culedonia, of 120 gims, and several other ships close at hand, all of which had pointed con- ductors up at the time. A powder magazine on shore, at no great distance, had also a conductor attached to it ; and the report says, that this conductor was in the direction from which the thunder-gust proceeded. In this case the damaged ship was lying in ordinary, without conductors, in Plymouth harbour. 'Die Caledonia had a conductor at each mast ; but neither that ship, nor any others which had their conductors in place, received any portion of the lightning.* 20. Case F. — H. M. ship Phaeton, 46 guns, whilst in Gibraltar Bay, in the year 1824, was much damaged by lightning, at a cable's length from the Warrior^ the latter having pointed conductors up at the time.f 21. Case G. — H. M. ship Pelican, 18 guns, was struck and * Nautical Magazine — Harris on Thander Storms. f Ibid. 66 MR. WILLIAM STUEGEON ON LIGHTNING much damaged by lightning, whilst on the coast of Africa, and at a short distance from the Waterwitch, the latter ves- sel having her pointed conductors in place at the time.* 22. Case H. — H. M. ship Ceylon was struck by Ughtning in the year 1838, whilst lying in Malta harbour, and at a short distance from the Talavera, BelUrophon, and Hastings, three line-of-battle ships, and fully rigged and equipped with conductors. The Ceylon, as a receiving-ship, had only a short pole above her fore-mast, whereas the other ships being fully rigged, their masts and conductors were above 150 feet up into the air.f 23. Case I. — In 1815, H. M. ship iVb?-_^e was severely damaged by lightning ; whilst the Warrior, 74 guns, with a pointed conductor, lying close to the Norge, received no injury. Many other ships with conductors were in the same harbour at the time ; they all escaped but the Norge, which had no conductor.^ 24. Case K.— On the 25th of March, 1840, H. M. ship Powerful, of 84 guns, \7as struck by lightning whilst at anchor at a short distance from the Asia, also an 84 gun ship, and furnished with fixed conductors in her masts. The Powerful had no conductors* || 25. The four following cases show that lightning occa- sionally falls into the sea close to tall-masted vessels, not- withstanding their being armed with pointed conductors ; which, according to the views of Franklin and his followers, ought to prevent such vicinal explosions. 26. Case L. — On the 21st of January, 1840, a discharge of lightning fell into the sea so near to the Neptune, a small revenue cutter at anchor in Ely Bay, as to cause the ves- sel fairly to reel by the concussion.§ * Nautical Magazine — Harris on Thunder Storms, ■f Ibid. J Ibid. |I Ibid. Also, Parliamentary Return of Sbipi struck by lightning. § Harris on Thunder Storms. AND LIGHTNING CONDUCTOBS. 67 27. Case M. — In the month of June, 1840, a discharge of lightning fell so near to H. M. ship Southampton, of 50 guns, that it appeared to strike the main-chains.* This vessel had fixed pointed conductors in all her masts at the time of the occurrence, they having been applied two years previously, f 28. Case N. — In the year 1840, a dense explosion of lightning fell close upon the quarter of H. M. ship Van- guard, of 80 guns, whilst proceeding from Portsmouth to the Mediterranean.^ 29. Case O.— A discharge of lightning fell close to the Dart, a steam-packet, whilst on her passage from London to Margate. [| 30. The most interesting of the last four cases is that of the Southampton (Case M.), because of the certainty we have of the presence of three of those conductors upon which so much confidence is now placed for protection ; and of the fact that the lightning, notwithstanding its near ap- proach, disregarded their conduction, their points, and their prominence, and found an easier transit to its destination close to the side of the ship. It is one of those events that would lead to the inference, that tall pointed conductors may facilitate discharges of lightning in their own direction, though incapable of preventing their taking another route when within a certain distance of them. Moreover, analogy would lead to the inference, that although the Southampton fortunately escaped, it is possible that such conductors might be the means of lightning falling on the deck of the vessel to which they were attached. Nor does such an inference rest on mere probability, it being already verified by the following well-authenticated facts : — 31. Case P.— In March 1848, at Calcutta, H. M. ship • Harris on Thunder Storms. f Parliamentary Return for 1849. X Harris on Thunder Storms. || Ibid. m ME. WILLIAM STURGEON ON LIGHTNING Endynnoii was struck by lightning on the fore -topgallant mast, at the distance of about 50 feet from a pointed con- ductor attached to the main -mast, reaching from the top- gallant-mast's head to the water.* 32. Case Q. — H. M. ship uEtna was struck by lightning near the bow, which exploded about 12 feet above the fore- castle, close to the fore-mast, whilst a chain conductor was attached to her main-mast in the usual way.f 33. Case R In May, 1835, at the Cape of Good Hope, H. M. brig Bucer was struck by lightning on the fore-topgallant-mast, a chain conductor being in its place on the main-mast at the time.:]: 34. Although more instances of this kind might be ad- duced, those already cited are sufficient of themselves to show the fallacy of that doctrine, which embraces the idea that tall pointed conductors will prevent violent explosions of lightning from falling on vicinal objects. They, more- over, prove that lightning does not invariably select the tallest objects for its transit to the earth ; which is another fact at variance with the views of Franklin, and the pre- vailing opinion at the present day. 35. I am not aware that oblique discharges of lightning had ever been noticed by writers on electricity previously to the appearance of my Memoir on Marine Lightning Con-' cZwc EAILWAT GBADIENTS. 161 engines still further increased, that gradients of 1 in 30, or probably 1 in 20, may be surmounted. The means necessary for the working of steep gradients, appear to consist in the power of the engine and the amount of the load which it has to overcome ; and, pro- vided the latter is duly apportioned to that of the former at a given velocity, there can be no doubt as to the work- ing of steep gradients with considerable certainty and effect. The power of the engine required in such cases will vary according to the nature of the traffic ; but in every instance where the distance is short and the transit is frequent, light trains may be used and a less powerful engine employed. At other times, where the traffic is less frequent and the transit of heavy trains cannot be dispensed with, it then becomes imperative to employ the most powerful engines, so as to ensure certainty in surmounting the gradients, and that, if possible, without the aid of an assistant engine. Should the gradients, however, be long, and any of them exceed 1 in 60, it may then be necessary under these circumstances to employ an assistant engine as an auxi- liary.* In every case of this kind, the generative power of the engine at the minimum velocity, becomes a question of considerable importance, as steep gradients cannot be effi- ciently worked excepting under circumstances where a plentiful supply of steam is at hand ; and hence arises the necessity of employing engines of greater power, and boil- ers of more than ordinary capacity, in the area of their tubular surface, than those in general use, accompanied * Since the above was written, a gradient of nearly two miles in extent has been opened on the East I^ncasbire Railway, between Accrington and Hasliiigden, with a rise of 1 in 40. This gradient is worked with one engine to the passenger trains at the rate of nearly 20 miles per hour, and also with ordinary goods trains, excepting only in cases of wet weather and heavy trains, when the pilot engine renders assistance. 1 52 MB. WILLIAM FAIEBAIEN ON THE with a large fire-box and increased powers of vaporiza- tion. In the discussion of this subject, it may be necessary to enquire into the laws which regulate the different elements of resistance to which the locomotive engine is subjected ; and, subsequently, to determine how and in what manner those resistances are to be overcome. It is well known from practical experience, and also from the experiments of Dr. Lardner, the Comte de Pambour, and Mr. Woods, on Railway Constants, that the resistances are — 1st. The resistance due to friction in the working parts of the engine, and the engine itself considered as a carriage. 2d. The resistance of the carriages, waggons, &c., com- posing a train. And, Lastly. The resistance of the air. In calculating the friction of a locomotive engine, two considerations present themselves ; first, the friction of the mechanical organs of the engine considered as a machine ; and, secondly, the friction of the engine when considered as a carriage. From a series of experiments by Pambour these elements are separated ; but, taking the friction of the whole engine at 104 lbs., and the average weight at 8 tons, we then have on the datum of 6 lbs. per ton for carriages, a resistance of 56 lbs. for the mechanism of the machinery, and 48 lbs. for the engine when considered as a carriage. Taking, therefore, the united powers of resistance at 13 lbs. per ton as a constant, we cannot be far wrong in estimating the friction of the engine alone at 13 to 15 lbs. per ton, or about two and a half times the friction of a railway waggon. From the above, it will be observed that the resistance of the motive powers being given, we have next to consider the resistance of a train of carriages and waggons. This retarding force is variously stated by different authors; but LOCOMOnVE ENGINE AND EAILTVAY GRADIENTS. 153 taking the mean of Lardner's, Wood's, and Pambour's experiments, it will be found to approximate to nearly 6 lbs. per ton : and, in the absence of more detailed and more extended experiments, it will not be improper to cal- culate the forces necessary for ascending steep gradients on the supposition of 6 lbs. per ton being the resisting force per ton of a railway carriage. Lastly. The resistance of the air, which is again variously estimated. By Pambour, the resistances are given on every square foot of surface, as follows :— At 20 miles aa hour 107 lbs. 22 „ ,y ....i, ...... ..,.., x'oO ,f 24 „ „ 1-65 „ 26 „- „ 1-82 „ 28 „ ,) ......... ^ 2"11 ,f .. 30 "„ „ 2-42 ,* „ 32 „ „ 2-75 „t. 84 ,/ „ 311 ^.. 36 „ „ 3-48 „ . 38 „ „ .... 3'88 „ 40 „ „ 4-30 „ 42 „ „ 4-74,, 44 „ „ 5-20 „ 46 „ „ 5-69 „ 48 „ „ 619 „ 50 >. » 6-72 „ Mr. Woods makes the resistance, on a calm day, at a velocity of 33 miles an hour, equal to ^ of the whole weight, or 25'16 lbs. per ton; and taking 6 lbs. for friction, we then have 25*16 — 6 n: 19 lbs. per ton for the resis- tance of the air at 33 miles an hour. In Pambour, we have at 33 miles a resistance of 2*93 lbs. per square foot ; and, supposing the area of surface exposed to the action of the atmosphere to be 40 feet, it then fol- lows, that 40 X 2-93 =: 117-2 lbs., the resistance of the air X 154 MR. WILLIAM FAIEBAIRN ON THE against the train ; which, in a train of 60 tons, gives a re- sistance of 19 lbs. per ton. Assuming, therefore, the resis- tance due to friction, exclusive of the motive powers of the engine, to be 6 lbs., and the resistance of the air 19 lbs., we then have an antagonist force of 25 lbs. per ton in con- stant operation against the tractive power of a railway train at 33 miles an hour. From these data, I have endeavoured to calculate the power and size of engines necessary to overcome these re- sistances (which, it must be borne in mind, are due to a level plain) on different rates of inclination, or on gradients varying from 1 in 20 to 1 in 200. Taking, therefore, 25 lbs. as the measure of resistance on a horizontal plane, the following will exhibit, in a tabular form, the elements of resistance to which a locomotive engine is subjected when ascending gradients varying from 1 in 20 to 1 in 200, at 33 miles an hour : — TABLE OF RESISTANCES ON RAILWAY GRADIENTS, Gradients. Resistance in lbs. per ton. Force of resistance due to gravity in Iba. per ton. Total resis- tance in Iba. per ton. Remftrks. 1 in 20 25 11200 137 00 1 „ 30 25 74-66 9966 1 „ 40 25 66-00 8100 1 „ 50 25 44-80 69-80 1 „ GO 25 37-33 62-33 1 „ 70 25 32-00 67 00 1 „ 80 25 28-00 63-00 1 „ 90 1 „ 100 1 „ 110 25 25 25 24-88 2240 20-36 49-88 47-40 45-36 Rate of travelling, 33 miles an hour. 1 „ 120 25 18-66 43-66 1 „ 130 26 17-23 42-23 1 „ 140 26 16-00 41.-00 1 „ 150 25 14-93 39-93 1 ., 160 25 14-00 39-00 1 „ 170 25 13-17 38-17 1 „ 180 25 12-44 37-44 1 „ 190 26 11-78 36-78 1 „ 200 26 11-20 36-20 LOCOMOTIVE ENGINE AND KAILWAY GRADIENTS. 155 Now, if we take the last column of the table, comprising the sum of the total resistance due to the different retarding forces, it will not only be easy to compute the force in lbs., or horses' power necessary to overcome the resistance, but it will also be easy to determine the load which a well- constructed locomotive engine, having 16 inch cylinders, will drag up the differently elevated gradients at the rate of 33 miles an hour. Before entering upon these calculations, it may however be necessary to state the properties and conditions of the engine on which they are founded ; and, in order to ensure sufficient accuracy as regards the ppwer, I have taken the pressure upon the piston at 40 lbs. on the square inch, the cylinders 16 inches diameter, 18 inches stroke, 5 feet driv- ing wheels, and travelling at the rate of 33 miles an hour. Now, an engine of those dimensions, acting with an effective pressure of 40 lbs. on the square inch, will exert a force (the piston moving at a velocity of 555 feet per minute) of 8,924,400 lbs., or 270 horses* power. This taken as a mea- sure of motive power, clearly exhibits the immense force given out by the locomotive engine at high velocities. In the first column of the table we have the inclination of the gradients; in the second, the resistance per ton on a horizontal plane ; in the third, the resistance due to gravity ; and in the fourth the total resistance per ton, or the retarding force which the engine has to overcome upon every ton raised on gradi- ents varying from 1 in 20 to 1 in 200.* * On consulting the experiments, it will be found that a less powerful engine (the Baltic), with only 14 inch cylinders, carried a greater load up the Hunt's Bank gradients. This may, however, be accounted for by the increased effective pressure upon the piston, which in this case was from 60 to 65 lbs. on the inch, considerably more than that given above. 156 ME. WILLIAM FArRBAIRN ON THE TABLE OF MOTIVE FORCES, Appucabu: "vo Kailwat Trains, om GRADiEMrFS at 33 Milbs an Houb. GrUisntl. Total restat- auceln hLoTsea!' poorer Load in tons for a locomo- lbs. pur ton. per ton. tive engine. 1 in 20 13700 12.05 22-4 1 „ 30 99-66 8-76 30-8 1 „ 40 81-00 7-13 37-8 1 „ 50 69-80 6-15 43-9 1 „ 60 62-83 5-50 490 1 „ 70 57-00 501 540 1 „ 80 5300 4-66 57-9 1 „ 90 49-88 4-40 61-3 1 „ 100 47-40 4-17 64-7 1 „ 110 45-36 3-98 67-8 1 „ 120 43-66 3-83 70-5 1 „ 130 42-23 3-73 72-4 1 „ 140 41-00 3-58 76-4 1 „ 150 39-93 3-51 76-9 1 „ 160 3900 3-43 78-7 1 „ 170 38-17 3-36 80-3 1 „ 180 37-44 3-29 820 I „ 190 36-78 3-23 836 1 „ 200 36-20 3-18 84-9 Level 25-W 2-20 122-7 From the above results it is obvious that the working of steep gradients is only circumscribed by the power of the engine; and, considering the- enormous expense of qoji- structing easy gradients in a mountainous district, it be- comes a question of deep interest to the community, in having lines formed at a moderate cost, and that only at the expense of a proportional increase of power. It cannot be doubted that the locomotive engine of the present day is more than commensurate for the attainment of these ob- jects; and, provided we carefully adjust the weight and powers of the engines to the work they have to perform, we may safely calculate on a great saving of expense to the community, increased dividends to the shareholders, and an equally efficient tractive power to overcome the resistances LOCOMOTITE ENGINE AND EAILWAT GEADIENTS, 157 of retardation in all the elements of gradients varying from X in 40 to 1 in 400. As a proof of what can be accom- plished in this way, I have to refer to a series of well- conducted experiments, made a few years since on the Hunt's Bank and Halifax inclines, with engines inferior in power, and also of construction, to those now in use. They were probably the best of their kind at that period, but con- siderably inferior as to weight and power to those which have since been constructed. The following experiments were undertaken at the request of Mr. Hawkshaw, C.E., and the Directors of the Lancashire and Yorkshire Railway Company, for the purpose of ascer- taining whether or not gradienti, not exceeding those on the Hunt's Bank incline, could be efficiently worked by the locomotive engine, and whether, and to what extent, im- provements could be eflfected for that purpose. The results are as follows : — ExPERiafENTS MADE ON THE LANCASHIRE AND YORK- SHIRE Railway, to determine the PRACTiOABiLiTr OF WORKING THE HdNT's BaNK INCLINE BY LOCOMO- TIVE POWER, INSTEAD OF THE FIXED ENGINES PREVI- OUSLY ERECTED FOR THAT PURPOSE. ^xp. 1. — With the locomotive engine, "London," 14 inch cylinders, 20 inches stroke, and 6 wheels, each 4 feet 6 inches diameter, coupled. The load, exclusive of the engine and tender, was composed of 10 waggons, 1 carriage, and 15 passengers. Weights. Wogooi. Load Oraai. Tons. Cwts. Tona. Cwts. Tons. Cwts. 31 12 50 10 82 2 Of engine and tender 25 13 Total weight 107 16 158 MB. WILLIAM FAIRBAIRN ON THE With the above load the engine, with steam at 75 to 90 lbs. on the inch, ascended the incline, which varied in the gradients or rates of inclination, as per sketch, as follows:— ^ - Il^'f-YAIIO* ;.-.. 702 YARDS - >rM>l CRADIEHTlIN 60 ; CRADICNT IIN46.ilHr8} LSVBL -2050 VARDa ' HOWIZOMTAL LIME In making the ascent, the engine and train were started from nearly the middle of the station ; and, having run a distance of about 160 yards, they entered upon the lower gradient of 1 in 60, with a momentum of nearly 14 to 15 miles an hour. Unfortunately, however, the wheels slipped, owing to the moist state of the rails at the entrance of the curve on the gradient of 1 in 46, half way up the incline ; the result was a repetition of the experiment. Ex'p. 2. — The same load in this experiment was carried up the incline, a distance of 2050 yards, in 6 minutes and 4 seconds, being an average rate of travelling of 11'2, or 11^ miles an hour. Exp. 3. — The " Scheldt" engine, 14 inch cylinder, 18 inch stroke, four wheels coupled, 4 feet 8 inches diame- ter, and two trailing wheels, each 3 feet 6 inches diameter, starting as before, with steam at 65 lbs. on the inch, took the same load, 82 tons 2 cwt., up the incHne, a distance of 2050 yards, in 5 minutes and 30 seconds, being at the rate of 12*7 or 12| miles an hour. Ex-p. 4. — In this experiment, with the same load and same engine, 6 minutes and 10 seconds were expended in the ascent, owing to the weighing-machine having broken over which the engine and train had to pass, which pre- vented the train from starting with the same momentum as before. From these experiments it will be seen, that gradients of considerable elevation can be worked by powerful engines XOCOMOTIVE ENGINE AND EAILWAY GRADIENTS. 159 with heavy trains, at velocities varying in the ratio of the powers of the engines and the loads respectively. A slight drizzling rain was prevalent during the last two experi- ments, which kept the rails wet, and consequently proved unfavourable for the experiment. The engine, however, retained its full power from the commencement to the top of the incline, without slipping. The next experiment was on the Halifax incline, which contains three distinct gradients, varying, in a distance of nearly two miles, according to the following longitudinal section. Experiment 5. K SSOYARDS ^ lOlO YARDS ^-—880 YAH DS -» I CRADIBNT I IM 44^ ; CRABICNT I IN £3^ | CRADIEMT I IN 73 L \ LIVKU -•J370 YAftDS. With the same engine (the Scheldt) the ascent was ac- complished, with a load of 11 tons 1 cwt, exclusive of the engine and tender, in 4 minutes, being at the rate of upwards of 28 miles an hour. The performance of the Scheldt engine obviously shows that a considerable saving may be effected in the original outlay of great numbers of railways, by the introduction of a class of engines calculated to work the different gradients at a rate of speed corresponding with the nature of the traffic ; and, notwithstanding the sacrifice of time, and the increased expenditure of fuel that would have to be made in making the ascent of the gradients, that loss and expen- diture would, nevertheless, be compensated, to a consider- able extent, by the increased velocity, and consequent saving of coke in the descent. But in fact, the extra locomotive power which under 160 MB. WILLIAM FAIRBAIEN ON THE such circumstances would be required, is not to be com- pared to the dead weight of the enormously increased out- lay in the first instance, which, in many cases, has been incnrred for the purpose of attaining easy gradients, and which, if properly and judiciously applied, would more than supply the motive power in perpetuity for working the whole of the line. In addition to the experiments on the Hunt's Bank and Halifax inclines, a laborious series of experiments were instituted on the Accrington incline, by Mr. Perring, the talented engineer of the East Lancashire Railway Com- pany; and to that gentleman I am indebted for the follow- ing important results.*'^ These experiments have been conducted with great care, and occupied a period of three months, from the 4th of February 1850 to the 2nd of May, in the worst time of the year. They become the more interesting, from the circumstance that they are taken from the company's register of the duty performed by each engine, the precise condition under which the ascents of the gradients were made, the weight carried, and the time occupied by each train from the time of starting from the bottom till its arrival at the top of the incline. Another circumstance which renders the expe- riments valuable, is the fact of them being records of the regular working duty of the engines for three consecutive months, and the great advantages derived from a regular system of working both goods and passenger trains on one of the most difficult inclines in the kingdom. The goods trainSi it will be observed, are worked under different cir- * The Accrington incline is t^o miles long, having gradients as follow :— Bottom 1 in 40, 90 chains'] Middle,.... 1 „ 38, 48 „ ^ = 160 chains, or 2 miles. Top 1 ,,47,22 „ on a mean incline of 1 in 41'6. LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 161 cumstances; the lighter trains with a single engine, and the heavier ones with a double or assistant engine. In the latter case, it is curious to observe the comparatively small power which in many instances was given out by the assistant engine, and the increased quantity of work per- formed by the single one. In Table I. the single engine ascends the gradients, on an average duty of three months of 38 ascents, with 71*6 tonSj at the rate of 6 '31 miles an hour ; whereas the average duty of the two engines — Table II. — in 10 ascents with 11 1*9 tons, was at precisely the same speed, or 6'31 miles an hour. Again, in 15 ascents of the double engines conveying a load of 123*9 tons, the speed was only 5*9 miles per hour, showing an evident saving in the use of the single engine and light trains to a very considerable amount. The re- sults of the experiments are therefore in favour of one engine, which carried 71*6 tons up the incline at the rate of 6*31 miles an hour, whilst the assistant engine carried only 40*3 tons at the same rate. This gives an excess of duty of 31 '3 tons in favour of working the gradients by the single engine. In this comparison we must, however, assume the leading engine of the heavy trains to be equally power- ful with that of the lighter one. The following tables are however more explicit,- and exhibit some curious and im- portant facts in the working of railway gradients. 162 ME. Wn-UAM FAIKBAIRN ON THE Experiments made on the Acceington Incline, ttith Goods and Passenger Trains, from the 4th Fe»- RUARY, 1850, TO THE 2nD OF MaT INCLUSIVE. TABLE I. ascent of merchandise trains up tue accrington incline. Gross Time occu- Particu- lars of Particn- larsof Totalheight weit^ht of pied in of ^» 13 60 18 Bails wet. March 2 72 18 78 19 » h 84 25 § ^ -April .9 ,72 78 60 24 25 22 Ratla 8llppi>en''. v< «s 10 70 20 o 00 66 20 .^a 90 26 1 IN n 84 72 23 23 Ba!l8 sUppperjr. 1 tM J, 72 24 « » O ,^ 60 19 »» « .- ■*J 60 18 , "5) IS 78 SO .a ft 80 66 22 ■20 o •a •2 66 23 78 26 f^ 90 28 as t: 16 70 20 .JT C3 17 72 23 .a OS 72 90 23 30 -S s S 18 66 20 ^ o 00 18 "*^ «? 66 18 "^ « 1 rH 68 22 .S5; ■* 19 72 25 >a II a> a ,, 66 20 .2 I a "^ 67 20 S '"' 78 25 Average dntjr performed = 71 tons, at a so W SI 54 15 the rate of 5-8 miles an hour. 55 15 ,M ^ 54 14 •c 22 72 20 "t^ £3 f3 66 20 ^1 o " 60 18 Greatest load,: 90 tons, at 4-61 mUes C^ 0) ,, 60 19 an hour. "3. ■si "S 15 126 25 i 1, 16 78 17 ^ 111 •s 17 18 128 168 22 30 Ayera^re duty performed = "2 S'^'X, "a 120-8 tons, at the rate of ^ . >.a52 19 132 20 5-68 miles an hour. Great- 1 s So 03 21 62 15 est load, 168 ton» at 4 uiilos jpor hour. a -i» •S'iS' 22 156 25 ■2 =■< 2|t s 23. 66 15 •3^ •a.5 2 24 25 114 126 20 25 Sci -T ■ .s 28 120 19 O ■!> Oi ** ''^ ll "Is rH 29 120 18 , ^""^ , 'S May 1 134 20 §3g '— ^^r^^ a 2 132 20 Mean 120-8 21-1 «'^ a .;-■ 9> /-.^-A .> 03 Feb. 28 126 22 a c-k'S a o March 1 84 12 9 108 20 Average duty performed = iU9'l tons, at the rate of CO 12 138 30 5-86 miles an hour. Great- est load, 138 tons, at i -? !N 13 84 20 miles per hour. a •a i 14 16 120 133 22 25 > CO s 16 84 15 1 S Mean 109-1 20-7 These experiments present the same uniformity in the force applied by the double engines No. 43 and 39, and also by 43 and 45, as exhibited in the preceding tables. The duty performed is still inferior to that attained by the single engine; but in other respects the performance is remarkably consistent. 166 ME. WILLIAM PAIRBAIBN ON THE TABLE V. GOODS TEAINS. Particu- lars of leading Partica- lars of assisting Total height of gTELdient, and mean Date*. 1850. Gross wt. of Train, exclusiTc of Engine Time oc- cupied in travelling from the Kemarks. Engine. Engine. rate of in- andTen- the top of incline. clination. dcr,intons FeD. 14 61 36 42 23 15 18 KaUs »lippei7. .. ^ 15 48 20 " wet "S t " 48 20 >i a (J ^ 54 22 bO g 'e 60 20 a 72 25 ^ S *« 46 19 Engine out of order. -S 30 22 i> •• 1 1 "d 60 54 20 20 30 57 60 72 20 20 23 Greatest load 72 tons, at D.22 miles an hour. Meaa 51.4 19.7 Comparing the above experiments with those in Tables I. and in., where the single engine is employed, a con- siderable diminution of speed is observable to the weight of the load carried. This is accounted for by the reduced powers of the engine, which had 15 instead of 18 inches cylinders, only two-thirds of the large engines. The loads carried in both cases will therefore be found nearly propor- tional to the powers of the engines and loads respectively. LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 167- TABLE VI. CL c\ r\ DS TRAINS \X KJ KJ Total Gross wt Time oc- cupied in travellini? from the bottom to the top of incline. Particu- lars of leading Particu- lars of assisting height of gradients, and mean Dates. laWk of Train, excloeive of Engine Bcmarks. Engine Engine. rate of in- and Ten- clination. der,iu tons Feb. 14 120 22 .. ^i 15 114 20 ■a ■p-3 «1 16 84 15 1 ii° 18 102 18 O !F c» 19 120 21 Rails dkperf.^ AVragjedutyper- 13 ^.§11 c^ 23 108 23 ) formed = 113-6 £■2 ■s.l.§ o Mar. 2 114 15 f tons, at the rate V of 4-87 miles an 1 18 102 20 hour. Greatest 1 load rjS tons, at III .a April 3 108 25 n wet ' 461 nils. pr. hour :-ia- 1 4 120 24 1 6 8 30 138 115 132 26 20 25 IS a Mean 113-6 24.6 a) -a a 15 S-Sfl Feb. 28 78 14 ■^1 t»ll s Mar. 1 132 30 ^ Av'ragedutypMT- ) formed = lor« 1 9 114 22 r tons, at the rate 14 100 20 Bails s^ppet7. V of 57 railaa an 1 hour. Oreatut ^w |s5-a5 s g 15 114 20 1 load 132 tons, at .S"3 li «S|^ II 16 108 20 .' imiiss per hoar. • Mean 107-6 21 %0 I >*-v^ Feb. 25 78 25 1 27 60 20 Average duty performed .^ S4-2 a tf O Mar. 3 60 30 tons, at tlie rate of i-7 miles an Ui 9 g hour. Greatest load 78 tons, at o a 1 15 36 15 4-8 miles p«r hour. i 1 Feb, 21 42 16 55-2 21-2 In the last five experiments contained in this Table with, the single engine, the duty performed approximates closely to that of the double engine, the rate of travelling and the load carried being nearly the same. It will, however, be observed, that the cylinders of the single engine. No. 39, are only 15 inches, whereas the assistant engine in the six pre- vious experiments, No. 45, had cylinders of 16 inches, which reduces the duty of the double engines to nearly the same paroportion. as that recorded in the previous experiments. 168 MB. WILLIAM FAIKBAIEN ON THE TABLE vn. GOODS TRAINS ParaoTilara of leading Particu- lars of assisting Total height of gradient, and mean Gross wt Time oc- cupied in travelling ftom the bottom to the top of incline. BemarkB. Dates. 1850. of Train, exclusive ofEngine Engine. Engine. rate of and Ten- inclination der, in tons ^ 09 OT Mar. 6 54 20 'Under, 1. whee f engir ins. J, .')7 22 54 20 "7 42 20 Ralls slippery. €■- <=S "S 48 20 ^2^^ «Sh jj 48 22 i> » g*: ttco OO » 64 23 still «.2 ^ to 8 46 20 "o Feb. 27 54 64 20 20 Average duty performed = 526 tons, .5 ■a ■6" -4J 48 19 at the rate of 5'6 miles an hour. Eng strok ouple endei ■a s Mar. 11 49 54 19 20 Greatest load, 57 tons, at 5'45 miles per boor. ^^sl: ■3 ,j G6 25 0^*3 « T3 „ 55 20 >i5 1 03 ., 60 22 Mean... 52-6 21-7 Feb. 28 36 20 ^ 37 23 .#.^ ■«' 42 24 •^ s § _c 42 24 "5 c 1 «w Mar. "1 48 20 o 48 20 •§"« .2 48 19 _o ° 'C "9 36 18 ifc s 42 20. 4J *3 0 C3 42- 20 1^ 12 .3.3 36 18 15 a o 48 26 .11 oT 1 13 42 38 20 15 Rla.sUppery.N^:-^ttor; =?.° 42 20 » i> r at the rate of 6-9 o'rt . es 42 20 20 „ „ y miles an hour. — J3-S « a-2 g 14 40 ( Greatest load, 48 " 1 tons, at 6-81 luUes •Sg^ ^ 36 20 " " ■' per boor. o|« 8- 36 18 " w II 1 15 42 20 §1- 34 18 •a.S2 S 48 22 1 • 16 42 20 30 14 •9^ a „ 48 22 |2i April 16 42 20 Mean... 40-8 19-8 The same results are indicated in the experiments here recorded as in those already obtained by the single engine. They appear to follow the same law as respects the weight moved, the velocity obtained, and the powers of the engine employed ; and, assuming the engines to be in good work- ing order, it will be found that the duty performed by each is nearly in that ratio. We now come to the Passenger Trains at increased velo- cities, which indicate some interesting results. LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 1G9 TABLE VIII. PASSENGER TRAINS. Particu- lars of leading Engine. Farticalara of assisting Engine. Total ' height of gradients, and n>ean rate of in- clination. Dates. 1850. Gmss wt of Train, exclusive of Engine and Ten- der,in tons Time 00- onpied In travelling fVoni the bottom to the top of incKno. Remarks. No. 36 Engine, 15 inch cylinder, 1 foot 8 inch stroke ; six 5 feet 6 inch wheels, 4 hind coupled Weight of Engine, ISJ, Tender, 12} = 31 tons. No 45 Eniiine, 16 Inch cylinder, 1 fool 10 inch stroke ; six 5 feet wheels, 4 hind couplwL Weight of Engine, 2U, Ten- der, 12} = 34 tons. Distance run, 2 miles, on a main rise of 1 In 41-6, or a vertical height of 262-8 feet. Feb. '27 28 Mar. 9 11 I'v! 13 14 16 18 25 2IS 25 25 30 35 25 30 20 25 25 25 25 26 20 30 5 5 5 6 7 5^ 7 5 5 5 6 5 6 5 7 5-G2 Average duty performed = 26 tons, Bt'jl'A'i miies an hour. Greatest loud, 35 tons, at 30 miles per hour. Mean . . 26 TABLE IX. PASSENGER TRAI> rs. No. 86 Engine, 15 inch CTiindcr 1 foot 8 inch stroke i six 6 feet 6 inch wheels, 4 hind coupled. Weight of En- gine, 18}, Tender, 12} = 31 tons. §1 5 Feb. 21 25 25 25 25 26 7- 1 6 5 R'ls wet f Av'rage duty per \ formed = 2525 " J tns., at 19-2 mis. ■\ an hour. Gr'test i load =3 316 tns., \ 24 ml9. per hoar. Average duty performed = 2616 tons, at 20'3 miles an hour. Greatest load, 30 tons, at 17-14 miles an hour. Average d iity performed = 28 tons, at 2U miles an hour. Greatest load. 25 tons, at 20 miles per hour. Mean . 25-25 6-25 o2::-Ss^sii ft > d 1 Mar. 2 4 5 15 25 21 25 30 25 25 7. 6i 5 7 5 6 Mean.. 2516 .591 Feb. 22 27 28 Mar. 2, 4 6 Mean.. 25 20 20 20 15 15 25 9 5 6 6 5 5 6 20 6 In the transit of passenger trains up long and steep gra- dients, assistant engines are almost invariably employed. They consist of the regular engines for the conveyance of the train, and an auxiliary engine, which is generally in waiting at the bottom of the incline, to assist by pushing at the tail end of the train till the summit is attained. The mean of the load carried and the speed obtained, in Table IX., with the z 170 ME. WILLIAM FAIEBAIRN ON THE assistant engine, is 23"5 tons, at 19'8 miles an hour; whereas, in taking the mean of seven experiments with the leading engine alone, we find that nearly the same weight is raised at about the same velocity, or 20 tons at 20 miles an hour ; and, what is still more extraordinary, on March 6th, a load of 25 tons is raised to the top of the incline at the rate of 20 miles an hour by the single engine. TABLE X. PASSENGER TRAINS. ■ . - Total Gross -wt Time oc- cupied in travelling from the bottom to the top of incline. Particu- lars of leading Particulars of assiating height of graiUentg, and mean Date*. 1850. of Train, exclusive ofEnglne Remarks. Engine. Engine. rate of in- and Ten- clination. der, in tons II ■ShS Feb. 27 35 8 ^2 „ 25 6 «>5 1 11 S '■2 35 25 8 5 1^ > 1*1 25 35 5 7 — is •g s s ,, 25 5 u? 12 35 40 8 8 .\Terage duty performed = 1: .s ,, 25 5 29-1.3 tons, at 1944 miles 00 a 13 32 7 an hour. Greatest load 40 if 36 26 7 5 tons, at 13-33 milci per hour. ti 2? « 14 25 30 5 7 ■g§- , 25 5 1 o .ss 1° ,J3 16 26 5 if 1^ 2'3 18 30 25 25 6 6 6 n Is c |«5 8 1 „ 35 25 7 6 Rails wet Average duty performed = Mean 2913 617 Feb. 21 15 35 25 8 6 Tf oT 22 25 38 45 7 8 31-C tons, at 16-6 miles an WH3 .a hour. Greatest load 45 it o tons, at 15 miles per liour. Mean 31-6 7-2 The duty performed respectively by the engines is nearly equal in this Table, and will compare with the trips in Tables VIII. and IX., which, taken in the aggregate, indicate nearly the same results, with the exception only of the performance of the single engine in Table IX., which, as before noticed, greatly exceeds the duty of the double -engines. LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 171 TABLE XI. PASSENGEH TRAINS. ParUcu- Parttctt- Total height of Dates. GhMis wt. of 'IVain, time oc- cupied in travelling from the l)ottoni to lars of leading larsof assisting gradients, and mean 1850. exclusive of Engine 'Remarks. Engine. Engine. rate of In- clination. and Ten- der, In tons the top of inclino. 6 rf |gs Mar. 2 20 "3 j> 35 8 1 ■o 2 0. 1 4 45 20 9 5 I** i >7 50 25 9 6 Average duty performed = 32-08 tons, at 17'14 miles an hour. g'«"Sc 6 30 7 Greatest load 50 tons, at 13-33 •s2 •5 K ^ o )> 45 30 10 6 miles per hour. a 2 .- 1! "P 16 S5 6 ^,0 0.-' ^ C 0 tj 35 25 7 5 1- .... 5 0) ? 00 2 o to 5z; Mean 3208 7 Feb. 22 15 2& 7 5 e o 23 15 4 I-^ « '25 15 6 . -g^' ^ ?> 20 6 Awrage dutj- performed = 80-8 e^j ^ "27 16 6 tons, at 23 miles an hour. Great- Cfl 28 15 5 est load 40 tuns, at 17-11 miles per hour. £^ g 40 7 S s >> 15 5 w:; s Mar. 6 20 5 s jj 35 8 1 j> 25 6 Mean 20-8 5-58 The anomalous condition of the performance of the single and double engines presents a, difficulty not easily accounted for. From the 22nd of February to the 6th of March in- clusive, the single engine conveyed to the top of the incline twelve trains, with a mean load of 20*8 tons, at 23 miles an hour, and the greatest weight carried on the 28th February, was 40 tons at the rate of 17*14 miles per hour, a higher rate than in any of the former experiments by the single engine. With the same engine, and an assistant of equal power, the greatest load raised is only 172 MR. WILLIAM FAIRBAIRN 6N THE 50 tons, at the rate of 13'33 miles per hour, a greatly in- ferior duty to that obtained by the leading engine itself on the 28th of February. The only reason for this discrepancy would be the im- perfect state of the assistant engine, which, in this and the former Tables, appears to have given out a very small pro- portion of the power, probably little more than sufficient to carry its own vi^eight up the incline. On the other 'hand, the leading engine must have generated steam rapidly, and, as usually happens in surmounting gradients, at a high pressure, which at once accounts for the great difference which exists in the duty performed by the single and double engines respectively. TABLE XII. PASSENGER TRAINS. Total Gross wt 'Jime iH-- cupied in travelling &ora tlie bottom to the top of incline. ■ Particu- lars of leading Engine. Particulars of assisting Engine. height of gradient, and mean rate of in- clination. Bates. isau. of Train, exclusive of Entrine and Ton- der,iaton9 Bemarks. a J ■^"Sf^ll Feb. 11 20 H .5 S S p 21 15 6 =.^s •-■^^T.^- *^ 30 6 Average duty performed = S8 57 C4 « . bin i 5 ii H» i 22 23 35 40 35 5 6 7 6 tons, at 20*76 miles an hour. tJreatest load carried, 40 tons. 'A 1 s ih 27 at 17"14 miles par hour. 28-67 rrS LOCOMCrtrVE ENGINE ANt) RAILWAY GEADIENTS. 173 In the above experiments we have a still further exem- plification of the inefficacy of the assisting engine, as the ratio of the load carried to the speed attained is remarkably consistent whenever the assistant engine is employed ; and this is the more strikingly apparent as the greatest load, 40 tons, raised in February 25th, is only equal to that drawn by the single engine recorded in the preceding Tables, the speed being reduced to its equivalent of the load. TABLE XIIL PASSENGEil TEAINS. Dates. 1850, Mar. 2 » 4 i> 5 15 Mean. Feb.2u 27 Mar. 6 Mean . Gro«»wt of Train, exclusive ofEo^ne and Ten- der, in tons 20 25 20 40 30 45 25 30 29-4 20 30 25 35 35 29-0 Time o copied travelling from ttie l»ottoni to tlie tup of incline. 6 6 6 9 7 H 712 6 9 11 8-0 Average duty performed = X9-4, at I6'86 miles an hoar. €h«ate*t load,' iS> tons, at 109 miles per hour. Average duty peribrmed = 29 tons, at 1&- miles an honr. Greatest load, 35 tons, at 13'33 miles per hour. The mean duty performed by the double engines in the first experiments, is similar to that attained by the leading and assistant engines in the conveyance of the trains at different periods of time. They are all of them in their united capacity defective ; and again, computing the work done with the power employed in the above experiments, it is evidently in favour of the single engine. Twenty-nine tons, it wuU be observed, were conveyed to the top of the incline at the rate of 1 5 miles an hour by the single engine ; 174 ME. WILLIAM FAIRBAIRN ON THE whereas two engines of equal power raised the same weight, or nearly so, at the rate of 16*85 miles per hour, which gives an increase of speed of only l'S5, little more than 1| miles an hour. These facts are confirmatory of the advantages of working trains on steep gradients by the single engine, whenever that can be accomplished, with light loads at a moderate rate of speed. Having effected the arrangement and classification of the experiments, it now becomes necessary to collect them in such form as will enable us to arrive at correct results, and to offer such remarks as may prove useful in determining in what situations, and under what circumstances, the various gradients on railways can be surmounted by the single en- gine, and when the assistant engine can be applied with advantage. To effect these objects the following summary of results, first, of the goods, and then of the passenger traffic, will be found sufficiently explicit to enable the en- gineer to judge as to the amount of power required, and whether or not the assistant engine would be conducive to the interests of proprietors and the service of the public. SUMMARY OF RESULTS, As obtained from 270 Experimental Trips in the transit of Goods Trains on the Accrington Incline. SINGLE ENGINE. DOUBLE ENGINE. 1 N°- 1 Mean rre?/|'BnT- No. Mean „ .. ,1 Power of 1 of Trips. weight carried in Toos. hour. I""""'^'*"' |cyUndcr«- of Trips. weight carried in Tons. ling in miles pr hour. ed by area of cylin- ders. 38 71-6 6-31 490-8 10 111-9 625 844-2 65 710 6-80 400-8 15 123-9 5-90 844-2 45 51-4 6-09 353-4 4 1335 4-25 844-2 5 55-2 4-80 353-4 18 1-208 5-68 844-2 16 52-6 ! 5-50 353-4 8 1091 5-86 892-8 27 40-8 5-90 402-0 13 6 113-6 107-0 4-80 : 706-8 5-70 755-4 ... 571 j 5-73 427-2 Mean. ... 117-2 5-49 818-8 In the above summary, it will be observed, that assum- ing the area of the cylinders of the engines employed to be LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 175 the measure of the force exerted at a given velocity, and supposing the power of the single engine to be represented by 427*2, we thus have a load of 57 "1 tons carried up the different gradients of the incline at the rate of 5*73 miles per hour. This duty, when compared with the double engine, whose representative of force is 818'8, is rather more than twice that of the single engine, a load of 117*2 tons being transmitted over the same gradients at the rate of 5*49 miles an hour. From these results we must infer, that in the transit of heavy trains at a low rate of speed, the advantages are not much in favour of the single engine; yet in cases of light trains, at a higher rate of speed, the benefits arising from the use of the single engine are sufficiently evident to secure it the preference, in every instance where the load is duly and properly regulated to the power of the engine. These facts are, however, more clearly developed in the experiments with the passenger trains, as follows : — SUMMARY OF RESULTS, As obtained from 123 Experimental Trips in the transit of Passenger Trains on the Accrington Incline. 8IKGLE ENGINE. 1 { DODBLE ENGINE. 1 No. of rrips. 7 1 Moan 1 Rate of weight ItraveUiug carried in; in miles Tons. 1 per lionr. Power of Erfrine represent- ed hy the area of cylinders. No. of Trips Mean weight can-ied in Tons. Rate of travelling in miles per hoxir. Power of Engine represent- ed by the area of cylinders. 20 00 2000 353-4 16 26-00 21-35 755-4 12 20-83 23-00 353-4 4 25-25 19-20 706-8 5 29 00 1500 353*4 6 23 5 12 18 7 8 26-16 2913 31*60 3208 28-30 28-57 29-60 20-30 19-44 16-60 17-14 18-40 20-76 16-85 706-8 755-4 706-8 706-8 755-4 706-8 706-8 23.26 19.33 353.4 Mean. ... 28-41 18-89 722-8 17§ M«. WILLIAM FAIBBAIRN ON THE The Accrington incline is composed of three gradients and five curves, as follows :— On this incline the curves are — 1. Bottom straight for 21 chains. 2. Curve to the right, 50 chains radius, for ... 19^ „ 3. Curve to the left, 50 chains radius, for 14 „ 4. Curve to the left, 56 chains radius, for 41 „ 5. Straight for 33 ,, 6. Curve to the right, 40 chains radius, for ... 8 „ 7. Curve to the right, 54 chains radius, for ... 23^ „ ToUl 160 = 2 milra. On a careful examination of the returns as indicated in the above summary, very different results will be found to those contained in the preceding Tables, as derived from the goods trains. There the duty performed by the leading and assistant engines bears nearly a direct comparison with the single engine, both with respect to the weight carried and the speed attained. In passenger trains, where the load is light, and more within the power of the engine, the superi- ority of the single engine is strikingly apparent, and will not bear a comparison. It appears, when the load does not ex- ceed 20 tons, the assistant engine is of little value ; and it is only in monster trains, or in cases where the load exceeds the maximum power of the engine, that the assistant engine proves advantageous. In the experiments herein recorded, we have for the mean of three trips a load of 23-^ tons, drawn up an incline of nearly 1 in 40, by a single engine, whose powers are represented by 353, at the rate of 19*3 miles an hour; whereas, on a mean of 9 trips, with two engines, where the maximum power is represented by 722-6, a load of 28*4 tons (little more than the other) is carried the same distance, and to the same elevation, at the rate of only 18*89, rather under 19 miles an hour. The result of these experiments is obvious. First ; that in heavy trains, where the load approximates or exceeds the LOCOMOTIVE ENGINE AND RAILWAY GRADIENTS. 177 power of the engine, the assistant engine under these cir- cumstances becomes absolutely necessary ; on the other hand, where the load is duly proportioned to the power, and that power so nicely balanced as to have full command over the resistance, we may then with great benefit, and no inconsiderable economy, dispense with every description of auxiliary force. In railway traffic these facts are worth knowing, and are fully borne out by the results thus ob- tained in working one of the largest and most difficult inclines in the kingdom. In the experiments thus recorded^ it may. be interesting to ascertain how far they correspond with the results arrived at on the Hunt's Bank and Halifax inclines, and the com- puted resistances as given in the preceding tables. In cases of the latter it is estimated, that a locomotive engine of 16 inch cylinders would, at a speed of 33 miles an hour, raise a load of from 35 to 37 tons up an incline of 1 in 40. Now, in the case of the Accrington incline, if we compare the power of the single engine with 15 inch cylinders, which raised a load of 40 tons at the rate of upwards of 17 miles an hour up gradients, some of them steeper than 1 in 40, and allow for the retarding influence of curves, we shall then have a duty nearly equivalent to that in the table, if not approaching to the experiments made on the Hunt's Bank incline, where a much greater load was carried at a considerably reduced rate of speed.. Viewing the subject in this light, and allowing for the increased pressure attained in experimental trials, compared to that of the regular working of the trains, we may reasonably conclude, that the diffijrences are not so great as appearances at first sight would indicate. In conclusion I would observe, that, notwithstanding the great improvements and increased powers which for the last fifteen years have been introduced into the locomotive en- gine, it would appear that we have not as yet arrived at that 2a lis MR. IT. FAIRBAIRN ON THE LOCOMOTIVE ENGINE. Sx. maximum state of perfection which is calculated to ensure the conditions and meet all the requirements of surmounting steep gradients. These improvements are yet before us, and the object of the investigations which I have ventured to submit, will be fully attained if they lead to further en- quiry into the conditions calculated to increase the powers and extend the resources of our locomotive traffic. 179 r XIII. — On the Security and Limit of Strength of Tubula (xirder Bridges constructed of Wrouffht Iron. By WlL- LiAM Fairbairn, Esq., V.P. Read April 2, ISSO. Bridges have been in use from remote antiquity, and have received in all ages that consideration which the im- portance of the structure, and their great public utility, so justly entitle them to. They form the connecting link between one part of the earth's surface and another ; allow of a continuous coijimunication, by connecting the opposite banks of rivers and deep ravines, and overcome various obstacles which might otherwise be considered impassable. They, in fact, form a very important element in that system of communication by which the civilized nations of the world hold intercourse with each other, and which constitutes the jnedium of commercial interchange between the different districts of a country. They add facilities for the enjoy- ment of social life — for the easy direction of the necessary political supervision — and for that invaluable interchange of intellectual and physical relations, which contributes so largely to the wealth and intelligence of a nation. They, moreover, in modern times (associated with that wonderful development of iron " highways," which now traverse in every direction the surface of the country) constitute a medium of concentration in that union of distant objects, which is productive of so much benefit, and by which — through the aid of the locomotive engine — the remotest districts of the empire are now united. 180 MK. WILLIAM FAIEBAIRN ON These advantages are common to all countries; and now that rapidity of transit has become an essential part of our existence, it naturally follows that every discovery and «very improvement which tends to the extension and en- largement of these facilities, must prove beneficial to the public, as well as interesting to the philosopher and the engineer. Impressed with these views, I have endeavoured to collect the results of a long series of experiments, and to narrow within the compass of a few pages those labours which have occupied no small share of my time and thought, whilst devising means for the construction and proportion- ing of the parts of a new system of bridge-building. I now propose to submit these results for the consideration of the Society, prefacing them by a few remarks relative to the construction and other matters connected with the security and permanency of this description of bridge. In a paper given' to- the Institution of* Civil Engineers I have stated, that " every erection of this kind, having for its objects public convenience and a public thoroughfare, should have within itself the elements of undeniable secu- rity. Bridges above all other structures should contain those elements: they are the most liable to accident; and, from whatever cause such accident may arise, the commu- nity are equally interested in the strength and durability of the structure. In attempting the introduction of a new system of construction, comprising the use of a new and untried material, it behoves the projector, therefore, on public grounds, to be careful and attentive to all the minu- tiae directly or indirectly affecting its security. In bridges of the tubular construction, considerations of this kind are of primary importance, as much depends not only upon a Correct application of the principle, but upon the quality of the material and the workmanship introduced, which, in fevery case, should be of the very best description. In the construction of Tubular Girder Bridges, I have endeavoured. TTIBULAB GIRDER BRIDGEl 181 as correctly as possible, to apply those principles; and having a strong conviction of the great superiority of strength^ durability, and cheapness which the system offers in com- passing large spans, I have not hesitated to advocate its extension. It, however, becomes necessary, from time to time, to submit the bridges to a rigid examination ; and, before opening any one of them as a public thoroughfare, it is essential to submit them to severe and satisfactory tests. These tests and examinations have been various and fre- quent ; and I believe we may venture to affirm, that in no case where the Tubular Girder Bridge has been duly pro- portioned and well executed, has there been the least reason to doubt its security. " The first idea of a Tubular Girdier Btidge originated in the long preliminary experimental research which I con- ducted, in connection with the great bridges on the line of the Chester and Holyhead Railway ; and, during its first application to railway constructions, the utmost precaution was observed in the due and perfect proportion of the parts. These proportions were deduced from the experiments made upon the model of the Britannia Tubular Bridge at Mill- wall, London ; and after repeated tests upon a large scale (full size), the resisting powers, and t)ther properties of the bridge, were fully established. From these experiments, a formula was deduced for calculating the ultimate strength of this tubular description of bridge, having spans of from 30 up to 300, or even 1000 feet; and, as that formula is now before the public, I believe it may be relied upon as practically correct To relieve it, however, from any thing like ambi- guity, I shall endeavour to state as briefly as possible, certain points which, in my opinion, should be taken into consideration in its application." Experiments were made on a large scale to determine the accuracy of my views, and to ascertain the best and strongest form of tube as a means of supporting the Chester 182 >IR. WILLIAM FAIRBAIBN ON and Holytbead Railway, across the wide spans of the estuary of the Conway and the Menai Straits. The original cooception of a huge wrought-iron tube, of a circular or elliptical sectional form, suspended in mid-air, and of dimen- sions calculated to allow of the passage of the locomotive and its accompanying train through its interior, yielded before the facts which these experiments brought to light, to the still more extraordinary nud daring project of 4 colossal hollow beam, having within itself not only self- supporting powers, but a sufficient excess of strength to carry the weight of nearly a dozen railway trains. Beyond this, the experiments gave the rough outline to the system now under consideration, and which has already received, in an extended application, the sanction and approval of practically scientific men, and the confidence of the public. The Millwall experiments not only successfully realized those objects, but they made us acquainted with other constructions of equal if not even greater importance, in the jievelopment of the tubular girder system, which is admi- rably adapted for almost every description of bridge ; and, beyond comparison, infinitely more extended and more general in its application, than the form of tube which now spans the depths of the Menai Straits. It is this girder con- struction which I am anxious to bring before the meeting, in order to explain its peculiar adaptations, and to receive those suggestions for its improvement, which, I am satisfied, will be freely given by the members of this Society. It waiS determined by the experiments, that, in order to balance the two resisting fwces of tension and compression in a wrought-iron tubular girder having a cellular top {as shown in the plate)^ that the sectional area of the bottom should be to the sectional area of the top, as 11 : 12; and the proportional of these parts being thus established, it therefore follows, that any increase to one or other of them will not materially affect the strength of the bridge. C ,, :i ' I X "^ rl % S t a ui It SgS:^f ^ §33 ^ 0 H "•J ■5 & . . 4 <^ *" s* s ' 1 02 i 2 •?' St>^^ TUBULAE GIRDER BKEDGES. isa On the contrary, if additions be made to the one (assum- ing the ratio to be correct) without a proportional addition to the other, if the girder does not become absohitely weaker, it is evidently not increased in strength ; inasmuch as increased dead-weight is given to the girder by the introduction of a quantity of material which is totally inoperative.* This being the case, it is of import- ance to preserve as nearly as possible the correct pro- portion of the parts, in order to ensure the maximum of strength in the two resisting forces of tension and compres- sion, an arrangement essentially important in those struc- tures r and atlso in the application of the formula to determine the utmost strength of the girder.f If, for example, an excess of material were given to the bottom of the girder shown in the plate, the formula (W= — f would not apply, inasmuch as the top and bottom areas would be- disproportionate to each other, and the girder would fail from the yielding of the top before the stronger bottomii * It may be said that an increase of material to either top or bottom will increase its stiffness, and— a fortiori — its strength. I do not, how- ever, admit this dootrine, as there is no telling to what extent these dis- crepancies may be carried, and the consequence of a disproportion of the parts, if once allowed, might lead to dangerous error. Besides, these proportions must either be correct or incorrect — if the former, any devia- tion from them i& inadmissible. f It is important to bear in mind, that in devising the formula for calculating the utmost strength of a tubular girder — which formula i?, that the breaking weight is equal to the sectional area of the bottom multiplied by the depth, and by a constant derived from experiment for' the particular form of girder rmder consideration, and tl^ whole divided by the length — I have invariably assumed that the proportions which I have announced, and which were arrived at by frequent and direct ex- periment, are maintained ; and further, that the constant which I have given is for a tubular girder constructed after these proportions, and with a cellular top. Other constructions would require other constants to bo derived from experiment 184 ME. WILLIAM FAIEBAIRN ON exerted its full resistance to the tensile strain. In estimating the dimensions for the application of the formula, the excess therefore would have to be reduced to the due proportion of 11:12; or, in other words, the additional strength must be left out of the calculation in computing the strength of the bridge* The same reasoning will appl j when the excess of area happens to be in the cellular top, although in this case the formula (W,^^ . --^/ does apply, as the excess (in my opinion) goes for nothing in the cal- culation of the strength of the girder. In every case, however, where these proportions are main-» tained, we have, in the above formula, a nearly correct prin- ciple on which to estimate the strength of similar wrought-iron tubular girders, whatever may be their relative dimensions.* It must further be noticed, that in calculating the strength of bridges of this description, it is always assumed, that in addition to the proportions of the top and bottom of the girder being maintained, the vertical sides are suffi- ciently rigid to retain the girders in shape ; and it is further assumed, that the whole of the plates, angle iron, &c., are in the line of the forces, and that the workmanship as well as the riveting is well executed. ♦ Mr. Tate, an eminent mathematician, remarks upon the formula — 1st. With respect to (W_. 11, where A is the area of the section of the bottom, and C = 80, the constant deduced on this supposition will apply to all depths of the tube within short limits of error where such depths, or A, are large in proportion to the depths of the cells and the thickness of the plates. 2nd. With red|>ect to the formula (W=^^-^-, when A is the area of the whole section, and C = 267, then the tubes should be similar in all respects j but a slight variation in depth from that of similar form will not produce much error, especially where tlie depth is considerable. At the same time, it must be observed that both formulae apply with great exact- ness where the tubes are similar. TUBULAE GIRDER BRIDGES. 185 At a recent discussion on this subject, which occupied two successive meetings of the Institution of Civil En- gineers, Westminster, it was maintained that m^^ formula was not correctly applicable in cases of girders of more than one span, and that I had neglected in the calculations the great increase of strength which was derived from the gir- ders being continuous. This continuity of the girder was estimated by some to add not less than one-third, and by others one-fourth, to the ultimate strength of that part of it which formed a single span, when viewed simply as a beam supported at the ends, as exhibited in the model now before you. On this question, I observe in a note appended to ray paper read at the Insti- tution of Civil Engineers, " that the doctrine of continuity is doubtless true to a certain extent ; and, although I admit the fact, I have purposely neglected in the calculation any auxiliary support of that kind as a counterpoise," &c. — I think it safer to do so, as any admission of increased strength in that direction, might lead to serious practical inconveni- ence, if not dangerous results. I have therefore freely given, as additional security, those advantages of strength, what- ever they may be, rather than adopt refinements in the cal- culation, which, if exercised by the general practitioner, might lead to serious error in reducing the ultimate strength of the bridge. To give to a tubular girder bridge, of more than one span, the full benefit of the extra strength derived from the counterpoise of the girders on the opposite side, the girders would require to be differently constructed ; and, in place of the joinings of the plates being prepared to resist compression throughout the whole length of the girders, the cellular top would require to be constructed for about two- thirds of the span in the middle of each girder on the principle of compression — and for a distance of one-sixth on each side of the pier on the principle of tension. In fact, it would require a complex series of constructive operations, 2 B 186 MR. WILLIAM FAIRBAIKN ON in order to meet all the requirements of varied strain to which horizontal girders of this kind are exposed. Viewing the question in this light, it appears preferable to adhere to a general formula, and to give to the artificer a simple rule of extensive application, such as he may safely use without entering upon theoretical investigation, which more properly belongs to the mathematician than the man of practical science. In offering these remarks, I am far from underrating the manifold advantages which we derive from the theoretical disquisitions of the mathematician. Every investigation for the elucidation or correction of existing formulae by the test of the exact sciences must be highly valuable ; but having corroborated certain facts by repeated trials and experiments on a large scale, and having found the formula from which the calculations were made, apply with remarkable precision to almost every extent of span, I am strongly inclined to adhere to its truth, and to place implicit confidence in the construction obtained from such a source. I hope, however, that the time is not far distant when we may receive from some able mathematician a preferable and more accurate formula, if such can be obtained. It may, however, prove instructive if we examine this question more closely, and endeavour to ascertain the real value of the additional strength thus imparted to each suc- cessive span by the continuous girder, and, for the sake of illustration, let us take the design of the bridge before us,* which has three spans, the middle being double the width of the two end ones, and consequently required to support double the weight. Now, it is evident that any considerable weight laid upon the centre of the large span of 250 feet, will cause a deflection ; and, supposing the depth of the girder at the pier to be 14 feet, we then have 125 feet, or half the span, as the distance of the point of greatest de- * The design for a Tubular Girder Bridge for supporting the Dublin and Belfast Junction Railway across the Bojae at Drogheda. TUBULAR GIRDER BRIDGES. 187 flection on one side of the pier, which, acting as the fulcrum, or support of the beam, has a tendency to raise, or tilt up the end of the land girder to the same height exactly from the abutment pier. Assuming this to be the fact, and the girder to be perfectly rigid, we should then have a ten- sile strain along the top side of the girder over the pier in the ratio of 125 : 14, nearly as 9:1. This is one of the advantages peculiar to the wrought-iron tubular girder, as, in every bridge having more than one span, the girders have always been made continuous ; but as repeated changes are continually going forward from the passing trains, and as these changes, producing a severe strain, have a tendency to destroy the elasticity of the material, and the soundness of the workmanship at that part, I have considered it essen- tial for the public safety to neglect it in the calculation, and to give in any additional strength which may arise from that source. Should it, however, be determined to take these advantages into account, a new formula must be de- duced, and a new system of construction must be adopted over the piers, in order to attain the full benefit of this new element of strength. The excess of strength that should be given to Girder Bridges, has received considerable attention not only from the profession, but also from the general public. The various accidents which have occurred in the failure of bridges of different constructions, have created of late years consider- able alarm as to the stability of those important structures; and when the enormous weight of a railway train, and the momentum of that train moving at fifty miles an hour, are taken into consideration, it requires the utmost foresight, and the greatest possible care, to have the bridge sufficiently strong. These are considerations of deep importance to the engineer as well as the public; and although great differ- ence of opinion exists as to the exact multiplier that should be given to the maximum load, to obtain the load which would produce rupture, I am of opinion that it should 188 ME, WILLIAM FAIKBAIKN ON never be less than four times the greatest load that can be brought upon the bridge. In the wrought-iron Tubu- lar Girder Bridge, I have computed the breaking weight at twelve tons to the lineal foot, inclusive of the weight of the Bridge, which is equivalent to about six times the maximum load than can practically be brought upon it On this calculation, the following Table exhibits the strengths and proportions of Girder Bridges, from 30 up to 300 feet span. It has been computed from experiments on previously constructed Tubular Girder Bridges. The first column gives the length of the span clear from pier to pier. The second, the breaking weight of the bridge in the middle. The third, the area of the plates and angle iron of the bottom of the girder. The fourth, the area of the cellular top. And the last, the depth of the girder in the middle. TABLE SHEWINa THE PEOPORXIONS OF TUBULAR GIRDEB BRIDGES, FROM 30 TO 150 FEET SPAN. SPAN. Centre Broak- Intr Weight of Bridge. Bee. Area of bottom of one Girder. See. Area of ton of ona Gilder. Depth at the Girder in the middle. Fest. Id. Tons. Inches. Inches. Feet In. 30 0 180 14-63 1706 2 4 35 0 210 17-06 19-91 2 8 40 0 240 19-50 22-76 3 1 45 0 270 21-94 25-59 3 6 50 0 300 24-38 28-44 3 10 55 0 330 26:81 31-28 4 3 60 0 360 29-26 34-13 4 7 65 0 390 31-69 36-97 6 0 70 0 420 3413 39-81 5 5 75 0 450 36 56 42-67 5 9 80 0 480 3900 45-60 6 2 85 0 610 41-44 48-34 6 7 90 0 540 43-88 6119 6 11 95 0 570 46-31 64-03 7 4 100 0 600 48-75 56-88 7 8 110 0 660 63-63 62-56 8 6 120 0 720 68-50 68-25 9 3 130 0 780 63-38 73-94 10 0 140 0 840 68-26 7963 10 9 160 0 900 73-13 85-31 11 6 TtBULAE GIRDEE BRIDGES. 189 TABLE SHEWING THE PROPORTIONS OF TUBULAR GIRDEE BRIDGES. FROM i60 TO 300 FEET SPAN.* SPAN. Centre Break- ing Weight of Bridge. Sec Area of bottom of one Girder. Sec. Area of top of one Girder. Depth at the Girder in the middle. FteL In. Tons. Inches. Inches. Feet. In. 160 0 960 9000 10500 10 8 170 0 1020 95-63 111-56 11 4 180 0 1080 101-25 118-13 12 0 190 0 1140 106-88 12469 12 8 200 0 1200 112-50 131-25 13 4 210 0 1260 11813 137-81 14 0 220 0 1320 123-75 144-38 14 8 230 0 1380 129-38 15094 15 4 240 0 1440 135-00 157-50 16 0 250 0 1500 140-63 16406 16 8 260 0 1560 146-25 17063 17 4 270 0 1620 151-88 17719 18 0 280 0 1680 167-50 183-75 18 8 290 0 1740 163-13 190-31 19 4 300 0 1800 168-75 196-88 20 0 In the above Table it will be seen that I have adopted a large multiplier for the excess of strength which I conceive necessary to be observed in the construction of a railway bridge. Twelve tons per lineal foot, equally distributed over the surface of the bridge, is a heavy load as a measure of strength; and although I differ with some of my professional brethren in this question, I am nevertheless of opinion, that the difference of cost in effecting this object is inconsider- able when weighed against the additional security obtained. In the wrought-iron tubular girder, the difference in the weight of the bridge itself is proportionally less than in any * I have generally taken the depth of the girders at J^. of the span; but in cases where the span does not exceed 150 feet, I have found it more economical to adopt 1 of the span. With upwards of 1 50 feet span it is, how- ever, more convenient, on account of the great height of the girder, to adhere to the original proportion of J^, in order to keep the centre of gravity of the girder low, and in order to {»*event oscillation to the passing load. In situations where it is objectionable to increase the depth of the girders, it tken becomes essential to increase the sectional areas of the bottom and the cellular top in the ratio of the depths. 190 MR. WILLIAM FALRBAIRN ON Other construction ; and, considering the risk of oxidation arising from neglect in attending periodically to the clean- ing and painting of the girders, I am satisfied I am not wrong in making such a provision, and in substituting this large power of resistance for the strength of the principal parts of the structure. It is for these reasons that I have assumed for a double line of rails 12 tons per lineal foot as the ultimate strength of a Tubular Girder Bridge, calculated to ensure permanency, and to meet all the requirements of railway traffic. I have done so in order to meet the various contingent forces of the weight of the bridge itself, the maximum rolling load, and the various other conditions to which railway bridges are subjected, such as vibration or the force of impact acting injuriously upon the bridge. Amongst other considerations which have engaged the attention of the commissioners on railway structures, is that of impact, and the effect of vibration upon bridges composed of cast-iron, either in the shape of the single or the com- pound trussed girder. The elaborate investigations on this subject, recently published, are exceedingly valuable ; and, although they indicate several new and important proper- ties in the strength of materials, they do not, so far as my own investigations extend, give the correct law as respects the effect of the impinging forces by which these structures are assailed. I believe Professor Willis (whose high stand- ing as an acute mathematician is a sufficient guarantee for the accuracy of the experiments) is perfectly aware of this fact, and has qualified the experiments made at Portsmouth on cast-iron beams, nine feet long, by others upon existing bridges of not less than 50 feet span. These latter experi- ments are more satisfactory than those at Portsmouth, and approximate much nearer to those made by myself, and other experiments of a similar character. The effects produced upon a girder bridge by a heavy body, such as a locomotive engine rolling over its surface TUBULAR GreDER BRIDGES. 191 at a high velocity, is a subject of such vital importance to the permanency and stability of the structure, as to require the most careful investigation. It cannot therefore be sur- prising that it should have occupied a considerable portion of the time of the commissioners, and that it should have found a prominent position in their report. It must, however, be observed, that the deflection of a girder bridge arises from one of two causes, or from both. First, from the weight of the bridge itself, which is a constant producing a permanent deflection ; and, secondly, from the passing load, whether viewed as a dead or a rolling weight, acting as an antagonistic force to the resisting power of the bridge. In some parts of the commissioners' report, the experi- ments do not appear to me to bear out the facts of increased deflection produced by a body, such as a railway train moving at great velocity, and the same body remaining stationary, upon the bridge. In several carefully conducted experi- ments on tubular girder bridges of different spans, some of them upwards of 150 feet, I found the deflection as nearly as possible the same at all velocities; and, although the experiments recorded by the commissioners are highly valuable, they do not afford to the general practitioner those conclusive results which seem to be essential for the attainment of sound principles of construction. It is true, the commissioners in their report have qualified the results obtained from these experiments by others upon existing cast-iron railway girder bridges, where the deflection was reduced from an increase of the statical deflection, amount- ing to T^ths of an inch, as produced upon the nine feet bars, at 30 miles an hour, to | upon a bridge of 48 feet span, at 50 miles an hour, clearly showing that the larger the bridge, and the greater the rigidity and inertia of the girders, the greater will be the reduction of deflection to the passing load. In the tubular girder bridges composed of riveted 192 ME. WILLIAM FAIEBAIEN ON plates, it must be observed that the commissioners had no experience, nor were they acquainted with the strength, rigidity, and other properties of girders composed of wrought- iron riveted plates. In these, the deflection due to the passing load is nearly the siame at all velocities; and unless there exist irregularities and inequalities on the rails, so as to cause a series of impacts, we may reasonably conclude that the deflections are not seriously, if at all, increased at high velocities. The questionable security of a great number of horizon- tal bridges which, of late years, have been introduced for the support of railways, or common roads, has not only called for legislative interference, but the appointment of a commission to watch over the public interests and public safety in railway constructions. This commission, or the inspectors under their direction, I believe, have instructions to pass no bridge or other structure upon any line of rail- way, until carefully tested as to its security, and other conditions calculated to meet all the requirements of general traflSc. These inspectors are employed for the exclusive purpose of examining every new Une of railway, and report- ing upon its efiiciency before it is opened to the public; and, in Older to assure themselves of the security of the bridges, cuttings, tunnels, embankments, &c, upon the line, these are generally submitted to severe tests, in order to ascertain their condition and fitness for securing to the public a safe and agreeable transit from one end to the other. Bridges, above all other structures, are regarded with sus- picion, and, in order that the lives and limbs of the public should be duly protected, are submitted to a certain proof, which generally consists of a double train of locomotive en- gines and tenders being run over the bridge at diflerent velocities. A train of locomotive engines is considered the greatest load that can be placed upon a bridge; and, having ascertained the deflection of the girders fi-om their own TUBULAR GIEDEK BRIDGES. 193 weight, and that of the roadway, the experiment generally proceeds as follows : — First — To ascertain the increased deflection due to the heaviest load, as a dead weight placed upon the bridge. Secondly — The amount of vibration produced by the passage of the same load at different velocities. Thirdly — The amount of deflection due to the rolling load, and the variations, if any, when the trains are retarded or accelerated ; and, Lastly — The principle of construction is taken into con- sideration, and the excess of strength which a bridge should have over the greatest load, in order to declare it safe for general trafiic. On the first, second, and third, there appears to be little, if any, difiference of opinion ; but on the latter, the greatest and most opposite views are entertained. Some contend- ing for three, and others for four, five, and six times the greatest load; whilst others again, more timid than the rest, insist upon eight or ten times the greatest load in order to be safe. Such appear to be the present views entertained by the profession, and such they will continue to be, unless decided by some high authority, from which there is no appeal, as to what should be the resisting powers of a bridge. I make no doubt we have now in existence several bridges which, to all appearance, are duly performing the important functions of supporting heavily loaded trains within the nan*ow limits of probably half the weight that would lead to destruction ; and others again are of such enormous strength as to bid defiance, for ages to come, to the heaviest load that can by possibiHty assail them. Such I believe to be the present state of a considerable number of our railway structures, and such are the widely spread notions which have taken possession of some of our railway engineers. Under these discrepancies, it becomes a question of deep importance as to what should be the 2 c 194 MR. WILLIAM FAIBBAIRN ON exact measure of strength, and what excess should be given to a bridge beyond the load it is called upon to support. It appears to be the opinion o^i|ftie railway commis- sioners, that the flexure of girl^R should never exceed one-third of the ultimate deflection; and, although I concur in that opinion, I would venture to afiirm that in wrought-iron tubular girders, such as are now in use, the effects of reiterated flexure is only one-sixth, and conse- quently they present a larger margin of security than girders composed of any other material. On the effects of impact I entertain the same views as the commissioners, that the deflection produced by the striking body on wrought-iron is nearly as the velocity of impact, and those on cast-iron greater in proportion to the velocity. These facts have, however, been strikingly ex- emplified by experiments made on the first tubular girder bridges constructed for the support of a railway. Two bridges of this kind were erected near Blackburn, over the canal and turnpike road. Both bridges were 60 feet span, and before they were opened to the public they were sub- jected to the following tests :— A train of three locomotive engines, weighing 60 tons, occupied the entire span of the bridge, and, having ascer- tained the deflection in their quiescent state, they were started at different rates of velocity, varying from 5 to 20 miles an hour, which produced a deflection of i^ths of an inch. Two long wedges of the height of one inch were then placed upon the rails in the middle of the span, and the fall of the engines from this, when moving at a speed of 8 to 10 miles an hour, caused a deflection of only '42 inch, which was increased to '54 inch, or about half an inch, when wedges 1^ inch in thickness were substituted. These were severe tests, such as should not again be re- commended, as the enormous strength of these girders is nov/ well understood, and they may safely be considered fit for TUBULAR GIBDEK BRIDGES. 195 service after being submitted to the heaviest rolling load, or one-sixth of the hf.eaking weight. In closing these remarks, I would observe that, after these experiments, I came to the conclusion that the tests for bridges of this kind should not exceed the greatest roll- ing load, and that load to be one-sixth of the breaking weight of the bridge. I may be wrong in this conclusion, which, with great deference, I submit for correction ; but in this I am fully persuaded, that in order to give the necessary security, and to provide for all the contingencies consequent upon railway traffic, it will not injure the in- terests of a railway proprietary to have the bridges of suf- ficient strength to resist six times the greatest load. TORKSEY BRIDGE OVER THE RIVER TRENT. The two main girders extpnd over the middle pier, on which they rest, ■with expansion rollers on each abutment. Feet Inches. Total length of each girder 282 0 Clear span of each opening 130 0 Depth of main girder 10 0 Breadth 2 9 Depth of cross girders 1 2 Spsin of „ 25 0 Distance of „ from centre to centre 2 0 Total length of bridge, including masonry 342 0 Tona. owt. Total weight of iron in main and cross girders ... 252 14 PERMANENT I.0AD FOB ONE OPENING. Tona. cwt Weight of main girders 91 12 Weight of cross girders 27 1 Timber 18 6 Ballast (2 inches thick) 19 10 Rails, chairs, and fastenings 7 18 164 6 laches. Sectional area of top, in inches 60*24 „ „ bottom „ 49-68 The bridge has been tested with six locomotive engines in steam, equally distributed over one opening, of the aggregate weight of 222 tons, when the deflection was found to be 1*26 inch in the middle. On the removal of the load the bridge returned to its original level. 196 XIV. — On the Cause of Unequal Falls of Rain in Cumber- land. By Alderman Thomas Hopkins. Read October 1, 1850. Mr. Millar, of Whitehaven, in a paper read to the Royal Society, on May 18, 1848, gives a number of important meteorological facts relating to the Lake district of Cum- berland and Westmoreland. The statement of the falls of rain that take place in many parts of this locality are very valuable, on account of the different heights of the parts above the level of the sea where the rain gauges were placed, and the particular shape of the face of the country. For a long time it had been known that the fall of rain became greater, as the ground rose from the low level of Lancashire to the top of the ridge that separates that county from Yorkshire ; and it appears that the same general fact is, to a certain extent, observable in Cumberland, Mr. Millar having found that the fall was small at Whitehaven and other places in the low country near the sea, compared with that which took place up the valleys and on the mountains of the interior country. And that gentleman, after stating many facts, attempts to exhibit a law which determines that the amount of rain shall increase up to a certain height, and decrease above that height. He says, " It seems probable that in mountainous districts the amount of rain increases from the valley upwards to an altitude of about 2,000 feet, where it reaches a maximum, and that above this elevation it rapidly decreases." — P. 85. Now, although the facts thus given are important in themselves, and afford a certain de- gree of countenance to the hypothesis advanced; yet neither MR. T. HOPKINS ON THE FALLS OF KAIN IN CUMBEELAND. 197 the facts nor the reasonings founded on them are sufficient to warrant the general conclusion drawn from them by Mr. Millar. A return is given of the quantities of rain that fell in twenty different places in Cumberland in the four years 1845-6-7 and 1848 ; and of these places we may, in the first instance, take three as a sufficient number to shew how far the facts harmonize with the law laid down, namely, White- haven, Wastdale-head, and Seathwaite — the first being on the sea-coast, the second, inland, at the mouth of the moun- tain pass of Sty-head, and the third, beyond that pass, and in the valley of Borrodale. In these three places there fell in the years named the following quantities of rain, namely : — At Whitehaven, 90 Feet above the Sea. At Wastdale Hend, 166 Feet ubove tlie Sea. At Seathwaite, 240 Feet alwve the Sea. Inchea. Inches. In 1845 49-207 108-55 151-87 „ 1846 49-134 106-93 143-51 „ 1847 42-921 96-34 129-24 „ 1848 47-344 116-32 160-80 Mean 46-589 106-60 145-63 Now, the differences in the heights of these three places are not very great, but the differences in the quantities of rain that fell are enormous — quite enough to warrant a sus- picion, that the very large amount that fell at Seathwaite is not attributable to the height of that place above the sea. But in addition to those three places, there is the Pass of Sty-head, 1,290 feet high, situated between Wastdale-head and Seathwaite, on which a rain-gauge was placed ; it is, however, so cold there in the winter, and the gauge is so much affected by snow and ice at that season, as to prevent reliance being placed on it during that portion of the year. Yet we may compare the quantities of rain that fell in the summer months only at Sty-head and Seathwaite, as given 198 MR. T. HOPKINS ON THE by Mr. Millar — they are for the six summer months of 1848— Inehei. Seathwaite, 240 feet above the sarface of the sea,... 68'96 Sty-head, 1,290 feet above the surface of the sea,... 60-35 Here we find no increase in the quantity of rain that falls above 240 feet pf height where the gauge is placed in Sea- thwaite. On the contrary, the quantity is greater there than at Sty-head, 1,050 above it. This fact furnishes rather strong presumptive evidence, that the quantity of rain that is received in a gauge, at any particular elevation, is not proportioned to the height at which the gauge is placed. In comparing the quantities of rain that fall at various heights, including great elevations, it is obviously necessary to compare them during the summer months alone, as has been done when comparing Seathwaite and Sty-head Pass; and the facts that are principally relied upon by Mr. Millar, and from which he draws his general conclusions, are the quantities of rain that fell in twenty-one months in 1846 and 1847 in six places, namely: — Incbes. The Valley (Wastdale) ... 160 feet above the sea, 17055 Sty-head, 1,290 „ 185-74 SeatoUer, 1,344 „ 180-23 Sparkling Tarn, 1,900 „ 207-91 Great Gable, 2,925 „ 136-98 Sca-fell, 3,166 „ 128-15 But these facts, although they countenance the hypo- thesis advanced, do not afford conclusive, or even strong, evidence upon the subject. The Valley we see, 160 feet high, has 170*55 inches; whilst SeatoUer, 1,344 feet high, and consequently 1,184 feet higher than the Valley, has only 180*23 inches, not 10 inches more of rain ; whilst Sty-head, 54 feet below SeatoUer, has 5^ inches more of rain than that place. There is another place noticed by Mr. Millar, called Brant Rigg, 500 feet high, between the Valley and Sty-head, which received 12|^ per cent, less of FALLS OF RAIN IN CUMBERLAND. 199 rain than the Valley, that is only 160 feet high, showing that here less rain fell in the higher than in the lower parts; and there are other anomalies that might be pointed out. It is, however, such a place as Seathwaite that shews, in the most palpable and striking way, that the amount of rain that is received by the ground in a particular locality is not de- termined by its height. Seathwaite, not faore than 240 feet above the sea, receives more rain than any of the places having a greater elevation ; and Mr. Millar candidly admits that he is *' unable to offer any satisfactory reason for the great excess of rain at Seathwaite over all other valleys ;" and he might have said, over all other places in the locality, high as well as low. In order to account for the great and unequal quantities of rain that fall in different parts of this district, it is neces- sary that we should briefly advert to the causes which de- termine the formation of rain at various heights in our atmosphere. The first, is the progressive diminution of temperature from the surface upwards, which is 1° for every 300 feet of height ; and it follows from this, that any mass of the atmosphere saturated with aqueous vapour that is forced to ascend 300 feet will be cooled 1°— 600 feet 2°— 900 feet 3°, and so on progressively to greater heights, and the aqueous vapour that is intermingled with the air will be condensed in proportion to that cooling. If at any time condensation was slight on the low ground near the sea, it would become greater should the air be forced to ascend the valleys, and climb the sloping sides of mountains ; and the greatest amount of condensation of vapour, and conse- quent formation of rain, would be at some certain height determined by the extent to which the air was saturated with vapour. The following figures will shew the heights at which vapour would be condensed under certain circum- stances ; that is to say, with the air and dew-point of the vapour at the surface both at 59^ when the tension of 200 sm. T. HOPKINS ON THE vapour is equal to half an inch of mercury, the wind at the time blowing up a valley and sloping sides of a mountain. The lowest stratum of air being 59", the temperature and dew-point would be reduced at a height of 300 feet to SS" 600 „ 67" 900 „ 56° 1,200 feet to 55° 1,500 „ 54° 1,800 „ 53° 2,100 feet to 52" 2,400 „ 51° 2,700 „ 50° And all the vapour that existed in the air between the dew-points of 59° and 50° would be successively condensed by the time that the air and vapour reached the height of 2,700 feet, and rain, the product of that amount of conden- sation, would be produced at the various heights as the cooling proceeded. There is, however, a second process going on under such circumstances as those just described, which, as it modifies the first, it is necessary to notice. When condensation of vapour takes place heat is liberated, and the temperature of the locality is raised. The gases in the part are then warmed, and they expand and ascend to a greater height, where they are further cooled, and where they condense more vapour. So that the vapour is condensed in the first place by the atmospheric mass being forced up the inclined plane of the land, mechanically — as a wind — and, secondly, by the as- cent produced by the heating power of condensing vapour ; and whilst the mass of air and vapour is carried up from both these causes, it is moving forward horizontal as a wind. In the locality, then, the wind moves mechanically towards the upper part of the valley, whilst, from the heating effects of condensation, it is ascending above that part; the condensing vapour will therefore be liable to be carried above the highest part of the land, and the greatest quantity of rain may fall beyond that part. And, further — after the vapour has been condensed, and the rain formed at a certain con- siderable height in the atmosphere, it has to descend from that height, and will be liable, while so descending, to be FALLS OF RAIN IN CUMBERLAND. 201 carried forward horizontally by the wind, and will reach the earth at a part beyond that over which it was formed. Now, to apply this general statement and reasoning to the case under consideration, let us suppose that air satu- rated with vapour of the temperature of 59" passes from the sea-coast, near Ravenglass, towards the mountains as a south-west wind. When this wind reaches land 300 feet high, it will be cooled by ascent P, and will have all the vapour condensed that is contained between a dew-point of 59** and one of 58°. When it reached land 600 feet high, 2° of vapour would be condensed ; 900 feet 3°, and so on in succession, 1° more for every 300 feet of height up which the air was forced mechanically. Add to this the vertical ascent produced by heat from condensation, and the actual progressive motion of the condensing vapour will be inter- mediate between the two. Supposing the two forces to be equal, the mass would proceed forward, ascending at the angle of 45". Sty-head Pass is 1,290 feet high — the atmo- spheric mass, therefore, when it reached Sty-head would be cooled, say 4°, and it would be liable to be carried over the head of the Pass, rising at an angle of 45°. We might therefore expect, from the known laws of condensation of vapour, and of the action of wind, that, under the circum- stances described, a larger amount of rain would fall beyond Sty-head, than either in the approach to it, or on the top of it ; and accordingly it is the fact that a larger quantity of rain falls in Seathwaite, which is a little beyond the Pass, than in any part between Seathwaite and the sea. In such a locality the saturated air, forced by the rise of the land to ascend, entering the wide mouth of the Valley, which contracts in breadth as it proceeds, rushes through the narrow gorge in the upper part, and over the top of the mountain pass, with great velocity and force ; condensation therefore will take place to a greater extent along this par- ticular and comparatively low line, than where the ridge 2 D 202 MB. T. HOPKINS ON THE of the mountain is higher by 2,000 feet. The higher parts stop the passage of the wind, which makes its way where there is the least resistance, and this is over the pass of the mountain ridge ; and as the horizontal rush of air is here particularly strong, any rain that is there formed, or that has been carried thither, will be liable to be borne forward until the air loses some of its velocity in the comparatively open space beyond the Pass where the rain is likely to be deposited — just as nmning water deposits sand when it reaches a wider and comparatively still part of a river. In the case stated, the air was supposed to be saturated with vapour, but if it should not be fully saturated, but have a dew-point of, say 1° below the temperature, the only difference would be that condensation would not begin until the mass of air climbed 300 feet. When the dew-point is two degrees below the temperature, the air must ascend 600 feet before condensation begins ; and the more the dew-point is below the temperature the higher must the air ascend before condensation will commence; but when it does begin, the process will be of the nature that has been described when the temperature and the dew-point were the same. And it should be recollected that it is not alone the vapour that is near the surface of the earth that may be condensed in the way described, but the whole vertical column may be affected in the same way as that part which is near to the surface. The vapour that is 300 or 600 feet distant from the surface of the low ground, may be equally raised and condensed with that which rests on the surface, seeing that the whole vertical column of the atmosphere may be raised by the obstruction presented by the moun- tain to the passage of the wind. And the rain that is formed in the higher, as well as that in the lower regions, is during its descent always liable to be carried forward by the wind. We see, then, why the largest quantity of rain should fall in Seathwaite when a south-west wind blows from the sea FALLS ON BAIN IN CUMBERLAND. 203 over Sty-head, as Seathwaite is favourably placed to receive much of the rain brought by that wind ; but other winds blow in this district during a large portion of the year, and as much more rain falls at Seathwaite than in any other part, these other winds must, we presume, also bring rain to that place. To see how this is effected, we have to examine the shape of the neighbouring country, and par- ticularly in the directions from which rainy winds come; and we may perhaps obtain a tolerably good idea of what that shape is from an account given in Hudson's Guide to the Lakes. In this work, page 118, it is said, "I know not how to give the reader a distinct image of the main outlines of the country, more readily than by requesting him to place himself with me in imagination upon some given point, let it be the top of either of the mountains, Great Gable or Scaw-fell; or rather, let us suppose our station to be a cloud hanging midway between those two jgnountains, at not more tlian half a mile's distance from the summit of each, and not many yards above their highest elevation, we shall then see strerched at our feet a number of valleys, not fewer than eight, diverging from the point on which we are supposed to stand, like spokes from the nave of a wheel." Now, this imaginary point in the air is nearly over Sty-head Pass. The writer then proceeds to describe Langdale, the Vale of Coniston, the Vale of Duddon, Esk- dale, Wastdale, Ennerdale, and the Vale of Crummock water, and Buttermerc. And he goes on to say, that " such is the general topographical view of the country of the lakes ; and it may be observed that, from the circum- ference to the centre, that is, from the sea or plain country to the mountains specified. Great Gable and Scaw-fell, there is in the several ridges that inclose these vales, and divide them from each other — I mean, in the forms and surfaces- first, of the swelling ground, next, of the hills and rocks, and, lastly, of the mountains — an ascent of almost regular 204 MR. T. HOPKINS ON THE gradation from elegance and richness to their highest point of grandeur and sublimity." Nearly all these eight valleys, in the low flat country, present wide openings to receive any wind that may be blowing towards theni^ — they con- tract towards the centre where the ground rises ; and the wind, whether it blow from, say, the south, south-west, the west, or the north-west, will force its way over the lowest points of the central chain, and be disposed to discharge rain on the country a little beyond those points. Borrodale is just in this situation, and must therefore receive rain from every moist wind that comes from a southern or western quarter, in the way that has been described ; and Seathwaite seems to be in that part of Borrodale which receives the largest quantity of that rain.* The large fall of rain in this village is then to be con- sidered a result of various Tsdny winds blowing up the different valleys, and particularly those which lie to the south and west of it, as those winds force the mixed masses of air and vapour to rise to the lower parts of the elevated ridges that are at the headf of these valleys. At or above these parts the vapour is largely condensed, and the rain that is formed is carried forwards and deposited on the low ground beyond the ridge ; but though deposited there it evidently descends from a great height. That the height at which a rain-gauge is placed does not alone determine the amount of rain that falls into it, up to an elevation of 2,000 feet, is strikingly shown in the tables given by Mr. Millar for the three years ending with 1848. In 1846 October was the wettest month — in 1847 Novem- ber, and in 1 848 February was the wettest ; and the mean quantities of rain received into the gauges at the following places in the wet months, to which I have added the mean of the years, are as follows : — * Since this was written, another part that is contiguous has been found to receive more rain. FALLS OP RAIN IN CUMBEBLAND. 205 Months ... Years ...... In WhitthaTen, tro places, namelf, In Borrodale. Hiffh-stmt, 9« feet nboTc the aea. Round Close, 480 f»'. aboTe the sea. Seathwaite, 240 bet above the sea. Inehes. 7-923 46-466 Inches. 8-077 45-329 Inches. 25-94 144-63 Here we see that whilst Round Close, which is 384 feet higher than High-street, receives only about the same quan- tity of rain that that place doe.s, either in the wet months or in the year — -Seathwaite, which is only 146 feet higher than High-street, receives on an average of both periods more than three times the quantity. And this particular fact is in harmony with the returns generally, showing that elevation alone towards 2,000 feet does not determine the amount of rain that shall fall into a rain-gauge. Speaking in more general language, it may be said that the largest quantities of rain fall from warm and moist atmospheres, as such atmospheres contain the largest quantities of aqueous vapour ; and the rain is formed by the condensation of a part of the vapour, at a height dependent on the elevation that is attained by the atmospheric mass when forced to ascend, and the difference between the temperature and the dew-point in that mass. When the dew-point is near the temperature at the surface, the largest quantity of rain will be formed at a moderate height. When the dew-point is more below the temperature, the largest quantity will be produced at a greater elevation; and when there is a great difference, or, in other words, when the air below is dry, should any rain be formed it will be at a great height, the particular locality in which the largest quantity of rain falls, being always more or less de- termined by the shape of the slopes of the land up which the air ascends. If the rise of the land is great and abrupt, approaching a vertical cliff, the larger part of the rain might possibly fall on the low ground in front of the cliff, the 206 MB. T. HOPKINS ON THE FALLS OF BAIN IN CUMBEBLAND. mass of air being unable to pass over it, until such a height was attained as would leave little uncondensed vapour ex- isting in the air. In sach a situation it is evident, that one gauge placed at a low level in front of the cliff, might receive more rain than another fixed at any height above it. And it is equally clear, that when rain is formed whilst pass- ing over an elevated ridge, that rain might be received either in a gauge placed beyond it, only a little lower, or in one not farther beyond it, but fixed in a deep valley below, as is, in fact, the case with the gauge at Seathwaite. We may therefore conclude, that in a country containing lofty mountains and deep valleys, with much irregularity of surface, the height of the gauge into which rain falls does not indicate the elevation at which it was formed — that elevation being determined by the laws of cooling of the aqueous vapour that is contained in our mixed atmosphere, whilst the vapour is diflfused through the gases. 207 XV. — On Impossible and certain other Surd Equations. By Robert Harley, Esq. Read January 7, 1851. 1. The ordinary method of resolving a given surd equation proceeds on the assumption, that the symbol of radicality which enters into it may sustain indifferently either a posi- tive or negative interpretation. It almost invariably happens, however, that from the nature of the enquiry whence such an equation originates, the sign of the radical is necessarily restricted to a plus signification ; and that, therefore, every value of " the unknown," which, with this limitation, will not satisfy the given equation, is inadmissible as a root. 2. Now it is frequently found, that when the symbol V is thus restricted in its signification, all the roots obtained by the ordinary method of solution, are rejective ; that in fact they are foreign roots, belonging to one or more other equations which, when cleared of radicals, produce the very equation that results from the rationalization of the given one. These foreign roots are introduced by the elimina- tion of the symbol of radicality from the proposed equation ; ^ for that elimination is effected either by multiplying by a factor or factors involving " the unknown," or by an evi- dently equivalent process. 3. That the removal of radicals does not eliminate fi-om the given equation any of its roots, is easily proved. For let / / be any rational functions whatever of ss, connected 1 2 by the equation V7+ V7 = 0, (1); 208 MR. R. HAELEY ON IMPOSSIBLE Then, multiplying by v/ — v/j we get /-/= 0, (2) 12 a rational equation. Now, for whatever value of x (1) ob- tains, (2) must likewise obtain ; otherwise we should have v7- v7= (/-/) -^ (v7+ v7) = (/-/) -T- 0 n «, ; 18 12 is 12 and V7+ V7= 0; 1 1 which is absurd. Hence the proposition in relation to (I) is estabhshed. Similar reasoning will evidently apply to any surd equa- tion whatever. Thus, if we take the equation V7+ V/+ V7=: 0, (3), where // f are rational functions of ar, we shall have 113 (v7+ v7+ v7) (v7+ v7~ v7) (v7- v7+ v7) (- v^+ v7+ v7) = 0,..' ! (4), or,/» +/^ 4-/^ - 2 (//+// -f //) = 0, (5), 133 121323 a rational equation. Now, if there be a value of x which will satisfy (3), and which will not also satisfy (5), or, which is the same thing (4), one at least of the factors -v7+ ^7 — V7 V7— V7 + V/ — ^7+ ^4- v7must be infiLite. 3 1 3 3 I t 3 Suppose V7+ ^7— v7= °o ; 12 3 then, subtracting this equation from (3), we get 2\7= — <» » ■*•/= 00 ; .-. a; = 00 ; 3 3 which is absurd. Cor. 1. Every value of ^ which will satisfy (2), will also satisfy either (1) or its congener. For, v/-/=0; 1 2 •••(^7+v7)(^7~v7) = 0; AND OTHER SURD EQUATIOXS. 209 hence either '7f-\' v^^^ 0, 1 3 or, J^— -v^f = 0, for all roots of (2). 1 s Cor. 2. In like manner it may be shown, that every value oi a which will satisfy the equation will also satisfy either one or other of the equations v7+v7+v7=o, 1 3 3 v7+ v7— s/f— 0, 1 2 3 v7— v7-}- V7=: 0, 1 % 3 — 'v7+^7+^7— 0. 1 3 3 Cor. 3. If, when a surd equation is rationalized, and all the roots of the resulting equation are obtained, none of these roots are found to satisfy the proposed equation, that equation has no root whatever. For if such an equation had any root, that root would necessarily satisfy (and there- fore be also a root of) the rational equation. 4. Definition. — An equation which has no root what- ever, is designated as impossible. This definition is pro- posed by Mr. Cockle in the Mechanics' Magazine, Vol. xlix. (p. 365), where it is clearly demonstrated that the very supposition of the existence of such equations involves an arithmetical contradiction. It is needless, therefore, to argue the propriety of the definition, which we adopt with- out any hesitation. 5. For the purpose of clearly illustrating the preceding principles, let us consider the particular equation 4 -f Va; — 3 -{- Va; -f 21 iz 0. Multiplying hj (4 -{- a/x — 3 — V^; + 21) (1 + Va^ — 3> in order to eliminate radicals, we get 8 (x — 4) =: 0 ; .-. a; — 4. 2 E 210 MR. R. HARLEY ON IMPOSSIBLE Now on trial we find that this value will not satisfy the proposed equation, unless the sign (+) prefixed to the radicals be taken to mean the algebraic addition of a root of the quantities Qc — 3) and {x -{- 21) ; that is (to be more explicit), unless we take the plus root of (x — 3) and the mintis root of (x -f- 21). This hypothesis, however, I regard as inadmissible, believing it to be at variance with that definite and distinct signification of symbols, which is so absolutely essential to the exact expression of the conditions of a problem. It virtually transforms the sign prefixed to the radical into a mere connecting link, exercising no real control over that radical. It is true, that, when the symbol V is introduced in the course of an algebraic investigation, it is always capable of sustaining both a plus and minus interpretation ; so that in every such case the double sign -|- is understood as involved ia iL But this does not seem by any means to justify an am' biguouSf or rather I should say a variable, interpretation of that symbol when it is necessarily employed with a par- ticular sign of operation before it, that particular sign being indispensable to the perfect symbolic representation of the given conditions. In the one case, the symbol having been introduced for the purpose of efPecling a transforma- tion, may be taken both positively and negatively without violating any stipulated conditions : in the other case, it is given with a specific sign of operation prefixed to it, and this sign cannot be altered or evaded consistently with the conditions of the problem. For these reasons, therefore, we cannot accept 4 as a root of the proposed equation ; but this is the only value of x which will satisfy the rational equation : hence (art, 3, cor. 3) 4 -|- Va; — 3 -f Vx 4- 21 =: 0 has no root whatever, in other words (art. 4), it is Art impossible equation. The foreign root 4 was introduced by AJfl) OTIIEK SURD EQUATIONS. 211 multiplying the equation by 4 -f- Va^ — 8 — V^; -j" 21> for it satisfies the equation 4 -4- -v/a* — 3 — VicH-2T zz 0. For the sake of further illustration, let the particular example 3a: + V30a; — 71 =: 5 be proposed. Eliminating the radical sign by the usual method, we get 9x2 __ gox + 96 — 0, which resolved, gives i» rr 4 or f . Now, neither of these values, when substituted for x, are found to satisfy the pro- posed equation ; the equation really satisfied by them is, 3a; — VSOir — 71 =r 5. We therefore conclude that 3a; -f v/30a; — 71 = 5 is an impossible equation. In Wood's Algebra by Lund (thirteenth edition), page 128, the equation we are now considering is discussed. After noticing the inadmissibility of the two values of x, above found as roots of the equation, the able Editor remarks, '' whether there be any values of ^ or not, which will satisfy the equation 3 .r + V^Oo; — 7 1 n: 5, we can- not say ; all that we know is, that the common method of solution will n,ot produce them." From what has been above shown, it is manifest that the doubt which is here expressed as to the possUnlity of the proposed equation, is altogether without foundation. It assumes, in fact, that the same value which satisfies the irrational equation does not necessarily satisfy also the rational one; but this assumption we know to be false. (Art. 3.) 6. In the algebraic solution of a certain class of problems, it is often of considerable importance to know a priori, whether the irrational equations which express the given conditions be possible or impossible, and (if possible) to 212 MR. R. HARLEY ON IMPOSSIBLE ascertain the exact number of roots belonging to each. A very little consideration will show, that for any surd equa- tion of a given form to be impossible, a certain determinate relation must obtain among the co-efficients of w; and to discover that relation becomes at once an interesting and important enquiry. To find also a method of solution, equally applicable to all irrational equations, by which the true roots (if any exist) may be exclusively evolved, is plainly a very desirable object. These, then, are the two main purposes of the present paper; how far they are accomplished I shall not pretend to say. 7. I shall not now attempt to give a general discussion of this subject, but shall confine attention to certain surd equations of a limited degree. To illustrate the method I propose for the attainment of the objects specified in the last article, let us consider the literal equations ax -\- \/bx -\- c ■=! d, (a), ax — i^bx -\- c zz d, (^). These are readily put under the more simple and conve- nient forms X -\~ »/2a X 4- b z=: 0, (a), 1 111 1 X — ^2ax-\-b ZZ 0 (/S), 1111 I where x zz x — --, a ZZ -^, and b zz —(ac-\- bd). _, a Ii: — , and b zz - 1 a 1 2a^ 1 a Equation (jS) may be written thus : — X -f V2^(— Ifx-^b (— 1)^ zz 0 ; 11 11 or, substituting n for — 1, X -\- »^2a ri? X -f- b n^ ZZL^\ 1 111 which agrees in form with (a). It hence appears that if X zzf{a, b) be the solution of (a), the solution of (/3) will 111 1 . 1 be X zzf{a «^ b w^). 1 1 I AND OTHER SURD EQUATIONS. 213 Let (a) be multiplied by a: -4- n V2a x -\- b; then, bear- 1 *^ 1 111 ing in mind that 1 -|- n z= 0, we shall have x^ -\- 2a nx -\- b 71 ziz 0 } 1 111 .'. x^ -j- 2anx in 6 n^ ; 11 1 I or, X z:z a n^ -\- n a/o' n^ 4- b, (1). II II The second and third steps of the above solution may need, perhaps, a little explanation. In transposing the quantity 6 n to the right-hand side of the equation, it will be observed, that instead of aflfecting it (according to the usual method) with the minus sign, we have multiplied it by n. That these operations are equivalent, is too evident to need demonstration; and it is easy to see also, that the introduction of the symbol n for the negative sign is indis- pensable, to prevent the ambiguity that would otherwise result from the obliteration of that sign by involution. But it may be asked, would not the same ends be answered equally well, were we (instead of multiplying) to divide by n? In replying to this question, it is important to observe, that in order to enable us to retrace the several steps of the solution with unerring certainty, the symbol n must always be employed in conformity with some invariable principle of operation ; so that, by adopting an inverse principle, we may return with confident correctness, firom any part of the in- vestigation, through the successive steps, to the original equation. Unless the operation be thus conducted, it is obvious that ambiguity and error will attach to our results. In fact, we assume as the great principle that should guide us in the solution of surd equations, that every successive trans- formation should be made to bear with it an unmistakeable index of its immediate origin ; for it is only by this means, we conceive, that those rejective roots (Art. 1) which enter into the ordinary solution may be excluded. Now, if we ^14 MR. B. HARLEY ON IMPOSSIBLE recur to the first introduction of the symbol n into the equation (a), we find that it was employed as a multiplier of a negative quantity ( — V 2aa; -f- b), in order that that quantity might be made to assume a positive form. Hence, therefore, in the foregoing transposition of b n, we must multiply (and not divide) it by n. So, in like manner, in extracting the root of the quadratic, half the co-efficient of JB is muUipUed by n, and the square of the result is then I added to the absolute term. Had we eliminated V from (a), by multiplying by a) -] — V 2ax -f- b, instead of ;c -j- ** V* 2a^ -}■ b, we should then have had to divide every trans^ 111 posed quantity by n, or (which is the same thing) to mul^ tiply it by n-^ : the value of x so evolved would be found to differ from that above given only in form. In fact, since «p in — 1, JO being any odd number, positive or negative, yffe might (if we chose) employ n" for n, as a midtiplier, throughout the investigation. Thus conducted, the opera- tion would be as below : — Multiplying (a) bv ^ + n^ *j2,cuc -4- ^> and bearing in 1 " 1 11 1 mind that 1 + ^^ ^= 0, we have x" -{- nP (2a X -\- b) z=i Q> ; \ 11 1 .', a? '\- ta nP X :zi b n^ ; .-. a; =: o n^ 4- V«* «*» -4- b »'p 11 1 1 11 II It is easily seen that, since p is odd^ (1) and (P) are vir- tually identical. We now proceed to verify these solutions — First, by (1,) V2aa;-j-6— -v/{2oV-|-2a» VoV + i+i} =: V {(aW -}-i) -f2ff« V«V+6-f aV) •=L »^a^n*-\-b -fan (2) 1 1 1 AND OTHER SUED EQUATIONS. 215 (1) ^ (2) gives, X -f- ^/2ax + 6 zi: ( 1 + «) (aw -fVaV-j-i); or, since 1 -j-w =z 0, a; -|- *^2ax-^b nz 0, which verifies (1). Secondly, by (IV) >/2ax-{^b=: V {2aVP-f 2a»'' V"'""^ -f-6+4} =: V {(«VP-fJ)+2awVa'w=*P+*4-a%'*} 1 f 1 (11) ^ (21) gives, X'\-\^2ax 4- J— (l+w^) (a/i'-j-V^'w^+S); or since 1 -^ n^ zz 6, x -{- \/2ax -4- 6 zr 0, which verifies (V). 1 111 The preceding solutions may be exhibited thus : — xzzn' (a + V«' + bn'') ... (3), tiM X zz n^ (a -{- V«' + *»-'") ... (3')> 1 111 (3) and (3') corresponding respectively to (1) and (1'). From (3,) */2ax 4-b — « (a + V«M^^~* ) ... (4) 11 1 1 1 1 „ (3S)V2ffa; ^ b zz nP (a ^ a/o,^ -\- brr'") ... (4') 111 1 1 1 (3) 4- (4), and (3') -f (4'), each give X -{- ^2ax -4- b zz 0, i 11 I Its it ought to be* If now we write — 1 for n, in (1) or (V), and bear in mind that nF zz -— Ij and n* zz 1, we get X zza — f/a^ -\- b (5) 11 11 But if we make the same substitutions in (3) or (3'), we S^* ______ xzza-^ Vo* + * (6). 1 11 I (5) and (6) are in fact the values of Xy which we should have found if we had solved (a) or (0) by the ordihary method. Now we know from principles established in a previous portion of this paper (Art. 3), that no values^ of a, other than (5) and (6) can satisfy (a) or (/3.) And, since 216 MR R. HARLEY ON IMPOSSIBLE (3) or (3^) is identical (except in form) with (1) or (P), it immediately follows that (1) or {V) embraces all possible solutions of (a) ; and that, therefore, no loss of generality has been sustained by the exclusion of the negative sign from before the introduced radical. Indeed, this is as might have been expected from the logical and consistent charac- ter of the operation: it accords with that exact compre- hensiveness of result, which must ever attach to those mathematical investigations which are conducted with due regard to the entire data of the problem discussed. When the foregoing verifications involve the violation of our symbolical conventions, it is clear that the roots indi- cated by the formula (1) or (1^) and (3) or (3^) are rejec- tive. But this can only be the case when the right-hand members of the equations (2) or (2^) and (4) or (4^) are negative (Art. 1). Now, still bearing in mind that the symbol */ is to be interpreted positively, we readily dis- cover, by mere inspection, that the right-hand member of (2) or (2^) is negative only when b is so; but that the right-hand member of (3) or (3^) is always negative. We hence conclude that (6) can never strictly satisfy (a), and that (5) does so only when h is positive. Combining these conclusions with what has been befjre demonstrated in this article, it is immediately seen that (6) is always a root of (/3,) and that (5) is so only when b is negative. These important conclusions may be deduced from other and more simple considerations. Thus, resolving (a) or (/3) by the common process, we obtain the relations marked (5) and (6.) Now from (5) we get 4^2ax -4- i rr « c/) V^^ -f- b (7) 5 111 1 1 1 And from (6) we get ^/2ax-\'b r: a + Va" + * (8) 1111 1 I A1SID OTHER SUUD EQUATIONS. 217 It is scarcely necessary to say, that in finding these values or < a. Hence, (o) + 0) 11 1 gives X -\- t^2ax -{- b z=z 0 OT 2a^ according as b is positive or 1 1 1 1 1 1 negative. In like manner, (/)) — (7) gives X — j^2ax •4-b zz2a or 0, according as 6 is positive or 1 1 1 1 I 1 ' negative. Again, (6) + (8)j and (6) — (8), give respectively, X 4- V^ooj 4- 6 rz 2a ; I 1 1 1 I and X — \/2ax 4- ^ in 0. 1 111 From these results it appears, that (5) is the solution of either (a) or (/3,) according as b is positive or negative ; I 1 I and that (6) is always a solution of (/S). Hence, also, by art. 3, cor. 3, and art. 4, when b is negative, (a) is impos- 1 1 sihUf and (j8) has two roots, viz., (5) and (6). We have already shown that (1), (P), (3), or (3^) is a rigid symbolical solution of the equation (a); and yet we now find that neither (5) nor (6), which were both imme- diately obtained from that solution — by merely substituting for the symbol n its arithmetical value — is necessai-ily a solu- tion of that equation. To the experienced analyst, this seeming incongruity will be no matter of surprise. In Professor Young's " General Principles of Analysis," Part I., art. 8, a somewhat analogous case is elegantly discussed. The principle on which such seeming discrepancies as the one above alluded to may be satisfactorily explained, is there developed with that clearness of illustration and logi- cal precision, for which that profound mathematician is so 2 F 218 MR. R. HARLEY ON IMrOSSIBLE deservedly celebrated. The general conclusion to which the Professor's reasoning tends, is thus elegantly expressed in the closing sentence of the article referred to : — " If, by any management or contrivance, we force, in a particular case, a violation of a general law, I need scarcely say that our result will be inadmissible." This remark is peculiarly applicable to the case now under consideration. For, by substituting for n its numerical value in (1), the law of generation is lost sight of, and consequently, (5) and (6) being severally substituted in the irrational equation (a), we violate, in a certain case already specified, the general law controlling the square root of expressions affected by the symbol n^. To illustrate this clearly, let us consider the simple surd equation 1 + V^=r0 (a), Transposing, ^/xzz — Izzn: .'.azzn^ (a'). Now (a}) evidently satisfies (a) ; for 1 -|- V^^ zz I -\-n zz 0 ; and yet, though n^ zz 1, /v zz 1 is not the root of (a), but of its congener, I — V^=0 (h). The reason is, that Vl and A,/n^ are not equal, the latter being n times the former.* So, in like manner, by substi- tuting from (3) and (6), we get respectively (4) and (8) ; * Possibly it may be objected that Vl is eitJier + 1 or — 1, and that, therefore, unity is the root of both (a) and (6). In answer to this, it might seem sufficient simply to refer to Art. 5, in which it is shown, we think, that such a conclusion is not consistent with rigorous reasoning ; that it involves, in fact, a virtual violation of the law of signs. As, how- ever, the entire theory of impossible equations depends for its existence on the non-identity of such equations as (a) and (b), we may further re- mark, that if these be treated as simultaneous equations, (a) -f- (b) will give 2 = 0, an arithmetical absurdity. Whether or not n* is philosophi- cally admissible as a root of (a), will be hereafter considered. AND OTHER SURD EQUATIONS. 219 but (6) was immediately derived from (3) by writing for n its numerical value, and yet if in (4) for n we substitute its value, and obliterate all even powers of unity, we get which does not agree with (8.) The fact is, the expression following the negative sign in the right-hand member of this equation being necessarily positive, the equation itself is a clear violation of our symbolical conventions, and is therefore inadmissible. This sufficiently explains the reason why certain expressions in terms of «, seem to satisfy (and algebraically do strictly satisfy) certain irrational equa- tions, which are nevertheless impossible. Thus, the equation (a) is easily shown to be impossible (see Art. 3, cor. 3), and yet it is strictly satisfied by a: rz n^. Further discussion of this part of the subject I leave until I come to speak of impossible eapressiont, with which, as will be seen, it is' closely and intimately connected. It has been demonstrated that the root of (/Sj) in terms of n, may be at once deduced from that of (aj) by writing flj n'^, bi Ji^, for tti hi respectively. We thus get rr :rZ f »?* -f- W* tJnSiS -j- />, , , , , , , .. (9\ 1 1 ' ' I ' r and.*. V2«^ -J- 6 rr a «^ -4- Jc^n^ -4- b. Ill 1 1 ' 1 .. (10), .-. X — V2a a: -f 6 — (re* — 1) (a 4- ^ahi"- -f i) z 1 111 1 1 ' 1 = 0; which verifies the solution (9). The invariable possi- bility of equation (/3i), and all the other conclusions which have been established with regard to that equation and its congener (a^), are immediately deducible from the above solution. Of course (9), like equation (1), from which it is derived, may be exhibited in a variety of forms ; these, however, it is needless to develop. Adapting the foregoing results to the original equations, we b *»ve the following conclusions : — 220 MB. K. HABLEY ON IMPOSSIBLE The equation (a) is possible or impossible according as —3 (jac -|- bd) is positive or negative, and (/3) is always possible. When —3 (ac -j- bd) is positive, (a) has one root, viz., a: z= ^ (6 4- 4ai — >/& -\- 4.abd + 4a'c), («')' and (;S) has one root, viz., x=~.{b -f 4a and, by (j8), x zz V^a x -{- b, 11 11 1 Xj in the first instance must be negative, and in the second positive, otherwise we should have to subject it to incom- patible conditions. 8. Tlie method of solution explained and illustrated in the last article, is evidently capable of application to any class of surd equations whatever, provided only that such equations, when rationalized by the common method, are capable of algebraical resolution. Whether we employ n or n" (p being any odd integral number, positive or negative,) for -1, we have seen that the results are virtually identical, and that nothing is gained as to generality by the use of one symbol rather than the other ; while the employment of n has the advantage over that of n" in point of simplicity and convenience. In practice, therefore, it will be found better ta employ n ex- clusively' 222 MR. R. HARLEV ON IMPOSSIBLE From what has been already done, it will have appeared evident that the chief value of w, as an element of operation, is this — that it enables us to discover certain expressions for Xy which, in every circumstance, seemingly satisfy the proposed equation. The consideration of these expressions will readily enable us further to determine the relation that must subsist among the several co-efficients of ar, in the given equation, in order to that equation being possible. This latter object, however, may be more easily effected, as we have seen, by simpler means. I have spoken of the roots of x, in terms of n, as satis- fying the proposed irrational equation only in appearance. I proceed to explain my meaning. In the last article it was shown, that when b is negative, the equation X 4- V2a x-\-h =z 0 1 11 I is impossible ; that is, it has no root whatever. And yet we have also shown, that the expression a n^ -\-n s/c^ n^ -f- 6, 111 being substituted for x, seems to satisfy the equation in 1 every circumstance. How are these conclusions to be har- fhonized ? If we recur to the verification of the solution (1), we shall be furnished with a satisfactory explanation of this difficulty. We there find that the substitution of the above root in the expression »^2a a; -}- 6, gives V2a X -\- b ^H a n •\' Va* W + b. Ill 1 1 I Now it has been before remarked, that when b is negative, the right-hand member of this equation is negative also ; but the left-hand member is always positive; hence, when b is negative, the above expresses an impossible relation; viz., the equality of two quantities, one of which is positive, and the other negative. The root, therefore, which.p sents AND OTHER SURD EQUATIONS. 223 this incongruous result, is not strictly receivable. In like manner the root marked (3), viz.. I 111 though it appears to satisfy the proposed equation, is re- jective, because it involves the acceptance of the impossible equality, Jlax 4- h rz w (a + V«* + hn-*\ 111 1 1 1 an equality which can have no more existence than the relation 1 zr — 1. On the same grounds we reject n* as being strictly a root of ] 4- s/x — 0. There can be no doubt, I think, that algebraically this value satisfies the equation, but arithmetically it does not ; and to accept it, seems to me to be nothing less than an evasion of the authority of the sign (-{-) prefixed to the radical. An eminent analyst, to whose researches we have already had occasion to refer, in an interesting paper * published in the Philosophical Magazine for October 1850, seems to take a very similar view of this subject to that which we have just been expounding. After giving an elegant dis- cussion of the equation 1 4" Va; — 4 — Vic — - 1 n 0, Mr. Cockle remarks, *' if, in the above instances, the diffi- culty is to be evaded, it is only by greatly refining our solution, and, as it has occurred to me, by using expressions of the form m (-}- \y -\-n ( — 1)', and by following certain rules respecting our reductions, and the signs to be affixed to the radicals. To those who would attempt such a com- plex and artificial system of solution, rather than admit the * " On impossible equations, on impossible quantities, and on tessarines. By James Cockle, Esq., M. A. of Trinity College, Cambridge ; Barrister- at-law of the Middle Temple." Phil. Mag., third series, vol 37, pp. 281-3. 224 MR. R. HARIiEY ON IMPOSSIBLE existence of an impossible equation, I may licreafter address some observations. They will, however, probably find, as I have done, that their attempts are unsatisfactory, and their results not philosophically admissible. But I shall here Content myself with remarking, that by any system of rules, however artificial, the difficulty is only thrown further back. Thus, the equation V^+ Va;4- 1 =: 0 is utterly intractable." After what I have already written, it is scarcely neces- sary to say, that in the opinion so elegantly expressed in the passage cited I entirely concur, and that I cannot but consider the existence of impossible equations an undoubted fact. With regard, however, to the equation proposed by Mr. Cockle, 1 may remark that, peculiar as it is, the method of solution explained in this paper, is applicable even to it. For, since »Jx -f V^+l ~ 0 (1) .'. hjx 1Z n \^x -j- 1 ; .'. X n: n^x -f- 7i^ ; •••" = 1-^:::^^ (2); substituting (2) in (1), we have I 1 Vl 4" ^ ^o 0, Vl— «^ Vl — ?? 1 — n 2 which verifies the solution (2). Still, it will be remarked, that the condition */x zn which ha^been admitted into the verification, is incompa- tible with the restriction imposed on the symbol of radicality ; and that, therefore, if the views which we have taken of the office of the signs {-\- and — ) prefixed to -\/ be correct, the numerical value of (2), viz., ^, cannot be accepted as a AND OTHER SURD EQUATIONa 225 root of (1). This numerical value, it may be interesting to observe, is otherwise obtainable, thus : — Multiplying both members of (1) by V ^ — V .J? -f* !> we have X — X — 1 — 0 ; .*. 0 ;p rz 1 ; .'.x-zz^ as before. This is evidently the root, however, of the equa- tion V* — Va: + 1 =: 0. 9. In a series of original essays, entitled, " Horas Alge- braicae," published in the Mechanics^ Magazine, Mr. Cockle has given a very interesting and general discussion of the theory of surd equations. Ho7'cb VIII., IX., and X., contain valuable disquisitions on the algebra of impossibles; tlie history of which is given in the last-mentioned Horce, To show that there is good ground for supposing that the existence of impossible equations was known, or at least suspected, by certain ancient philosophers, Mr. Cockle cites two very curious arid interesting solutions from the Vija-ganita. My own remarks on those solutions I re- serve until I have given Mr. Cockle's discussion, which is so interesting and instructive in all its parts, that I feel sure no apology will be deemed necessary for introducing it here entire. " I am inclined to think,"* says Mr. Cockle, " but I oflPer the opinion with great hesitation, that the existence of im- possible equations has been known for many ages — or, if known should seem too strong a word, I will state some circumstances which tend to indicate that the existence of such equations was at least suspected by those philosophers —whether Caucasian or Mongolian, Indo- German or Indian, or other, is not now the question — by those philo- sophers whose labours are preserved to us in the lAlavatif * See Mechanics' Magazine, Volume xlix., pp. 655 — 7. Some of tiie foot-notes we omit as comparatively imimportant ; and the rest, for obTi- 0U8 reasons, are transferred to the text, and bracketed. 2g 226 MB. R. HAELEY ON IMFOSSIBLE the Vijcu-ganitaj and the remaining records of ancient Oriental science. Of these circumstances, one is afforded by the first example of a quadratic equation, which occurs in the Vija-ganita. That example is as follows : — (see pp. 211-12 of Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara. Translated by Henry Thomas Colebrooke, Esq., F.R.S., &c. &c. &c. London: Murray, 1817. * * *) " * The square root of half the number of a swarm of bees is gone to a shrub of jasmin, and so are eight-ninths of the whole swarm : a female is buzzing to one remaining male that is humming within a lotus, in which he is con- fined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees ? ' ** To solve this problem, we are directed to * put the number of the swarm of bees ya, v, 2.' In this expression, ya is an abbreviation of yavat-tavat, of which the literal sig- nification is, ' so much as,' and of which the meaning is an unknown quantity ; v indicating that ya is to be squared, and 2, that twice the square is to be taken. In modern notation, this assumption would be represented by 2^' ; but, the qiccesitum of the problem being the number of bees in the swarm, why was not ya 1 (or a) taken to repre- sent that number ? It was not to avoid fractions, because, with that assumption, fractions occur in the statement of the question, and, moreover, such a purpose would only account for the occurrence of the co-efficient 2. As little was that assumption a capricious or accidental one, as we may see from the next example in the Vija-ganita. (Cole- brooke, p. 212, Art. 133; and pp. 30 — 1,) which is: — ** * The son of Pritha, exasperated in combat, shot a quiver of arrows to slay Carna. With half his arrows he parried those of his antagonist ; with four times the square root of the quiver full, he killed his horse ; with six arrows he slew Salya ; with three he demolished the umbrella. AND OTHER SURD EQUATIONS. 227 Standard, and bow ; and with one he cut off the head of the foe. How many were the arrows that Arjuna let fly ? * " The instructions which follow (p. 212) are : — * In this case, put the number of the whole of the arrows ya, v, 1.' In other words, assume that number to be x^. But why not X ? In this instance there appears to be but one answer for the Oriental investigator to give, viz., we must avoid surds. And this answer explains the peculiar form of the assumption in the first example. Had x^ been assumed as the total number of bees, we should have had the surd V2 introduced into the expression of the problem. But, why object to the introduction of surds ? It was not that the Orientals did not recognise surds ; on the contrary, their knowledge of their properties was extensive and accurate (pp. 145-155). It was not that they had not a convenient notation, for such a surd number as V2 would be denoted by ca, 2, or (adopting Mr. Colebrooke's variation) by c 2 ; and, if we say with Narayana (page 145, note 1), that * a quantity, the root of which is to be taken, is named Carani,'' I can- not see why ca or c should not have been applied to ya — thus, c, ya. This quantity would have corresponded to our V^. The solution of the problem would then have been effected by our supposing the sum of ya, ^, and c, ya, 4 and rw, 10 to be equal to ya, 1 ; and we should thence arrive at [c, ya, 1 ru, 4] [c, ya, 0 ru, 6], whence we obtain 100 as the value of ya ; and the further advantage that yavat-tavat is the very qucBsitum of the pro- blem, the number of arrows. But, even if we suppose the word carani to be exclusively applied to number, those who achieved in notation the results which we see in the Vija- ganita would not have had much difficulty in expressing the square root of ya. Is it improbable, then, that the 228 MB. E. HAELEY ON IMFOSSIBUB avoidance of surds, which, as we have seen, takes place in the above examples, became a rule of proceeding in conse- quence of the contradictory results to which surd equations sometimes lead us ? Bhascara was aware of the double sign which attaches to a square root (p. 135), and has used that double sign to obtain two positive roots of a quadratic (p. 216), and I believe that he also admitted, in all cases, two roots of a quadratic ; for we see him (p. 135), squaring a negative quantity, considered by itself, without reference to other quantities; and further, when wo see him rejecting the root 5 because it is 'incongruous ' (p. 217), he qualifies the rejection by, as I presume, assigning as a ground, that 'people do not approve negative absolute numbers,' and negative quantities are by means of this root 5, introduced into the conditions of the question. Now suppose for a moment, that, in the first attempts at the solution of the two problems given above, the qucesitum, had been taken as ya, then the algebraist would have had in the first example, 72 and |, and in the second 100 and 4, as the values of ya. It would have been seen (for, as in other cases, both values would certainly have been tried), that the second value would in neither case satisfy the required conditions. I think it also highly probable, that the reason of the failure would have been seen, as the double value of the radical in the enunciation would natu- rally offer itself as a mode of explanation. The question then is, whether ya was originally taken as the qucesitum in those examples; and I confess that I cannot help seeing, in the introduction of square roots into the enunciation of the first two examples of quadratic equations, and in the assumption that ya is something other than the qucesitxmij a marked desire to overcome a natural tendency to make such assumption — a tendency which the writer had found to lead to error. Add to this, that if the writer had con- sidered the error in question as a mere * incongruity,' he AJtD OTHER SURD EQUATIONS, 22f would probably have noticed it as such, as he has done other 'incongruities' further on (pp. 217 — 18); but perhaps he felt that he could not call that an * incongruity,' which was in fact a solution of the congeneric surd equation, cor- responding to the given one. Lastly, let me observe, that we can hardly suppose that enquiry was directed to the two examples given above, and only to them. And the dis- covery of one surd equation without any root, would ac- count for the studious avoidance of surd formulae, which we see in the above portion of the Vija-ganita — a magnificent work, which would be more generally studied, did the his- tory of science hold the position which it deserves in the estimation of the learned." In the passage above cited, Mr. Cockle has, I think, shown good reason for suspecting that the ancient Oriental algebraists were not altogether ignorant of the existence of impossible equations ^ or at least of the fact, that in the solution of surd equations foreign roots are frequently evolved. The system of solution employed is admirably adapted for the exclusive determination of the possible root; and it is difficult to conceive why that system was so uniformly adopted, if it were not to avoid the contradictory results above referred to. A very slight extension of the ancient system will enable us to compnehend within it every irrational equation; for by making suitable assump- tions, as we shall see, every such equation may be resolved into as many simultaneous rational ones as there are radicals and rational terms of a; in the proposed ; and by rejecting all negative roots, the true and only value, or values, of the unknown will be determined by the common process. This system of solution, as we shall also see, has the subsidiary advantage of preventing the necessity for adopting the method of experimental verification, in order to ascertain to what equations the foreign roots introduced by the operation for eliminating radicals severally belong. 230 ME. R. hahley on impossible Let us recur to the examples cited by Mr. Cockle from the Vija-ganita. And first, as it regards the bee question, if, as directed, we denote the number required by 2^% the conditions of the question will be expressed by the equa- tion X+-Q ^ + 2 = 231^^ which gives x zr 6, — the negative root being rejected ; and consequently 2^7* 1:1: 72, the true and only answer. But if x (instead of 2a!^) be taken as the qiccesitum of the question, the equation will be ViaJ-j- I a? + 2 = x; or, 2a; — 9 V2x rr 36 (A); whence, by the usual process, we find x zz 72 or |, the latter root being foreign, and belonging to the congener of (A), viz., _ 2a; + 9 ^2x = 86 (B), and the former root being alone the true answer. Again; to solve the second example from the Vija- ganitay let us assume, as instructed, a' for the whole number of arrows; then we shall have I- s^ -\- 4x -}- 10 =z x' ', whence « zz 10, — the negative root being rejected as before ; and /. «* zz 100, the number required. But taking x (instead of a;^) for the number sought, we shall have |a;+ 4 Vx_-\- 10 — z; .\x — 8 V« = 10 (C); whence, resolving as usual, we get x zz 100, or 4. The former root satisfies (C), and is therefore the answer to the question; the latter belongs to the congener of (C), viz., x-{-S \^x=z 10 (D). In passing, X may just observe that (A, B), (C, D), being put under the forms — (2x — 36) + 9 V2 a; — 36) -f- 36 n 0, ( a; — 10) + 8 V( a; — 10) -f 10 = 0, AND OTHER SURD EQUATIONS. 231 we immediately infer, from the fact of the last quantity imder each of the radicals being positive, that the equations (A), (B), (C), (D), have each one, and only owe, root. (See the latter portion of art. 7.) Now, in each of the foregoing cases, the advantage of the ancient over the modem method, if we may so distin- guish them, is too obvious to require argument. That method, as I have before intimated, is easily adapted so as to comprise within it every class of surd equations, and will always be found of important service when the sign of the radical is to be taken strictly as indicated. The considera^ tion of a few particular examples, will sufficiently show the general applicability of the ancient system of solution to irrational equations, and will tend to illustrate more clearly than any number of general observations, the peculiar value of that system. K the equation 4 + Va: -^ 3 + Va: + 21 = 0 (see art 5), be proposed for solution, the ancient method at once suggests the following assumptions : — put a? rz V ^ — 3, and £c zz 's/ a: -{- 21, I s then we shall have 4 -|- as 4- iK =i: 0, 1 a 24-4-a:* — a^ — 0; 1 s whence we readily find x m 1^ and a: :=z — 5. Now the 1 3 latter of these values, being negative, is rejective ; and, since a has only one root, we immediately infer that the equation is impossible. We likewise learn that the only possible equation analogous to the proposed one, is 4: -\' j^x — 3 — a/x -\- 21 zz 0, the root of which is [a; = ^ + 3 = (aP) — 21 =] 5. The 232 MR. R. HAELEY ON IMPOSSIBLE remaining corresponding equations, viz., 4 _ ^x — 3 + Vic + 21 :=z 0, and 4 — a/x — 3 — V^^ 4- 21 — 0, being also impossible. Again, let the equation Va;* + 9 + V25— x^ — 2x=z0 be proposed. Assume Va:" -^9 =i o!, and V25 — a?* zz a;; 1 s then a: -4- a; — 2x m 0, 1 2 a^ 4. ar" — 84 rz 0, 1 2 ar^ — x + 9—0. Eliminating j? between the first and third of these equa- tionS) we get (x + ar)* — 4 (.r^ — 9) zr 0 ; 1 2 1 whence, by means of the second equation, we obtain a: 3: 5, — 5, -^r, or :^ ; and K =: 3, — 3, — - ii, or 14. Now, since no corresponding pair of these roots, except the first, is positive, there is only one possible value of a? capable of satisfying the proposed equation, viz., (cc ~i~ ^ \ A !=::] 4. The information afforded by the signs of the other roots is, that the equation Va^ + 9 — ^25 — 3^-{'2x=zO has one root, and one only, viz., 2_ and that the equation V5, ^x» 4- 9 + ^25 — a^ — 2a: = 0 is impossible. To recur to the equation Va 4- i^x-{-l = 0, AND OIHER SURD EQUATIONS. 233 (Art. 8.) If we make s/a: n .r, and s/x + \ zzi x^ we shall have X ■\- X n: 0, x^—l? —— 1 ; 1 2 whence we get x zz. — » and x'zz — : the only value of I 2.0 2 2,0 X being rejective, the equation is impossible; x "=1 x^ (or x^ — 1) ^^ — > is the solution of the equation K^x -j- 1 — V^ ^^ 0. Examples of this kind might be multiplied indefinitely ; these, however, are sufficient to show, that the method of the ancient algebraists (adapted) is admirably suited for the ready exclusion of foreign roots from the solution, and likewise for enabling us at once to determine, without trial, to what equations those rejective roots severally belong. 10. Garnier seems to have been the first mathematician who distinctly affirmed the existence of impossible equa- tions. In his Ajialyse AlgSrique, p. 335, art. 92, he says, in speaking of the equation — \/x — 1 rz 1 — \/x — 4, it " cannot be satisfied when the radicals are taken with the sign plus;"* and in, the same place he further remarks, that " the operations by means of which the radicals are made to disappear, introduce roots foreign to the proposed" equation.* Subsequently the subject received some attention from the late Mr. Homer of Bath, a gentleman whose valuable * The above translations are taken from Mr. Cockle's ITorm, X., before referred to, and are followed by the subjoined remark : — " So that Gaknier must be understood as having distinctly asserted the existence of surd equations vi^ithout roots, and also that the appearance of roots which such equations present, are the roots introduced by the processes through which we seek to rationalize the equations." — Mechanics' Magazine, vol. xlix. p. 557. 2h 234 MR. R. HAftLEY ON IMPOSSIBLE contributions to Algebra will ever occupy a conspicuous position in the history of that science. An interesting and instructive letter on the subject of surd equations, from Mr. Homer to Professor T. S. Davies, was published in vol. viii. s. iii. (pp. 43 — 50) of the Philosophical Magazine. In that letter, the mode in which foreign roots are introduced by the elimination of radicals is very clearly explained, and the existence of rootless or impossible equations satisfactorily demonstrated. Some interesting remarks on the impossible equation 2 iF -I- V^' — 7 = 5, will be found on pp. 34, 35 of the Gentleman's Diary for 1837. This equation is also very elegantly discussed by Professor J. R. Young in the fourth edition of his valuable Elementary Treatise on Algebra* Assuming that the symbol V niay be always interpreted, either positively or negatively, as circumstances may require, Mr. W. S. B. Woolhouse seems to contend that the doc- trine of impossible equations is founded upon too restricted a view of that symbol. This question I have already suflSciently discussed, and only call attention now to Mr. Woolhouse's views, in order to make an opportunity of re- marking, that much as I differ in opinion from that highly accomplished mathematician in the present instance, I con- sider his views on every mathematical subject as entitled to our best attention. Of Mr. Cockle's researches I have already spoken. I believe he is the only mathematician who has taken any thing like a general or extended survey of the subject, and ♦ See pp. 131-2 of that admirable little work. It may be proper here to state, that Mr. Cockle, in the Mechanics' Magazine, vol. xlvii. p. 331, has taken objection to certain parts of Prof. Young's argument in relation to the equation noticed in the text, and that the Professor, with his cha- racteristic frankness, has admitted the validity of Mr. Cockle's objections. — See Mechanics* Magazine, vol. xlvii. p. 546. AND OTHER 9TJBD EQUATIONS. 235 who has bestowed on it that degree of attention which its importance demands. In concluding this paper, the author would adopt the language of the elegant writer last referred to in treating of the same subject : — " I hope that these investigations will not prove to be barren of results. At any rate, I trust that any efforts, however humble, to throw light on an anoma- lous— ^perhaps I may say mysterious — difficulty in algebra, will be regarded with toleration, if not with indulgence." 236 XVI. — On Impossible EquatioTis. By Professor Finlay. Bead February 4, 1851. The following paper is intended as a supplement to Mr. Harley's paper on the same subject, read before this Society about a month ago, in which the fundamental principles of the theory were established, and some of the simplest cases of irrational equations were solved, in a very elegant and direct manner. My paper is divided into five paragraphs. The first contains the definition of the new sense of the term " impossible," with some illustrations relative to that definition. The second and third para- graphs contain the discussion of an irrational equation con- taining a single radical of any order. The third and fourth paragraphs contain the discussion of an irrational equation containing two or more radicals of any order. The object of the discussion, in all cases, is to ascertain, a 'priori, the number of impossible roots which the equation contains, and to determine the possible roots exclusively of the im- possible ones. Although the paper is extremely short, I should hope that what it contains is sufficient to show the method of separating the possible from the impossible roots in any irrational equation. I. An impossible equation is one the roots of which are all impossible. In this definition, the term root, as applied to an equation, is used in the ordinary sense, and the PKOFESSOB FINLAY ON IMPOSSIBLE EQUATIONS. 237 whole question turns on the new sense in which it has been proposed to use the term impossible. To illustrate this point, let the equation X -{- V(4a; -f- 1) =: 5 be proposed. Clearing this equation of surds, and solving it by the ordinary process, the roots are found to be 2 and 12. Now, if 2 be substituted for x in the proposed equation, we obtain 2 + V9 = 5, which is obviously true; but if 12 be substituted for «, we get 12 + V49 rr 5, which is evidently false, provided that the radical be restricted to a positive signification, or to its arithmetical value. On these grounds, it has been proposed to call 2 a possible root of the proposed equation, while 12 has been designated as an impossible root. Thus we see, that an impossible root of an irrational equation is one which does not satisfy the equation when the radical which it involves is restricted to a positive signification. In extending the theory to imaginary roots, a difficulty occurs as to the positive signification of an expression of the form V (« — b V — 1). For if (a _ /3 V— 1? z=: a — b V— 1, we shall also have (— a-j./3V— 1)^ = « — *>/— 1; so that V ip, — b V — 1) may be equal either to -|- a — j3 V — 1> or to — a-f-i^V — Ij and it is not immediately evident which of these is to be taken as its positive signification. Now, in extending any algebraical rule to a case not originally contemplated, the extension must invariably be so framed that the new rule may include the original one as a particular case. According to this principle we must evidently take +a — /3 V — 1 as the positive signification of 238 PBOFESSOB FINLAY OK IMPOSSIBLE HQUATIONS. »/(a — 5V — 1); for if we took — a-f-/3 V — 1, when/3zr 0, we should have — a for the arithmetical value of V a, vvhich would be inconsistent with our original assumptions and restrictions. IL Let us now consider the equation X -^ m *^(2ax -\- b) zn c (1), where w, a, b, c, denote given numbers, which may be positive or negative, fractional or entire. If we assume V(2«x + i)=:y (2), and therefore 2ax -^ b :zz t/*, or X zz ^—5 — , the proposed equation becomes -2^ \-myz:zc,ov 1^ ^ 2amy =. 2ac -\- b (3)! Solving this equation by the ordinary rule for quadratics, we get y zz.-^ am -\- »^{a^m^ -|" 2ac -j- ^>) or y = — am -j-B (4); where, for the sake of brevity, we use R to denote the arithmetical square root of the quantity aW -{- 2ac + i- Now, if the radical in equation (1) be restricted to its arithmetical value, it is evident from (2) that y must be positive; and therefore all negative values oft/ must be re- jected as leading to values of x, which are impossible- Thus we see, that the roots of (1) will be hoth possible when the roots of (3) are both positive, and both impossible when the roots of (3) are both negative; but when one of the roots of (3) is positive and the other negative, one of the roots of (1) will be possible and the other impossible. Firstf Let m and a have the same signs; then — ma is negative, and the lower sign must be rejected in equation (4), as giving a value of y essentially negative. PROFEBSOB FriO:-AY ON IMPOSSIBLE EQUATIONS. 239 (a.) When 2ac + J is negative, we have R < ma ; hence, in this case, the second value of y is also negative, and equation (1) is impossible. (j8.) When 2ac -\- h is positive, we have R > ma ; hence the second value of y is positive, and points to a possible root of equation (1), which may be found as follows : — Substituting — ma -j- R for ?/ in equation (2), we get V (2 ax -|" ^) ^^ — "^ 4" ^» .*. 2ax -\- b zn ni'd — 2/wa R -|- n^c^ -[- lac -|- h^ or cc zz rr^a -\' c 1- w R. Secoiidlyy Let m and a have contrary signs ; then — ma is positive, and if the upper sign be taken in equation (4), the corresponding value of?/ will be positive. In this case, therefore, one of the roots of equation (1) is always possible, and may be found as above. (a.) When 2ac '\- b v& positive, R > »ia; hence the second root of equation (3) is negative, and must be rejected as leading to an impossible value of .». (jS.) When 2^ -f- 6 is negative, R < ma; hence the values of y given by the formula (4) are both positive, and equation (1) has two possible roots, which may be found as above, or by the ordinary rule for quadratics. It may be observed here, that when the roots of (3) ar« imaginary, the quantity 2ac + 6 is essentially negative; from which we see that when m and a have contrary signs, and the roots of (3) are imaginary, the roots of (1) are both possible in the new sense of this term, although they are both imaginary in the ordinary sense. By taking successively m-mly and w rz — 1, in equation (1), we obtain the two equations discussed by the author; so that this discussion of equation (1), appears to embrace all the results at which he had arrived up to the time of reading his paper. 240 PROFESSOR FINLAT ON IMPOSSIBLE EQUATIONS. III. The preceding method may be generalized with the ut- most facility. Let us consider, for instance, the equation X-{-WX'z=zO (1), where X and X' denote any rational and entire algebraic functions of x. If we assume WX'=i/ (2), and .-. J^' =: y" (20, equation (1) becomes X-\-2/ = 0 (10. Eliminating x between (!') and (2'), we obtain an equation of the form 9(y) = o (3), where f denotes a rational and entire algebraic function of the quantity ?/, to which it is applied. Now, if the radical in equation (1) be restricted to a positive signification, it is evident from (2) that y must be positive; and therefore the negative roots of (3) must be rejected, as giving impossible values of a;. Consequently, if the number of negative roots in equation (3) be found by means of the theorem of Sturm, the number of impossible roots of (1) may thence be readily ascertained. Thus, if p and q denote the de- grees of the functions X and X' respectively, it is evident from (2') that every negative root in equation (3) will give q impossible roots for equation (1). IV. The same method may be applied to equations contain- ing any number of radicals. For the sake of clearness, let us first consider the particular equation mjs/(ax -^-b) ■-\- nA/(cx -^ d) zzf (1> If we assume \/(ax -\-b)zzi/, 's/Cpx -}-d) zzz, (2), and.', ax -\- b zz i/^, ex -\- d zz i:^ (20, equation (1) becomes PEOFESSOB FINLAY ON IMPOSSIBLE EQUATIONS. 241 my-\-nz=:f (1') ; and by eliminating a' from equations (2') we get be — adrz cy' — az^ (3). Again, by eliminating y from equations (1) and (3), we obtain (cti^ — am*) 2^ — 2f/icz zz: (be — ad) m^ — c/^...(A.) Let Zi and z^ be the values of z deduced from this equation, i/i and ^2 the corresponding values of y deduced from (1'); then if each of the radicals in equation (1) be restricted to a positive signification, it is evident, that if either y or z be negative, the corresponding value of x will be an impossible root of equation (1), and that the root of (1) corresponding to t/j and Zi cannot be possible unless 1/2 and z, be both positive. To illustrate this theory by a numerical example, let the proposed equation be 3 V(2a; -|- 5) 4. 4V (Bx — 2) n 17. Comparing this with equation (1), we have ni=z3, n=zi, az=:2, b — 5,c=z8, dzz — 2,/= 17 ; hence equations (1') and (4) become 3y + 42r rr 17, 302^ — 408z =: — 696. From the latter equation we readily obtain and, by substituting these in the former, we get Now, since y^ is negative, the corresponding value of x will be an impossible root of the proposed equation; but since yi and Zi are both positive, we see that the equation has a possible root. To find the possible root, let z zzzi rz 2 in the second of equations (2) ; then ^(Sx — . 2) =: 2, and .-. a; =: 2, which will be found to satisfy the proposed equation. To find the impossible root, letz =: z^zz: «/ in the same equation; then V(3a: — 2) = V» and .-. x = 3^*, 2 I 242 PBOFESSOB FINLAY ON IMPOSSIBLE EQUATIONS. which will not satisfy the proposed equation, unless the first radical be taken with the positive, and the second with the negative sign. V. The theory which has just been given for the case of two radicals of the second order, may now be readily extended to the case of any number of radicals of any order. For, if t, u, V, &c., be assumed equal to the several radicals, we shall obtain, in all cases, a series of equations free from radicals; and the number of these equations being always equal to the number of the unknown quantities x, t, u, v, &c., the values of t, u, v, &c., may always be found by the ordinary methods of elimination, at least when the pro- posed equation is numerical. Let ti, m„ v,, &c., be any system of simultaneous values of t, u, v, &c., then it is evident that if mG THE FORMATION OF COAL, &c. 253 bon with 22 atoms of water. There can be no doubt, how- ever, that the elements are intimately united with each other, and do not exist as mere carbon and water; still this extreme simplicity of constitution was designed for an allwise pur- pose, and greatly facilitates the necessary changes which must take place on the death of an organic body. When such a body dies, and is exposed to the action of the air, it is probable that 2 atoms of the atmospheric oxy- gen unite with 2 atoms of the hydrogen of the plant, and form 2 atoms of water, which are eliminated, whilst 2 atoms of the oxygen of the organic body being thereby Uberated, immediately unite with one atom of the carbon of the plant, and form one atom of carbonic acid, which is also eliminated; so that, for every atom of carbon removed, 2 atoms of hydrogen and 2 atoms of oxygen are also displaced, and in time nothing but carbon would remain, were the process to go on in the atmosphere freely and undisturbed. When the process is about half completed, what is called vegetable mould, or humus, is formed. When a vegetable body decomposes under water, the circumstances in which it is placed being different to its condition when freely exposed to the atmosphere, the pro- cess and order of decomposition are also very different. It decomposes with limited access of air, and in contact with a body (water) itself susceptible of decomposition in the presence of decomposing organic matter. Water has the property of absorbing a certain quantity of oxygen from the atmosphere. 100 cubic inches of water at a temperature of 60° Faht., and under a pressure of 30 inches mercury, will absorb about Z^ cubic inches of oxygen. It is well known that all plants in a living state absorb oxygen united to carbon or carbonic acid; and hence it is, that the herbage on the brow of a hill, on which a rill of water trickles, looks so bright and green, and fresh and vivid, from the incessant supply of oxygen to its roots, in the best form for oxidizing 254 MR. JOHN LEIOH ON THE CHEMICAL CHANGES the carbon of the manure, and supplying it with the carbonic acid which constitutes its chief food. And hence it is, that in stagnant water, the oxygen soon being removed by the immersed plants, and a farther supply from the atmosphere being prevented by the still water, the herbage is dark and rank, sickly and unpalatable. We have seen, that by the absorbed oxygen of the water, and we may add, by the action of the wind and waves, dead plants, immersed in water, are not entirely, at least at first, excluded fi-om contact with oxygen, but that this is sup- pUed in very limited quantity. When the plants begin to decompose, the water participates in the change, its ele- ments unite with the decomposing matter, and the oxygen which it held in solution is absorbed by the decaying vege- table matter, carbonic acid being given off. It has been stated above, that the composition of wood may be repre- sented by carbon 36 ; hydrogen 22; oxygen 22. This is the actual composition ofperfectlypurified, well-dried oak wood, as determined by Gay Lussac and Thenard; and although different varieties of wood may afford minute deviations from these proportions, yet it may be assumed that these repre- sent the empirical formula, and may be taken without much risk of error as the groundwork of our reasoning and cal- culations. When oak wood is decomposed under water, a white mouldered matter is formed, which yields on analysis, carbon 33, hydrogen 27, oxygen 24. Now, if to the elements of oak wood, carbon 36, hydrogen 22, oxygen 22, we add the elements of 5 atoms of water, with 3 atoms of oxygen rz hydrogen 5, oxygen 5 -f- oxygen 3, and sub- tract 3 atoms carbonic acid, carbon 3, oxygen 6, we have the exact composition of the altered wood, or mouldered oak. And this is what must really take place ; 5 atoms of water or its elements, and 3 atoms of oxygen from the water and air, unite with the decomposing woody matter, and 3 atoms of carbonic acid are given off and escape. ATTENDING THE FOJRMATION OF COAL. &c. 255 Whenever free oxygen, as that of the atmosphere, has access to the decomposing matter, it is probable that no hydrogen is removed from the plants, except in union with it, and therefore in the form of water. In the change of wood into wood coal, in which the whole ligneous structure is preserved, this would appear to have been pretty con- stantly the case ; for in one specimen of wood coal analysed in Liebig's laboratory, the composition differed from oak wood merely in the loss of 3 atoms of carbonic acid and 1 atom of hydrogen ; and in another, which had undergone farther decomposition, by 4 atoms carbonic acid, 5 atoms water, and 2 atoms hydrogen. It is most probable that in both these cases the hydrogen had been removed by free oxygen, and the carbon by the oxygen of decomposed water. A beautiful illustration of the formation of wood coal, by the united action of water, and a limited supply of air, accelerated by a high temperature, was afforded in the analysis of a piece of wood, which had been long kept in the boiler of a steam-engine, and had acquired the appearance of wood coal. It had exactly the composition of the first of the wood coals spoken of above, viz., carbon 33, hydro- gen 21, oxygen 16, having lost 3 atoms of carbonic acid, and 1 atom hydrogen. When vegetable matter decomposes under an entire exclusion of air, or nearly so, as must take place in deep or still water, or when imbedded in such masses of rock as we find the coal formation to be, the changes, no longer influenced by free oxygen, must vary from those already described; and some products of a different character be eliminated as the result of the decom- position. The carbon, still seeking oxygen from every available source, obtains some from the decomposition of the water in contact with the decaying vegetable matter; some from the vegetable itself, whose relative quantity is continually diminishing; and a portion from the salts originally existing within the decaying mass, and in the 256 MR. JOHN LEIGH ON THE CHEMICAL CHANGES water in which these changes take place, reducing the sulphates, phosphates, &c., which chiefly constitute them, to sulphurets, phosphurets, &c. The metallic bases with which these are ultimately left in combination, being finally oxidized by the oxygen of decomposed water, unite with the carbonic acid eliminated by the decaying matter, forming carbonates of the earths, alkalies, &c., which origin- ally existed in other forms in the vegetable fabrics, and in the water. The hydrogen, liberated from the decomposed water, seizes on the sulphur, phosphorus, and carbon, with which it is in contact at the moment of liberation, and escapes as sulphuretted hydrogen, phosphuretted hydrogen, and carburetted hydrogen. A portion of the hydrogen also of the decomposing plants now enters into new com- binations ; part uniting with the oxygen of the plant, form- ing water, and another part with its carbon, forming light carburetted hydrogen, or marsh gas, or fire damp, a gas composed of 2 atoms of hydrogen, and 1 atom of carbon, or of carbon 2, hydrogen 4. Instead of the mere elimination of carbonic acid and water, then, as takes place when vege- table matter decomposes or decays with free access of air, we have, when occurring under water, or in contact with water, with exclusion of air, the formation and escape of carbonic acid, light carburetted hydrogen, sulphuretted hydrogen, phosphm-etted hydrogen, which are all gaseous, and water, which remains behind. Whoever has stood over a marsh or a stagnant pool, or watched a foul drain drag its slow length along, has observed bubbles of gas to gurgle up to the surface, float awhile and burst. The gas con- tained in these bubbles, on analysis is found to consist of C£^bonic acid, light carburetted hydrogen (hence called marsh gas), and, when in considerable quantity, sulphuretted hydrogen also. The pale phosphoric light which seems to enwrap masses of decaying wood in the interior of trees, sometimes called ATTENDING THE FORMATION OF COAL, &c 257 phosfire, is due to phosphuretted hydrogen. In tropical countries, favoured by the warmth, the succulent vegeta- tion brought down by the large rivers into the stiller waters of their estuaries, together with the abundant maritime vege- tation naturally growing therein, decompose with a rapidity unknown in our colder climes, and pour off their gases in immense volumes, creating a pestiferous stench, and a de- structive miasm, fearfully fatal to the adventurous European who may visit these fetid waters. The production of carbonic acid and light carburetted hydrogen must often be sufficiently simple ; for 2 atoms of carbon uniting with the element of 2 atoms of water con- taining Oa H„ would produce 1 atom carbonic acid C. O5,, and 1 atom light carburetted hydrogen C. H^. It will not be necessary, in the present state of geological knowledge, to adduce any proofs of the vegetable origin of coal and cannel. Independently of the occurrence of the trunks of trees within the beds, and the abundant existence of fern-like organic remains in the roofs and floors of the coal scams, thin slices of coalj when examined under a microscope, exhibit a true ligneous cellular structure, and have all the appearance of wood. The very frequent occur- rence, not only of the scattered teeth, bones, and scales of fish, but of their entire skeletons, in cannel, show that this latter was either formed under water, as described above, or was submerged very soon after its formation. The great extent to which the original structure has been destroyed in cannel, also points out the subaqueous formation of it. The remains of fish are never found in ordinary coal, and the ligneous structure is much better preserved. As the accumulation of vegetable matter, by changes on the earth's surface, became covered with mineral deposits, and air and even water more effectually excluded from the changing mass, its own substance was compelled to furnish the oxygen to the hydrogen and carbon of the decaying plant, and thus 2 L 258 MR. JOHN LEIGH ON THE CHEMntCAL CHANGES in more rapid and unequal proportion, was this diminished in the forming coal. It has been shown by the products of the decomposition of vegetable matter and water, when air is excluded, that the elements of the decaying matter divide themselves amongst each other, so to speak, under these circumstances, and pass off"; one in combination with a por- tion of each of the rest, the proportion of the combination, as will be shown hereafter, varying with the temperature. In the earlier periods of decomposition and entombment, the carbonic acid evolved would be large, relatively to the carburetted hydrogen and water ; but as the process went on, and the oxygen became diminished in the decomposing matter, these proportions would become reversed, till scarcely any thing but carburetted hydrogen would be at length given off by the coal, and this even would finally cease on its con- version into anthracite. Let us see how far analysis w^ill carry out this reasoning. There is no reason for believing, however much the external form, and perhaps internal me- chanical structure, of the Flora of the ancient world may have differed from that of the present, that the composition of the woody fibre of that remote period differed in any material degree from its composition now. Prodigal as nature is in shapes, and forms, and hues; unsparingly as ar- rangements have been varied — the Almighty hand thatguides her operations works with the simplest means, and the most unchanging processes. His power displays itself in the in- finite variety accomplished with the most limited materials. It is reasonable to suppose, it is in accordance with all know- ledge of the subject, that the same formula that represents the composition of woody matter now, would exactly corre- spond with, or closely approximate to, that representing its composition in the vegetation of a past world. We have seen before, that the general formula for woody fibre now may be represented by Cgg H^j O^j. An analysis of the cannel coal of Lancashire, and of the splint coal of New- ATTENDrSG THE FOBMATfON OF COAL, &c. 259 castle, gave the formula C^ Hj, Oi. The analysis of coal from the Oakweli Gate colliery, near Gateshead, and from the Hebbum colliery, corresponds very nearly to this formula. Of course the greater number of coals and can- nels will vary from it more or less, but it may be taken as the expression of a general formula. Now 1 atom of wood C. 36 H. 22 0.22 Minus 9 atoms carbonic acid C. 9 0. 1 8 „ Satomswater H. 3 O. 3 „ 8 atoms carburetted hydr. C 3 H. 6 == C. 24, H. 13, O. the composition of co^ So that all the elements of the vegetable fibre have partici- pated in the decomposition; and the wood, in its conversion into coal, has evolved from its structure 9 atoms carbonic acid, 3 atoms water, and 3 atoms light carburetted hydrogen. It would appear, that in the earlier stages of decomposition, when the air had partial access to the decaying matter, and oxygen existed in the mass as a main constituent, that the carbon and hydrogen combined with this agent, in prefer- ence to uniting with each other, as might, from the immense combining or chemical energy of oxygen, have been antici- pated, and that thus, during these periods, only water and carbonic acid would be evolved, or vdth a very minute pro- portion of carburetted hydrogen. The wood coal or brown coal of Lavbach in Hesse-Darmstadt is composed of C. 33, H. 21, O. 16, and differs from fresh wood, therefore, by the elements of 3 atoms carbonic acid and 1 atom of hydro- gen. The wood coal (brown coal) of Ring Kuhl, near Cassel, is much further decomposed, and is losing the woody structure. It contains C. 32, H. 15, O. 9, and differs from wood by the loss of 4 atoms carbonic acid, 5 atoms water, and two atoms hydrogen. There can be no doubt that the surplus hydrogen has been removed in both these cases by external oxidation. The gas eliminated in mines of wood coal is invariably carbonic acid, and never contains carbu- 260 MR. JOHN LEIGH ON THE CHEMICAL CHANGES retted hydrogen. The avidity with which the decomposing vegetable mass seizes, even when converted into beds of coal, on every available source of oxygen, was lately ob- served in analysing the gas (consisting chiefly of fire-damp) from the mines of Newcastle ; the nitrogen forming from 14 to 21 per cent, whilst scarcely any oxygen remained. Now, the nitrogen -must have been derived from atmo- spheric air, admitted by the mine to the coal, which had re- moved the oxygen and combined with it. The constant pre- sence of sulphuret of iron in coal, which originally must have existed as sulphate of oxide of iron, and been converted by the removal of its oxygen into sulphuret, also shows the powerful deoxidizing power of the decomposing organic mass. When, after a long lapse of ages, the oxygen had been gradually' removed from the vegetable mass, in the form of carbonic acid and water, until at length the wood coal had lost its structure, and acquired the composition possessed by our own beds of more perfectly formed coal, in which 1 atom of oxygen only remains in union with 24 atoms of carbon and 13 atoms hydrogen, or approached this composition, it is evident that a new series of results must attend the changes going on witiiin the still altering coals; oxygen no longer existing for the formation of carbonic acid, the carbon and hydrogen now constituting almost the entire mass of the coal, must of necessity unite, and escape as car- buretted hydrogen. An analysis of the gas evolved in mines from coal, shows it to consist almost exclusively of light car- buretted hydrogen. The following analysis by Mr. Wiight- son, made in the laboratory of the Museum of Economic Geology, of the gas evolved from a seam in the Hebburn colliery, will show this : — Light carburetted hydrogen 91"8 Carbonic acid 07 Nitrogen 6*7 Oxygen ,.., 0-9 1001 ATTENDING THE FORMATION OF COAL, &c. 261 There can be no doubt, that the minute quantity of carbonic acid present, had been formed by the atmospheric oxygen, the large excess of nitrogen showing that the air had been robbed of its oxygen ; whilst it is further evident, that the coal itself was pouring out from its own materials pure carburetted hydrogen. As light carburetted hydrogen contains 2 atoms of hydrogen to.l atom of carbon; it is also evident, that our present beds of coal are hastening to the condition of anthracite, which consists almost entirely of carbon, and has gone through all tbe stages of forest, peat, and wood coal, to anthracite. Cannel appears to differ from coal in having been formed under water; its abun- dant remains of fish, interspersed through its substance, its layere of sulphate of lime, which it could have obtained from no other source, and which are not found in coal, all prove this ; whilst its conchoidal fracture and homogeneous texture, seem to indicate that it formerly existed in a softened muddy state. Where the temperature has been low, we have no evidence of the formation of any higher carburets of hydrogen than that of fire-damp, which contains 1 atom of carbon to 2 atoms of hydrogen, during the decomposition which precedes the formation of coal ; but under an ele- vated temperature, such as could be produced by an injec- tion of ignited matter into the adjacent super or sub jacent rocks, we have new affinities called forth, and compounds formed, in which the relation of carbon and hydrogen is altogether different, the atoms of carbon sometimes exceed- ing those of hydrogen in the compounds, and sometimes being of equal number. We know that no such injection of heated matter now takes place within the limits of the Lancashire coal-field, or the coal-fields of Northumber- land and Durham ; and in the fire-damp of the mines sunk there, we find no higher carburet of hydrogen than that so often spoken of. But where the coal-measures have been traversed by dykes of trap rock, which must have been 262 MB. JOHN LEIGH ON THE CHEMICAL CHANGES injected in a melted state, the neighbouring coal has been subjected to a true distillation, and products are found in the vicinity, the ordinary results of such action. Thus, in Derbyshire, where the measures have been traversed by dykes of trap, popularly called toadstone, springs of naph- tha are found, which must have distilled from the coal. Similar springs are found at Baku, near the Caspian ; at Ammiano in Italy, at Rangoon, and in some parts of Ger- many, &c. The analysis of the fire-damp which streamed out of clefts in the coal at Wallesweille, Luisenthal, and Lickwey, indicated the presence of from 6 to 16 per cent, of defiant gas, according to Bischoff. defiant gas con- tains 2 volumes of carbon and 2 of hydrogen, condensed into 1 volume. In some places, when the coal has been near to the heated matter, it has been found completely charred, and converted into coke. In reflecting on these decompositions, there are two cir- cumstances that strike us as remarkable, and which possess peculiar significance. The first and most remarkable fact, is the entire absence of pure hydrogen in any of the gases evolved by the decomposing vegetable niatter, or in any of the fire-damps issuing from the decomposing coal, although so constantly present in coal gas. The second is the equal absence of defiant gas, or of any other compound of carbon and hydrogen, except the light carburetted hydrogen, C H. 2, unless under circumstances that could lead us to believe that the coal had been subjected to a high tempera- ture, and that the higher carburets of hydrogen were true products of distillation, where defiant gas, naphtha, petro- leum, &c., are found as natural products. It will be appa- rent, then, if the foregoing reasonings and remarks be ai^itted as proo^ that it is a law of nature, that when organic masses decompose without access of air — that is, with exclusion of free oxygen — all the elements participate in tho change, and unite reciprocally with each other ; the ATTEKDING THE FORMATION OF COAI^ &a 263 mode of union, and the products of that union, being regu- lated by the temperature. That, including in the consider- ation the nitrogen, which, though not a constituent of the woody tissue, is invariably found in the juices and in many of the organs of a plant, and permeates every part of it, when vegetables decompose with free access of air, the products are carbonic acid, water, and ammonia; when under water, with very limited access, or total exclusion of air, carbonic acid, water, carburetted hydrc^en, ammonia, sulphuretted hydrogen, phosphuretted hydrogen, &c. ; with limited supply of water, and exclosicm of air, as in, wood coal, carbonic acid, water, and a little carburetted hydrc^en, till the process having nearly exhausted the oxygen, as in truly fossilized coal, the carburetted hydrogen exclusively takes the place of the carbonic acid ; and finally, that when the divellent affinities are exalted by a high temperature, other compounds are formed, in which some of the elements are united with each other in increased proportions, (olefiant gas, naphtha, petroleum, &c.) It ia worthy of remark, that when olefiant gas is found in the fire-damp of mines, when the coal is supposed to have been subjected to heat, its proportion varies from 1*5 to 16 per cent. The most usual proportion was about 6 per cent. These numbers represent the whole amount of illuminating gases existing in the gas here formed by nature's operations. The analyst is BischofF; and nowhere but in Germany has tliis gas hitherto been found in fire-damp. When vege- table bodies, or bodies of vegetable origin, as coal, cannel, &c., are subjected to distillation in close vessels, without access of air, as in the process of gas-making, manufacture of pyroligneous acid, &c., the products will vary with the composition of the substance employed, and with the tempe- rature. We have seen that all the elements participate in the change, and form new combinations. The proportion and relation of the elements of fresh wood, and of coal. 264 MB. JOHN LEIGH ON THE CHEMICAL CHANGES being then unlike, the results of the application of heat or other agents must also be unlike, analogous but yet unlike. Both give off gases, but those of ooal are richer in car- bon ; both give off oils, but those of coal are richer in carbon. Naphtha is the turpentine of coal. The products of coal are alkaline ; those of wood, acid^ arising from the large relative quantity of oxygen that the latter contains (C. 36, H. 22, O. 22, being the composition of wood); C. 24, H. 13, O. being that of coal. The oxygen in wood seizes the hydro- gen, and diminishes the production of illuminating gases — the gases found consisting of carbonic acid, light carburetted hydrogen, and very little defiant gas ; the oxygen also seizes on the combined hydrogen and carbon, forming acetic acid, a compound of C. 4, H. 3, O. 3, pyroxilic spirit (wood naphtha, wood spirit), a kind of alcohol containing (C. 2, H.4, 0.2), xylite (C. 12, H. 12, 0. 5),another liquid (C. 21, H. 23, O. 10); all compounds containing a large amount of oxygen, and from which it will be seen how important a part the large amount of this element contained in wood plays in the products of its destruction, and modifies the results of its distillation. The gases from wood and from peat, as well as from brown coal (wood coal), possess a very low illuminating power. Coal is a compound of carbon, hydrogen, a little oxygen, very little nitrogen, earthy con- stituents constituting its ashes, sulphur in the form of sulphuret of iron (iron pyrites), and in cannel occasional layers of sulphate and carbonate of lime. Its empirical formula is C. 34, H. 13, O. When distilled at a high tem- perature in close vessels, part of the hydrogen unites with carbon, forming light carburetted hydrogen, defiant gas, gaseous hydro-carbons, naphtha, and its associated oils. Another part unites with oxygen, forming water ; another with nitrogen, forming ammonia; a fourth with sulphur, forming sulphuretted hydrogen ; and a fifth with cyanogen, forming prussic acid. Of the carbon, part unites with hydro- ATTENDING THE FORMATION OF COAL, &c. 265 gen as above ; part with oxygqn, forming carbonic acid and carbonic oxide ; part with nitrogen, forming cyanogen ; part with sulphur, forming sulphuret of carbon. So that there is a perfect division of the elements amongst each other; and it can never be that the whole of the hydrogen shall unite with the carbon, and produce illuminating gases only. Here, however, we have no actual or necessary pro- duction of hydrogen, whose presence in the gas, therefore, must be the result of a defect in the process of gas-making. A little chlorine also is present in most canncls in the form of chlorine salts, and is given off as muriatic acid, but al- ways in combination with the ammonia. A little sulphu- rous acid is likewise found in combination with the am- monia. The muriatic and sulphurous acids never pass into the gas holder, and need not be considered, being con- densed with the ammonia. The cyanogen, sulphuretted hydrogen, ammonia, and carbonic acid, constituting a very minute proportion of the whole gas, ought to be all removed in the process of purification ; so that there remain as con- stant ingredients of the gas, as at present manufactured, hydrogen, light carburetted hydrogen, olefiant gas, volatile hydro-carbons, carbonic oxide, and a little nitrogen ; and of these, the olefiant gas, volatile hydro-carbons, and light car- buretted hydrogen, alone contribute to illumination. The water formed and condensed from the distillation (gas water, ammonia water), retains in solution carbonate, sulphite, muriate, hydro-sulphate, and prussiate of ammonia ; the tar which condenses from the distillation, consists of nume- rous oils, called naphtha, heavy oil of tar, &c., composed almost entirely of carbon and hydrogen. I add here a tabular view of the products of the distillation of coal, with the composition of each. 2 M 266 MB. JOHN LEIGH ON THE CHEMICAL CHANGES 1. GASEOUS. Hydrogen H. Light Carburetted Hydrogen H. 4 C. 2 Olefiant Gas H. 4 C. 4 Volatile Hydro-carbon H. 6 C. 6 probably Benzole C. 12H.6 Carbonic Oxide C. O. Cyanogen C. 2 N. Sulphuretted Hydrogen H. S. Ammonia H. 3 N. Aqueous Vapour H. O. Sulphurous Acid S. O. 2 Hydrochloric Acid H. CI. Carbonic Acid ....'..V«;..'..i...... C. O. 2 Sulphuret of Carbon...........'..'...'.'..:..'....... 0. S. 2 Nitrogen N. 2. AQUEOUS. Water holding in solution Carbonate of Ammonia. Hydro-sulphate of Ammonia. Prussiate of Ammonia. Sulphate of Ammonia. Muriate of Axomonia. 3. OILY. /■Benzole \ C. 12 H. 6 Liquid-5 Toluol VNeutral C. 14 H: 8 (Cumol ) C. 18 H. 12 Aniline C. 12 H. 7 N. Picoline C.12 H. 7 N. Leucoline C. 18 H. 8 N. Hydrate of Phenyle C. 12 H. 6 O, 2 acid. Naphthaline C. 20 H. 8 Paranaphthaline..!. C. 30 H. 12 Pyren C. 16 H. 3 Chrysen C.12 H. 4 Various undescribed oils. Of these oily products, which are all contained in the tar, it will be perceived that only one contains oxygen, and this possesses acid properties. The first thifiee are neutral. Liquid Solid • Constituents of Naphtha. Const!- tnents of heavy oU of tar. ATTENDING THE FORMATION OF COAL, &c. 267 and constitute rectified coal naphtha. Three are alkaline, and contain nitrogen. The other four are solid and neutral. It is worthy of remark, how few of the products of distilla- tion contain any oxygen, and how much they differ in this respect from the products of distillation of wood ; and that, where the oxygen does enter into combination, it produces compounds having no illuminating properties, viz., carbonic oxide, carbonic acid, and water ; and that in one instance il; unites with a compound of carbon and hydrogen, producing an acid oil, that is found in very small quantity in the tar. The nitrogen of the coal forms ammonia, cyanogen, and a few alkaline oils, the latter found in small quantity in the tar. The products are nearly all compounds of carbon and hydrogen, and respecting these it is further worthy of re- mark, that when the hydrogen exists in the compound in greater quantity than the carbon, as in light carburetted hydrogen, which contains 2 atoms hydrogen to 1 of carbon, the compound is permanently gaseous ; this gas has been subjected to a pressure of 32 atmospheres, and to cold 166 degrees below zero, without liquefying. Olefiant gas, in which carbon and hydrogen exist in equal proportions, but in which 2 volumes of hydrogen and 2 of carbon are condensed into one volume, is permanently gaseous at the ordinary atmospheric temperature and pressure, but becomes liquid under a pressure of 27 atmospheres at zero of Fahrenheit. The volatile hydro-carbons in coal gas, the exact nature of which has not y«t been determined, and whose compo- sition is valuable, or rather, perhaps, whose proportions in the mixture are valuable, probably consist of propylene Ce He, Faraday's gas, Cg Hg, and benzole, Ci^ He, with perhaps a portion of Mansfield's allyle. In my earlier ex- periments on coal gas, which had been made from a differ- ent cannel to what is now employed at the Manchester Gas- Works, I found, pretty uniformly, that each volume of the gas condensible by sulphuric acid or chlorine re- 268 MB. JOHN LEIGH ON THE CHEMICAL CHANGES quired 4^ volumes of oxygen for combustion ; subsequently I found that, \«^hen richer cannels were used, the conden- sible gases required a still larger proportion of oxygen for combustion. The fact, that a greater amount of carbonic acid is produced on exploding the gas with oxygen, than would proceed from a body of the series C„ H^ shows that some other hydro-carbon, the carbon in which exists in a greater ratio to the hydrogen than in this series, is to be found in coal gas. When the number of atoms of carbon, in a compound, exceed those of the hydrogen, they are, within certain limits, liquid ; beyond those limits, solid ; thus benzole — car- bon 12, hydrogen 6, in which the carbon is double the hydro- gen ; toluol — carbon 1 4, hydrogen 8 ; and cumol — carbon 18, hydrogen 12, are liquid. Whilst naphthaline — carbon 20, hydrogen 8; paranaphthaline — carbon 30, hydrogen 12; pyren — carbon 15, hydrogen 3; and chrysen— carbon 12, hydrogen 4, are solid. It may be safely asserted, that where the carbon and hydrogen exist in an equal number of atoms, the compound will be gaseous, unless a very large number enter into the combination. Let us now see what arc the practical inferences to be drawn from the preceding postulatory statements. In the first place it will be manifest, that the greater the quantity of hydrogen, and the less oxygen and sulphur a cannel or coal may contain, the better it will be for gas-making; for the two latter rob the coal of a portion of its hydrogen, which is thereby prevented from uniting with a portion of carbon for the production of an illuminating gas. The coal should be selected as free from iron pyrites and sulphate of lime as possible, and lumps or masses of these should be thrown out, as they often occur in such a form in the coal. The coal should be moderately dry be- fore being used, which can only be secured by being stacked under cover, otherwise the rain would keep it saturated with ATTENDING THE FORMATION OF COAL, See. 269 moisture. The water, in its decomposition in the retorts, furnishes oxygen to the carbon of the coal, impoverishing the gas, whilst the hydrogen of the water does not combine with the carbon of the coal, but is liberated in the simple state. When the vapour of water is passed over red-hot coke and coal, and analysed, the resulting gas is found to consist in 100 volumes, of hydrogen 56, of carbonic oxide 29, car- bonic acid 1 5*8, and of light carburetted hydrogen only two- hundredths of 1 per cent It contains no olefiant gas whatever; this experiment is quite conclusive against the use of water or steam. It is evident that there are no products of the decomposition of water by red-hot coal or coke that possess any illuminating power. It has often been proposed to pass steam into the retorts during the distillation of coal, but such a proceeding could have no good effect, but the contrary. When it is considered that 50 per cent, of the whole of the gases pro- ceeding from the decomposition of water by red-hot car- bonaceous matter is hydrogen, another very formidable objection arises to its use ; viz., that it would not only diminish the light of gas with which it was mixed, but would give out such an amount of heat during the burning of it, as would render the use of such gas almost insupport- able. The coal then should be dry ; but we have also seen that air, passed between seams of coal, has been deprived of a portion of its oxygen, which must have combined with the carbon and hydrogen of the coal, and by as much have impaired its quality. We have seen that coal and cannel are continually giving off gas (fire-damp), and this teaches us that the coal should be dried quickly and then used. Let us now consider what takes place during the manu- facture of gas by the distillation of coal in red-hot retorts. The nitrogen in gas is entirely derived from atmospheric air, admitted into the retorts during the charges, and by 270 MR. JOHN LEIGH ON THE CHEMICAL CHANGES leakage in the apparatus, and is not a product of the de- composition of the coal at all ; it need not, therefore, enter into our consideration. The quahty and illuminating power of the gas will be affected, not only by the quality (composition) and cojidi- tion (wet or dry, old or recently got) of the coal or cannel, but by the degree of heat employed in its preparation, and the mode in which the operation is conducted. The chief products of the distillation are compounds of carbon and hydrogen, and these alone yield light; but of these we find that some are solids, some liquids, and some gaseous — ^the two first are valueless for the purpose of illumination, be- cause their physical condition (solid and liquid) precludes their use. The gaseous are three — one containing very little carbon (light carburetted hydrogen), and, therefore, giving very poor light ; the other two very rich in carbon (olefiant gas — carbon 4, hydrogen 4 ; and volatile hydro-car- bon— carbon 6, hydrogen 6, or carbon 8, hydrogen 8, &c.), and giving great light, though in small quantity. We find mixed with these, besides the necessary impurities (sul- phuretted hydrogen, ammonia, carbonic acid, &c.), two gases constituting the chief bulk of the mixed coal gas, which we have also seen are never given off in natural operations, viz., hydrogen and carbonic oxide. They are also not necessary products of the distillation, but result from the mode of distillation. The carburets of hydrogen may be conveniently divided into three classes ; in the first, the number of atoms exceeds that of the hydrogen — they are very rich in carbon, as benzole (carbon 12, hydrogen 6); these are either liquid or solid, and would give great light could they be burnt, but give great smoke; — in the second, the atoms of carbon and hydrogen are equal in the compounds, as olefiant gas (car- bon 4, hydrogen 4), volatile hydro-carbon (carbon 4, hydrogen 6, and carbon 8, hydrogen 8) ; these are gaseous, ATTENDING THE FORMATION OF COAL, &c. 271 but condensible by great pressure and intense cold, and give much light ; — in the third, the atoms of hydrogen ex- ceed those of the carbon; these are altogether unconden- sible, and give little light (carbon 2, hydrogen 4). When coal and similar organic matters are distilled at a compara- tively low temperature, the carbon has a disposition to pass off with little hydrogen; the liquid hydro-carbons are formed, there is much ta* and little gas, but the gas iarich. As the temperature rises, the liquid hydro-carbon diminishes in quantity, and gaseous hydro-carbon increases; there is more gas and less tar (olefiant gas and volatile hydro-car- bons). The temperature still rising, the gaseous products become richer in hydrogen, and poorer in carbon ; light carburetted hydrogen is formed in abundance; and at length, the temperature becoming still higher, pure hydro- gen is given off, as is always observed in the last hour's distillation in gas- making. It is a well-known law of organic chemistry, that the -higher the temperature, and the more advanced the decom- position of organic matter, the simpler are the products. When olefiant gas is passed through red-hot tubes, or over red-hot lime or crystal, or, in fact, over any red-hot surface, it deposits a portion of carbon on the red-hot mat- ter in a solid form, and escapes as a mixture of carburetted hydrogen, and hydrogen. The same thing I have proved also of naphtha, C. 12, H. 6, which deposits carbons in like circumstances, and is resolved into simple products. The affinity between carbon and hydrogen seems to di- minish with the temperature. Is it not probable, that in the distillation of masses of coal, compounds rich in carbon are first formed^ the carbon being in excess of the hydrogen ; as the product rises in temperature, it deposits a portion of its carbon, the atoms of hydrogen become equal, and a rich gas is formed ; but this, getting still hotter, deposits more carbon, the hydrogen 272 MR. JOHN LEIGH ON THE CHEMICAL CHANGES is now in excess, the gas is poor and gives little light ; the heat still increasing, the affinity between the hydrogen and carbon is altogether disrupted, the remaining carbon is de- posited, and pure hydrogen given off? Certainly all this can be effected artificially; and that, to a large extent, it is so in gas-making, is evident from the thick lining of almost pure carbon which soon forms in the interior of gas retorts, and which must proceed from the decomposition of the gas by the red-hot surface — must be deposited from it, in fact. Still it is not simply and entirely thus; there are probably three products, at least, of the decomposition of a liquid carbo-hydrogen, solid carbon, a gaseous product containing much hydrogen, and a solid hydro-carbon containing much carbon, the elements being divided amongst each other. When naphtha vapour is passed over red-hot crystal, it deposits carbon, gives off olefiant gas and light carburetted hydrogen, and forms a crystalline compound, naphthaline, composed of C. 20, H. 8* With these facts before us, is it not reasonable to conclude, that there is a temperature, a point at which, in the process of decomposition, olefiant gas and volatile hydro-carbon (which I would call trito or tetarto carburet of hydrogen) should be formed, and which yet should be unable to decompose these into compounds poorer in car- bon? for we have seen that the intensity of decomposition is proportionate to the intensity of heat. I think there cannot be a doubt that there is such a tejnperature ; but it must be far below that at present employed for the manu- facture of gas. Let us now examine the present system of gas-making, and I thiiik we shall soon see the true source of the hydrogen and carbonic oxide, so invariably found in gas, and constituting so large a portion of its bulk. I may premise, that when carbonic acid is passed over red-hot coke it is resolved into carbonic oxide, by taking up an additional atom of carbon. When compact masses of coal are thrown in heaps of a ATTESPniG THE FORMATION OF COAL, &c. 273 hundred-weight and a half, into retorts heated into a bright redness as is now done, it is exposed to two very different conditions; the surface of the mass, the exterior, in contact with the intensely hot retort is instantly decomposed, and charred ; hydro-carbons, as olefiant gas, &c., are eliminated, which also, at this high temperature, are partly decomposed, and resolved into light carburetted hydrogen and pure hy- drogen, with deposition of carbon, which, with some un- decomposed olefiant gas and volatile hydro-carbons, pass oj0f from the retort — ^the interior of the mass, on the contrary, is for some time exposed to a very moderate temperature, and a simple distillation is accomplished; those compounds which are formed at a comparatively low temperature, the heavy hydro-carbons, which would ordinarily be in a liquid state, are given off; a portion, rising into vapour as it reaches the hotter surface, passes off with the gases formed, and condenses again when it has left the retort in the form of tar; but that portion of the vapour which, in its passage, comes into contact with the red-hot surface of the exterior of the mass and of the sides of the retort, deposits a portion of its carbon, and is resolved into simple compounds, olefiant gas and volatile hydro-carbons, which themselves partly un- dergo the change already described. As the heat penetrates to the centre, and a red-hot mass of charred material of considerable thickness comes to surround the decomposing coal within, as happens towards the end of the distillation, the whole of the hydro-carbons, viz., light oils, volatile hydro- carbons, olefiant gas, and even light carburetted hydrogen itself, that are eliminated, are decomposed in passing over such an extent of red -hot surface, and pure hydrogen is almost alone evolved. The carbonic oxide, which is formed from the union of the oxygen oi the coal and of the air admitted with the carbon of the coal, is also partially decomposed during the whole of the process, but in an opposite direction ; not by depositing carbon, but by taking up more, and being 2n 274 MR. JOHN LEIGH ON THE CHEMICAL CHANGES converted into carbonic oxide, which is evolved with the gas. These are the true sources of the hydrogen and car- bonic oxide in gas ; they are not necessary results of the distillation, but products of the decomposition of the dis- tilled matter. It has been perfectly ascertained by myself, and by other chemists, that when olefiant gas is passed through a nearly white hot porcelain tube, it is entirely de- composed, depositing the whole of its carbon, and giving off pure hydrogen gas. The late Dr. Henry subjected cannel coal to distillation, beginning with a moderate heat, and gradually raising it — the degree of heat is not specified, but it was much inferior to that generally employed in gas manufacture. The operation lasted 10 hours, and he examined the gas at the beginning of the process, after 5 hours, and after 10 hours. At the beginning of the process, 100 parts of the gas contained 13 of olefiant gas, 82 of light carburetted hydrogen, 3 of carbonic oxide, and the rest nitrogen and hydrogen. After 5 hours, 100 parts contained 7 olefiant gas, 56 light carburetted hydrogen, 11 carbonic oxide, and 21 hydrogen. After 10 hourj, the gas contained no olefiant gas, 20 light carburetted hydrogen, 10 carbonic oxide, and 60 hydrogen. With these analyses my own entirely accord, except that, from the greater heat employed, I obtained hydrogen and carbonic oxide almost from the beginning. The decomposed matters occupy greater bulk than the original substances from who(>e decomposition they proceed. The greater the number of atoms of carbon and hydrogen combined together the less space they occupy (C. 20, H. 8, is a solid naphthaline); C 12, H. 6, is a liquid naphtha; C. 4, H. 4, is a gas (olefiant gas). This gas, on depositing a por- tion of its carbon, becoming C. 2, H. 4 (light carburetted hydrogen), retained its original bulk, which latter gas is, therefore, more voluminous for its composition than olefiant gas. Were a gas containing 1 atom carbon and 1 atom ATTENDING THE FOKMATION OF COAL, fcc. 275 hydrogen known, which at present is not, it would un- doubtedly occupy the space of olefiant gas. In other words, could olefiant gas, C. 4, H. 4, be resolved into 2 atoms of C. 2, H. 2, i| would occupy double the space of the olefiant gas itself. For olefiant gas itself occupies exactly the space of the volatile hydro-carbon, C.8, H. 8; and it is ascertained, as stated before, that when 1 volume of olefiant gas is passed through a nearly white hot porcelain tube, it is certainly decomposed, depositing all its carbon, and giving 2 volumes of hydrogen. In other words, its bulk is doubled by the decomposition; and 1 volume of the volatile hydro-carbon, C. 8, H. 8, on being decomposed by heat, and depositing carbon, forms 1 volume olefiant gas and 1 volume light carburetted hydrogen zn to 2 volumes. One volume carbonic acid, on becoming converted into carbonic oxide, occupies two volumes. The greater the heat employed, then, in the process of gas- making, above a certain limit — viz., that requisite for the decomposition of the liquid hydro-carbons — the greater will be the bulk of the gas, and the poorer its quality; the more light carburetted hydrogen, hydrogen, and carbonic oxide it will contain, and the less volatile hydro-carbon and olefiant gas. The analysis of the gas will therefore fur- nish a test of the excellence of the process employed in the manufacture, and a check on the workman, by exhibiting, in the relative amounts of hydrogen, and of the illuminating hydro-carbon, whether too great a heat has been employed. A great quantity of gas may be made from coal, and very badly made. The mere amount of gas produced is no proof of the excellence of the manufacture. Cannel yielding 1 1,000 feet of gas per ton, of specific gravity 600, would furnish for every 100 pounds distilled— about 376 MB. JOHN £EIGH ON THE OHEMICAL CHANGES lbs. Gas...., 22| Tar..: 8i Ammonia wateT ...< 9| Coke 591 100 These proportions will vary cc«isiderably, but still the numbers will represent a general average of produce. It is seen, from abOve, that considerably more than a third of the weight of the gas produced, is distilled from the cannel iii the form of tar, which contains, and is almost entirely composed of, the richest carbo-hydrogen, and very little oxygen ; whilst the gas, as it contains only about 45 per cent, of compounds of carbon and hydrogen, by measure amounting to about half its weight, really only contains about 1 1 pounds of carbo-hydrogen, and of this only about 4 pounds will be defiant and richly illuminating gases. So that in the tar is really contained as much illuminating matter^ or nearly so, as in the gas — not twice as much as would appear from the numbers ; for it must be borne in mind, that in the oils composing the tat the carbon exists in much greater proportion than the hydrogen, one of the lightest, benzole, being a compound of carbon 12, hydrogen 6 ; naphthaline and the solid carburets being represented by carbon 20, hydrogen 8, and even higher proportions of carbon. So that, in the decomposition into illuminating gases, much of the weight must be lost in the form of deposited carbon. I think it is now tolerably apparent, that in the form of distilled matters nearly one-half of the illuminating matter derivable from coal and cannel is lost to the gas* It is probable that a perfect system of gas-making would produce from good cannel a gas containing 20 per cent, of olefiant gas, or other illuminating gases. We have seen, moreover, that a great waste of illuminating material takes place in the present system, from the actual destruction of illuminating gases when formed, by the large quantity of ATTEWDIN© THE FOBMATION OT OOAL, fcc, 277 red-hot material through which they are obliged to pass, and by which they deposit their carbon, and are eliminated as pure hydrogen, a gas which forms from 30 to 40 per cent, of coal gas, and is utterly useless in the illumination. The production of hydrogen and tar are manifest evi- dence that the heat employed is too great at the surface of the coal, and too low in the centre of the mass. When gas is made from resin, or from oil, the melted resin or the oil is allowed to fall in thin steamlets on a red- hot plate, or to trickle over a .more extensively heated sur- face. The facility with which resin or oil is converted into volatile liquids, renders necessary a somewhat ample sur- face, in order that these liquids may be decomposed into gas. There are some furnaces to steam-boilers in which the fuel is burnt with limited access of air, and is supplied to the furnaces by means of hoppers, the coal, in a somewhat finely divided state, falling on a revolving horizontal plate, and being by this scattered in a thin layer on the incandescent matter within. This method of supplying the furnaces is not found to be profitable, and for a very sufficient reason: the coal, in the first instance, is simply distilled, and the gases eliminated meeting with little or no oxygen within the furnace, pass off undecomposed, and the heat that would arise from their combustion is not only lost, but the gases impinging on the bottom of the boiler reduce its temperature, and lower the tension of the steam. If no air whatever were admitted to the furnace, the coal would then be placed in exactly the same condition as the resin and oil which have been used for gas-making. The course of the research indicated by the analysis of the subject of gas-making so far completed is obvious. These are the determination of the constituents of the coals and cannels to be employed as materials ; a rigid examination of the products, gaseous, liquid, and solid, resulting firom 278 MR. JOHN LEIGH ON THE FORMATION OF COAL. their decomposition, with reference to their amount and their quality; an examination of the nature and bulk of the pro- ducts eliminated by distillation at different temperatures ; an examination of the results of distilling coal in aggregated masses and in thin leyers; an examination of the relative effects of heating coal in a thin layer on a sufficiently hot surface, with a comparatively cool arch above, as in a D shaped retort, with the bottom alone heated; and the effects of surrounding the material with a heated surface, as in the round retorts heated all round. 279 XIX. — On Linear Constructions^ hy Rev. Thos. Penyng- TON KlEKMAN, A.M., RectOT of Crofi-with-Southworth. Read March 18, 1851.. It is generally, known to mathematicians, and is stated by Professor Chasles in his " Apergu Historique, &c.," as well as by Professor Steiner in his " Systematische Entwickelungy u. s. ?o." (Anhang), that the following question was twice proposed as a prize question by the Academy of Brussels above twenty years ago, and received no answer : " What is the relation among ten points of a surface of the second degree?" One obvious answer is, that the equation to the surface, if the constants are made functions of nine given points, expresses the required relation among the co-ordi- nates of ten points. But it is plain that the solution re- qviired is to be purely geometrical, and such that it shall give a criterion independent of all properly analytical re- sults, computation of numbers, or measurement of distances, whereby it may be determined whether any tenth point lies on the surface which passes through a giveu nine, and where any line meets the surface a second time. Mr. Weddle, of the Royal Military College, Sandhurst, gave a construction of the tenth point in the November number 1850 of the Cambridge and Dublin Mathematical Journal; and his is the first answer, so far as I can learn, that the Brussels prize question has received. I am not sure, nor does Mr. Weddle seem to be certain, that his solution ■will be accepted as purely geometrical, for it is the con- struction of an analytical result. Further, I am not able to say whether the Brussels Academy will grant the use of the compasses in the required construction, and Mr. Weddle does not show that his can be effected without them. 280 REV. T. p. KIEKMAN ON LFNEAR CONSTRUCTIONS. What mathematicians have been so long looking for, is a property in solid geometry like the celebrated theorem of Pascal. By this, if any five points be given in aplane, and any line through one of them, we can determine by the joining of given points, that is, with the ruler only, where the given line cuts a second time the conic which passes through the five points. Pascal's theorem requires only a ruler; but it is important to observe, that there is no limit to the length of that ruler. If the given points are 12 3 4 5, and A 1 be the given line, Pascal's theorem teaches us, considering the hexagon 12 3 4 5 6 — 6 being the sought point in A 1 — to produce the line 34 to meet A 1, and 12 to meet 45 ; then, through the points (A 1, 34) and (12, 45) thus found, to draw a line cutting 23 in a point from which a Hne drawn through 5 will cut A 1 in the point 6 required. If now it happens, either that A 1 is parallel to 34, or that 12 is parallel to 45, the line cutting 23 cannot be drawn with a ruler of finite length, since one of its two points (A 1, 34) and (12, 45) passes off to an infinite dis- tance. Thus, Pascal's theorem itself can be shown to fail unless an infinite ruler be conceded, that is, unless it is granted that a line is given by its direction and one of its points, or that we have the power of drawing parallels. Nor is it any answer to this claim to say, that we can choose another hexagon. For how far are we to pursue the lines Al and 34, in order to convince ourselves that they never meet ? Let but a ruler of imlimited length be granted me, and I will show how to effect, by its aid only, and from purely geometrical data, the solution of a more general question than the Brussels prize question, namely, this problem of linear constructions : — Any locus of the N'^ order (or class) being given geometri' tally y by the requisite number of points (or tangents) of which n-\ are in a line (or pass through a point), to find the N"" point REV. T. P. KIRKMAN ON LINEAIl CONSTRUCTIONS. 281 in that line (or the iV'* tangent line or plane through that point), upon the locus, and this hy the aid of the ruler only. By the same method can be found the value of a given ex-local function F Qci yi z^y whether a?, 3/1 and z^ be the co- ordinates of any assigned point, or of any given plane: i. e., it can be determined whether or no {as\ yi z^ satis- fies the equation F (x y z^zizO. We shall, in the first place, exhibit our data in a conve- nient form. Any axes and origin being chosen, we can by hypothesis draw Cartesian co-ordinates through oiu" given points. Let us consider, by way of example, the surface of the second order. Putting x^y^ z^ x^ y^ z^, &c., for the co- ordinates of the nine points, 1^ 2, &c., and a^o ya Zq for those of any tenth point, we shall first frame a paradigm of the surface, which may be compendiously represented tlius— :2 ■\;^ x^ . y' . z^ . ocy . yz . zx . X . y . z , 1 zz 0 : the terms of this paradigm are* in number 10 . 9 . 8 . 7 . 6 . 5.4.3.2, and are all formed from the first term -^ x^ . if . 2^ , xy . yz . zx , X . y . z . 1, 0 t 2 3*3 4155 67 8 9 by permutation of the sub-indices alone ; the sign of any term being determined by the simple rule, that if it is made firom the first by an odd number of transpositions of single pairs of sub-indices, it shall be negative, and positive when that number is even. Thus the terms -^ a^ . tf . 2^ . xy , yz , zx , X . y . z. 1 3 I 2 00 44 65:6 7 » 8 •— 0^ . y^ . z* , xy . yz . zx , X . y , z . 1 3 I 8 4400 55 6 79 8 have the signs which are prefixed to them. The cyclical permutation of an even multiplet involves always a change of sign, this being effected by an odd number of transposi- tions of single pairs, while that of an odd multiplet leaves the sign unchanged. The former of the two last vTitten terms is made fi"om the first by the cyclical permuta- tion of two even multiplets, the quaternion 0 12 3, and the duad 8 9, either of which permutations alone would 20 282 REV. T. p. KIEKMAN ON LINEAR CONSTRUCTIONS. have changed the sign ; the latter is made from the former by the transposition of the pair 0 4, and it will be found deducible from the first term only by an odd number of single transpositions. The paradigm, being formed, is either a purely geome- trical datum, or an analytical one, according as we define ^1 yi ^u y . z . 1 rr 0 0 1 a 3314 55 678 9 is the equation to the surface of the second order which passes through the nine points 1, 2, 3,...9 ; for it is plainly of the second degree in x^y^z^, and it vanishes when for x^y^z^ we put the co-ordinates of any of the nine points. For let {x^ y^ Zq) and (a*, y, zs) be the same point. The first term is now identical in value, but not in sign, with the term — ^ • y? • -sf • ^q^o • yt^i . ^6^5 .Xa.yj.Za I9, which being formed fi*om the first by a transposition of the pair 0 3, stands in the paradigm with a negative sign. And thus the whole equation is reduced to a system of internecine pairs of terms. We are thus in possession of a certain and surprisingly simple method of writing down at once the equation to a locus of any order (or class), in terms of the defining points (or tangents), provided that their number does not exceed that of the constants in the general equation of the iV^ order (or class): and, what is of much importance, we have the equation not only without the labour of elimination, but free from all superfluous terms and factors ; while the most compen- dious expressions which modern geometry employs to re- present curves and surfaces, would, if written out at length in terms of the co-ordinates included in the factors, be generally found to contain an enormously disproportionate amount of superfluous terms. HBV. T. P. KIEKMAN ON LINEAR CONSTEUCTIONS. 2$3 Generally, if n be not greater than the number of con- stants m the general equation of the N* degree, the expres- sion • bo •'b'c' a"b"o" «, b, e, o o o 2 + xyz . «yz . xyz x y z , x y zznO 000 111 S28 n-I n>l n-1 n n n is the equation to a surface of the N*^ degree passing through the n points x^y-^z^^ XiViSn &c., where the indices ahc are any positive numbers, zero included, different or alike, as are also a! h' c', &c. ; provided first, that at least one set of three has a sum a -\-b -if c -^l^i secondly, that no set (except the last) be all three zeros ; thirdly, that no set has a sum greater than N ; fourthly, that in no pair of sets a'h'c', and aj),c,f we have at once a' zz a,, b' = b,, and c' zzc,; and, lastly, that the indices are not reduced to a a^ ... alon* — to 6 5^ ... alone — or to c c^ ... alone. In the same manner, we can write out at once the equa- tion to a locus of any degree for a geometry of four dimen- sions, the defining points being (x^ r/i Zi i/7,), (xs 2/2 ^2 W2), &c. ; or with equal facility for a geometry of any number of dimensions. Thus, for example, 2 + iK^ . y, . Ij ::r: 0, is a parabola having for diameter the axis of y, and passing through the points 1 and 2 ; 2 ^j a^o yo • ^i ^^ ^ is an hyperbola through 1 , whose asymptotes are the axes ; 2 + Xq 2/o • ^''i • 3/2 • I3 is an hyperbola through 1, 2, and 3, whose asymptotes are parallel to the axes; 2 ± (4+2/S) • a?i • ^2 • I3 ==0 and :s±(afo±fo).li=zO are conies referred to equal conjugate diameters which are parallel to the axes; these curves are either circles or equilateral hyperbolas, if the axes chosen are rectangular. The additional terms are in every case to be formed by permutation of the sub-indices So far as I am aware, this is a view of equations to geo- metrical loci which has not been given before. I know that the shape which the results of elimination must assume, is no secret to analysts; but has this simple mode of stating 284 .REV. T. p. KIPvKMAN ON LINEAR CONSTRUCTIONS. these geometrical results, so easy to remember, and so easy of demonstration, been previously laid down ? We return to our paradigm 2 + a?o . y? . ^^| . x^y^ . yiZi.Zf^oik . x^ yT .z^. I9 zr. 0, considering it as a purely geometrical datum. It is necessary that we should both interpret and prove this proposition (2 m 0), without borrowing any aid from arithmetic, and that we should show how a surface of the second order is given thereby, and can be therefrom con- structed. The interpretation is more easy to be given than to be understood. Every one can conceive the reality denoted by xy, a parallelogram having a certain angle, or that re- presented hjxy Zf a. prism whose edges have given inclina- tions. But what geometrical entity is x y z w, or a; x 2 y? It is assuredly a volume or solid of four dimensions ; for the product of four right lines can be only a figure of some kind, which does not straightway become an absurdity because the inhabitants of this planet find it difficult to imagine its existence. In like manner *•? . yl , z\ . Xzy^ . ytZ^ s^^ .Xe.yy. Zs, the product of fifteen right lines, represents a volume of fifteen dimensions, and the proposition before us (2 zn 0) asserts that a given number of such volumes, constructed of course in space of fifteen dimensions, and having edges equal to certain lines given in common space, viz., the co-ordinates of our ten points, 1 2 3... 9 0, have a sum equal to zero. If the reader feel distressed with the effort to imagine such transcendental volumes in space of more than three dimensions, that is no affair of mine ; my duty being, not to supply him with additional senses, but with sound argu- ments, of which he is competent to judge with even fewer than five. Had the reader been so unhappy as to enter this world deprived of the sense of touch, he would probably have been as much in the dark about the geometrical im' port o{ x . y . z, as he now is, being endowed with only five senses, concerning the real existence of these solids of fifteen KEV. T. P. KIRKMAN ON LINEAR CONSTEUCTIONS. 285 dimensions ; yet he might not have been less qualified to judge of the logic of tri-dimensional geometry, nor less pro- fited by the devices which it employs for the solution of problems in a plane. I beg the reader to believe, that to the mathematicians in the planet Mercury, the outward apperception of these entities is a very different affair Jfrom what he finds it now. All that is incumbent on me is to show, that the propo- sition 2 + al.yl.zl ,x^s . y^^ . z^x^ . ^g . y^ . xrg .I9 rz 0 ex- presses the law of the solid locus of the second degree. I proceed to prove geometrically that every point x^y^^Zo^ which by its co-ordinates satisfies the proposition, lies in a continu- ous locus, such that no right line can meet it in one point only, or in more than two. It is conceded readily, that in a product of lines, as in a product of numbers, the order of the factors is indifferent, so that x^ .y* . z^ .xy .yz .zx.x.y.zmx^.x.x.x.yz.yz.yz.yz. yzzziA. 0 1 a 33 44 55 6 7 8 0 3 6 6 I a 1 a 4 4 3 s 7 3 Let O be the origin, and taking any three points on the positive axes, ^v, y, z^ let Ox -I, Oy — n. Oz — l. The Cartesian co-ordinates being drawn, we have in the plane of yz the parallelogram y^z^, having an angle at O. Join the extremity of y, to z, the extremity of ^ ; and from that of z^ draw a parallel to this joining line^ cutting the axis of 2/ at a distance e from O. We have plainly Z, : z^ zzyjie, or the parallelograms Z^e and y^^s are equal, e being a length cut off from the origin on the axis of y, and positive or negative according as y^ and z^ have like or unlike signs. By drawing four more pairs of parallel lines, after the manner of the pair just drawn, we can reduce A to the form A zz a^ . yf . 4 . x^yz . y^i . z^x^ .x^.yj . Zs=z xl. X3.Xi.X6 , te . ^«i, . Z,e . Z,e3 . Zfit, e «i €2 €3 64, being lengths cut off from O on the axis of y, and positive or negative as the case may be, by the drawing of the described five pairs of parallels. 286 REV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. What we have done in the plane of y 2^ by the aid of our »-uuit t,, we can imitate by drawing pairs of parallels, in the plane o£ x y, by the aid either of jj or ^; for Xo ei, x^ «„ &c., are given rhomboids in that plane, having each an angle at O. Three pairs of parallels will effect the trans- formation, Azz H^xl . e . Ci . x^t . Xrje^ . x^i zz t^i^ . icj . ecj . XjXjXj, in which ofsCg zz 9}x„ Xffii zz JJX2, x^i zz 9)X3 ; and a fourth pair gives A zz ^jj* . a^fl . Xi Xs X3 ; which, by three pairs more, and the equations ex* zz ffj, ^5X3 ir ^e^, ^5X3 — ^y, becomes A = l'rtle,^,i^^o zz ^n'^e,x^l zz l^ri'^Ya^ i X1X2X3 being lengths cut off from O on the axis of a?, and e^ e^ and Y lengths on that of y, where Y is positive or nega- tive with A. Thus, by drawing twelve pairs of parallels, we effect the reduction Azzioi^,!/* ,!^ .xy ,yz.zx ,x .y,z .Izz Cv*^ . x*o, t a S3 44&S6;89 The term, B :r: — a;2 . y? . 2? • ara^s . y^^ . z^x^ - Xe-y^'Zj.l^ zz ^>3*^'Y^ a^. is reduced in the same manner, Y, being a length from O on the axis of y, and positive or negative with B, the sign of which depends on the co-ordinates by which that solid is determined. Let B be supposed positive,- then the addition A + B=:^^^f (Y + Y,)«i is to be performed. If c and c^ be the extremities of Y and Y, remote from O, and be the line intercepted between the axis that cuts off Y r;z Og ; draw cb, parallel to the axis of a? to b, in b b, parallel to that of y; join bc^j and parallel to this draw b^ e^ to c^, in the axis of y; the length Oc„ n Y -}- Yi. This requires the three lines bc^ cb, be,,, be and bb, hav- ing been drawn before. The subtraction ofY, from Y might have been performed with equal facility, giving Y-Y^, a length cut off from O. The sum of all the terms of the paradigm which contain xi» can thus be reduced to a single term ^|^»;* K wl, in which BBV, T. P. iilBKMAN ON LINEAR CONSTEUCTIONS. 287 K is a length found by the txihr only on the axis of y, and positive or negative with the sum of the solids of fifteen di- mensions, which is thus proved equivalent to one single solid of the same dimensions; an equivalence rigorously geometrical, and of the truth of which the reader can judge for himself, although he has my word only for the existence of such solids. The term C z:z:i^ .'i^-z^-xy.yz.zx.«.y.z.\ zzix.x.x.x.x-yz.yz-yz.yz.yz 618 3341A50789 «66IS>2iSt«3&78 can be reduced by drawing thirteen pairs of parallels, to the equivalent C n: ^^S* a*o Yj which is a solid equal in geometrical value to C. The whole of the terms contain- ing iCo can thus be collected into one term, X^^r^ %, Xq. Let us suppose that we are in quest of all the points (^oy© 2^0) in any line, passing through one of the given nine, whose co-ordinates x^ y^ and z^ satisfy the geometrical con- dition, 2 + 4 . yf . 2I . 3:3^3 . y^Zi . 2-5X5 . ar, . ^7 . sTg . I9 = 0. Let the line pass through the point 9; let this be taken fcMT the origin, and the hue for the axis of ar. We shall have ajg ^i ^9 n: 2^9 zz 0 n yo =^ ^jg in our paradigm, which will thus contain no terms free from X(^^ The proposition before us is then equivalent to a:oer^*{Kr„ — ^K,)=0, which is satisfied by all the points which we are seeking on the axis of a?, K and Ki being lengths on that of y from O, found with the ruler only, and ^ being the a:-iinit, arbi- trarily assumed 'on the axis of a?. There is plainly only one such point, which is found by joining the extremities of § and K, and then firom the extremity of K, drawing a parallel to the joining line: the parallel cuts the axis of » in the point required. If K, zz o, the axis of a? is a tangent of the surface at the origin. We have thus proved that our paradigm represents a surface, for it is the locus of a point which can be assigned more than once upon any line 28& REV. T. p. KIRKMAN ON LINEAR CONSTRUCTIONS. through one of the point's positions; it is also a surface of the second order, since no line can meet it once only, or more than twice. That this locus passes through all the nine given points, is proved by the same a' gument when the paradigm is treated as a purely geometrical datum, as that which has been adduced when we considered it as an analytical datum. The expression, namely, is zero, whenever the tenth point 0 coincides with any of the given nine; for the solids that form it destroy each other in pairs. If Xq 3/0 2-0 be any tenth point on the surface, the term A can be reduced by the drawing of fourteen pairs of parallels to the form A n: X^-i^^ x ; and every term in the paradigm can be by the same labour reduced to the same form ; the addition of the lines x x^ x-j, x^j &c., thus found on the axis of X must give the result. r^rf^' {x + a:, +. a:, + X3 + ... ) = 0, which is the condition, both necessary and sufficient, in order that the ten points, 1 2 3... 9 0, should lie in a surface of the second order. The addition of these lines x j^„ &c., is effected by drawing certain lines, in sets of three each, the last of which lines will pass through the origin. If, then, any eleven points in space be taken, the condition that any ten shall lie on a surface of the second order is, iliai a line, found by the ruler only, shall pass through the eleventh. This is a })urely geometrical solution, by the aid of the ruler only, of the Brussels Academy's prize question. The found line is one of an infinite number forming a pencil through the eleventh point — the particular line depending on the axes drawn through that point. The paradigm of the general surface of the n^^ order can always be transformed by drawing of given lines with the ruler — the axis of x being drawn through the point (xq, 0, 0), EEV. T. P. KIKKMAN ON LINEAK CONSTBUCTIONS. 289 whose relation to the surface we are examining, to the form H{Y^+^Y,a:-»H-fY,a--»+ 4. ^^Yx + fYJ=rO, where H is a factor that may be disregarded, and Y Yj Y,... are lengths from o on the axis of y, and ^ is an arbitrary length on that of oj. If now n-1 of the points that define the surface are on the axis of ^, the w* point in which that axis meets the surface, is found by drawing a small number of additional parallels. The expression before us — which, like the paradigm of which it is the reduction, is a purely geometrical datum and proposition — is an equation whose co-efficients and roots are not numbers but lines. Since the addition and multiplication of lines are subject to the same laws of aggre- gation and commutation with those of numbers, there is no reason in the world why the doctrine of equations, to a certain point at least, should not be property common both to arithmetic and geometry ; so far, namely, as the symbols in the theory of equations retain their perfect generality. By this theory, we know that S, the sum of the roots of the above equation, is given by the proportion ~S:|=:Y,:Y, and — S is found by the drawing of two parallels. It has been already shown that the addition and subtraction of lines from lines on the axis of x can be effected by the ruler; hence the sought point is obtained by subtracting from -j- S, thus constructed, the known sum of the other n-1 roots of the equation. All that is proved above concerning loci of the iV* order holds for constructions in loci of the N^ class ; if the co-ordi- nates, X y z, &c., determine not points, but lines or planes. If the axes of the line or plane co-ordinates are parallel axes, as they may be if we so choose, I think it will be found that all the constructions in question will be effected 2p 290 EEV. T. p. KIRKMAN ON LINEAB CONSTRUCTIONS. by the drawing of given converging lines ; so that a finite ruler will in general solve the problems. I readily allow that these linear constructions, although they will, as I flatter myself, be found rigorously geometri- cal, are far from being reduced to their most simple form ; and I could state, if space were allowed me, methods of abridging the operations indicated. If the locus under consideration be represented not by Cartesian co-ordinates, but by a system of terms, each being a product of linear functions of w y and 2:, which represent distances from given lines or planes measured in a determined direction, a slight modification of th& above method of using the ruler, will often bring out the required result with comparatively little labour. I shall content myself with one example of a more compendious method, forming the solution of a problem of remarkable interest; to find the ninth intersection of two curves of the third order through eight given points. The principal difficulty in the solution of the ninth point problem, lies in the finding linearly a fifth known point on each of the two conies through the points 12349 and 82349, (vide page 83 of the 6th vol of the Cambridge and Dublin Mathematical Journal), [123458] [123467] (67) (58) — [123457] [123468] (68) (57) = 0=z (12349), [123458] [823467] (67) (51) — [823457] [123468] (61) (57) zzOzz (82349) ; where [123458] is the integral function of the co-ordinates of the six points 123458, which vanishes if they are on a conic; the aconic function [12345 8'], as it has been denomi- nated by Sir W. R. Hamilton, and (67) =1 o, denotes the integral equation of the line through 6 and 7. A fifth point on the conic (12349) is the intersection of the two lines [123458] (58) — [123457] (57) = o rz m [128468] (68) — [123467] (67) = o = v. EEV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. 291 I shall proceed to draw these lines by the linear construc- tion of the aconic functions in their equations. Putting 0—3/ iyx—yx) — (y—y) (yx—yx) 1 8 45 54 i 5 12 2 1 __ ^IJ y — y . X — X — y — y x — x IS45 4512 and .2:" for what this becomes when yi is exchanged for as^ &c. ; the area of the Pascalian triangle, having its angles at the intersections of the opposite sides of the hexagon, 123458 is, A = y . (x—x) + y (x—x) -j- y (x—x) zz C : D, 13 23 84 23 34 12 34 13 23 45 58 81 68 81 45 81 45 68 C being the aconic function [123458] and D being the factor (y-y.x-x — y-y.x-x)(.y-y.x-x — y-y.x^(:,y-y.x-x — y-y.x^) 1246 4612 3368 6823 3481 8134 which vanishes whenever A becomes infinite, i. e., when any pair of opposite sides of the hexagon 123458 are pa- rallel. ITiat C rz 0 is the equation' to the conic when this hexagon lies in one, is evident from the consideration that it is of the second degree in (xgy^), and thafit vanishes if for (^sy«) you put any of the other five, («,y,) (^23/2) (x^ys) ... If (arsyg) is {xiyi) or (x^y^), this is instantly seen ; and if (xsy^) is (xiys), the triangle A has all its angles in (12) ; if (x^g) is (xtVi)i it has all its angles in (54) ; and if {x%y^ is (x^y^, two of its angles coincide with the point 3, in none of which three cases does D become zero. Wherefore, if the points (1245), (23*58), and (34-81), be a, c, e, [123458] =: \y ' x—x -\-y . x—x -f- y . x—xl D, La o e o e a a a cJ ri23457l — \y ■ x—x +y . x—x + y . x—xl D^, i-a c; e,' o, «, a ' «/ • ojJ where c^ e, Z>, differ from c e D only by the exchange of 8 for 7. Neglecting the common factor of D and Dj, which is free from 8 and 7, we have to draw the line { (y . x-a; + y . x-x + y . x-x) (y-y . x-x — y-y . x-x) {y-y . x-x — y-y x-x) . (58) i «eeoeaaae2368 68233481 8134 ( . — — — — r-o — (» i X-X + y . x-x + » , jT-ar) (y-y ■ x-x — y-^ . x-x) {jf-y • *-a' — V-y n^^) . (57) V a G/ e, e, •>»•/> e, ^8 367 6783 3471 7134 J 292 REV. T. p. KIBKMAN ON LINEAR CONSTRUCTIONS. or, taking for our axes of x and y, the lines (34) and (32), putting thus X3 =: 1/3 =z Xi =2 yiZZx^z=i a:J = ye = y? = Oj (y . x-x — yx) (ti . x^)(f^ , x) (58) — (y .x^—yx) (y . x^) (pg . x) (57) =r. « =- o, o a a ae 2 C 8 8 1 4 c/ e; a a e/ 2 S 7 7 1 4 is the line which is to be drawn ; where a is the point 12 . 45, c is the point 23 .58, e is 34 . 81 ci 23 . 57, ^1 is 34 . 71 Let the co-ordinates of the eight points be drawn, which amount to twelve lines besides the axes, and on these let any two positive lengths from 0 the origin, as y^ rz. k^ x^zn m, be assumed. Let that diagonal of any parallelogram made by the co- ordinates produced, which produced cuts from the axes seg- ments of like sign, be called positive, and negative when those signs are unlike. The rhomboid k . x^ — x^ is a determined portion of the figure ; and has a side w^ — x^ in the axis of a;. Draw that dia- gonal d of it which has the sign oixe-x^, and parallel to d from c, draw cAto A in the axis of a;; then is 3/0. (a?, — x^):=zk, OA. Draw the ordinate AAi, meeting a?, in Ai; draw the diagonal di of the rhomboid k x„ which differs in sign firom x^ and parallel to di draw A,X to X in the axis of x : then is X'. AX zz. — y^Xc and ^.OX zz y^ . {x^-x^ — y^ x^. Draw next J2 the diagonal of ^ . (x^ — x^) like-signed with x^ — Xs, and parallel to d.^ draw from 2 to Xi in the axis of x the line 2 X, : y, .{x,'-'X,)=.K.OXi. Parallel to 24 draw a line Oi B, meeting on the axis of y at O, one of the abscissoe x^ and Xi so as to cut the other in B on the positive or negative side of that axis, according as y^ — yi is positive or negative. The ordinate BX, will cut off OX2 such that K, OX2 zz y^ — y, . ^4. Thus by drawing the ten lines d^ c Ay AAxy di, AiX, d„ 2X1, 24, OiB, BXif we have effected the reduction (y . x — X — yx) (y . x — x) y — y . x zz ^ OX OX OX. ceaae3588Il 13 Join now 2 X„ and draw the lines Xk^^ k,X', X'k^ k„X" ; 'Xk, and X'k' both parallel to 24, and hX' and k^^X" in ordex fiEV. T. P. KIRBLMAN ON LINEAE CONSTRUCTIONS. 293 parallel to 2X, and 2X2; k, and k„ being in the axis of y, and X' and X" in that of a. Then since m: OX=z OX : OX', and m : OX' = OX, : OX ", OX . OX, . OX, = w» OX", and (3/0 . a?e — a?a — ^y» a^o) (^2 • a?6 — ^^g) ?/8 — '^i . a;* =: P m* OX"; a transformation effected by 15 applications of the ruler, when the Cartesian co-ordinates, and the four lines determining the points a e Cy have once been drawn. Let now 57 and 71 be drawn to c, and e, : let the diagonals of k cc,^ — x^^ h x^ — x^, like-signed with x^^ — x^ and x^—^Xj, and that ofk x^, of the sign contrary to that of a;,^ be drawn. Four parallels and two ordinates, after the manner of c A, AiX, 2X„ OiB, AAi BXi above, will suffice to determine the points X Xi X2 such that (y . X — X — yx) Cy . X — x) (y — y . x) 11: A'. Ox. Ox Ox ; 0/6/* a 0/867714 18 and by drawing five lines after the manner of 2X2, X^j, ^-jX^, X'^^,, It^Xl' above, we shall obtain the point x" on the axis of x such that (y . X — X — ya:) (y . x — x^ {ju — y . a;) rz J^m^OTi!'. 0 0% ae3S7714 II I consequently the line to be drawn is OX." (08) ~ Ox" 57 = o rz «. Join 78; let 8x" meet 7X" inp; let Op meet Xj in qi let qr parallel to 8x" meet 78 in r ; then 5r is the line required. This is constructed at the expense of 40 applications of the ruler, besides the drawing of the Cartesian co-ordinates of the eight points. It is necessary in the next place to find the line iV .x^ — yx)(p . x^) (f:^ .X) (6S) — (9'X^—yx){y.x^ (jPya:) (67; = o =» deb be368 814 deb be867 714 111 11 where 6 = 12.46; d= 23.68, e =z 34.81 d — 23.67, e, r= 34.71; By drawing, in addition to the lines, 46, 68, 67, the dia- gonals of k jc, — ajb, k Xe — Xa, with proper signs, then tliree par 294 REV. T. p. KIKKMAN ON LINEAR CONSTRUCTIONS. rallels and an ordinate, after the manner of c Ay AiX, 2Xi, AA|, yre obtain two points X3X4 such that it/ . x^^ — yx) {y . x—x) (y^ . x) =1 k" OX OX OX. deb bea68 814 41 3 Four lines more, setting out from X^, after the manner of the four Xk,f hX', Xk^^ k^^X", will determine the point S on the axis of x such that OXi OXz OX^ zz w' OS, or (y . X — X — yoc) {y . x — x) {y — y . x zzz khn'OS. deb be26 88 14 Next, the co-efficient of (67) in v :zi o can be reduced by drawing the diagonals of A (xl — x\), k (xe — a,), and then as above three parallels and an ordinate, to A' Os^ Ox^ Ox^, which four lines now will transform to A' m^ Os, s being a point found on the axis of x. Thus twenty-three new lines are expended in transforming the problem to the shape OS . (68) ~ Os (67) =:o=zv, and r in the sought line 6r, is obtained by drawing other four. Draw now 5r and 6r^ to meet in P; and we have found a fifth point of the conic ( 12349) by 67 applications of the ruler, if the Cartesian co-ordinates are drawn before- hand. We have yet to construct the pair of lines C234581] (51) _ [234587] (57) = o = u, , [234681] (61) — [234687] (67) =: o =: Vj . Now[234587]=zl>e(^e,-^aj+ye„(^a-^o;+yaX^o-a?e.)] .DS ai being (72*45), e„ being (34-87), and D^ being the quantity (y-v . x-x — y~y . x-x) . (y-y x-x — y^ x^) .{y~y . x^ -— y^ x^'), 83 58 68 2. 3 348 7 8734 45 72 7246 Omitting from D and D^ the first of these three factors, the line Mj zz 0 is p. x-x— yx) (y-y . x-x + yx) (f^ . x) (51) — {y . x^-r yx) (y^.x^ + yx) (j^-y . x) (57)==o c e a a e 1 2 4 5 5 1 8 I 4 o ei, ai a,e„ 7245 67874 Draw 72 and 87, to find a and e on 45 and 34. Draw the diagonals of ^ (Xi — iCs), kx^^ k Xc^^ — x^^, Iix„ having the signs of these areas, and that of ^a^e,/ having its sign opposite to that of Xeif. A parallel to the first of these five diagonals meeting the axis of y on 07ie of a7i, 0:2, as the case requires, will intercept ftEV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. 295 At F on the other an absciss having the sign of y, — y^. Draw the ordinate FF^ meeting a?5 in F'', and next F^'Xj parallel to the second diagonal to Xs in the axis of x. then is k . O X = k. (0F'» + F"X) =: ^ . ^^x +'^x, i .6 13 4 6 6 1 and (y • x—x—i/ x) (^yl^x^y x) (^y . x) — k' ox . OX OX Draw now from c a parallel to the third diagonal, meet- ing the axis of x in G, and from the intersection G, of the ordinate GGi, with oe^a draw Gjx' to x' in the axis of ar, parallel to the diagonal of Icx^,,, the last of the five. We have thus y^ . ce^,—x^, — Vf^x^,, =.h.{0 Gi+ Gix')=: ^. Ox*. Next let a parallel to the first diagonal meet either Xy or x^ on the axis of y, intercepting at H, on the other, an absciss having the sign of y? — yg > ^^^ from H', the intersection of the ordinate HH^ with a^j draw a parallel to the fourth dia- gonal, to x/ in the axis of x. Then is Jc . Oxx' zr k . (OH -f-H'xi') n: y^ — y^ . x^ — x^ + y* «?; and by drawing a parallel to 24, and an ordinate meeting the axis of x in x'a, we obtain (y,.a:e,-^a— ya/aJeXy?— ^2— ^4— ^5)_(y7— y.-«0=^- Ox', Ox',. Ox', By drawing, in addition to X ¥ before drawn, the four 2 Xs, ¥X,, X,k\y k\X,, XeA-l, parallel to 24 like Xk\ k'Xe to 2 Xj, andAjjXy to 2X2, drawn before, we effect the reduction, /P. OX . OX, OX, = F wi» OX; Drawing next 2x} and 2xJ, and a broken line of four strokes, beginning at x' and ending at Xj, we obtain ^. Ox\ Ox}. Oxa zr A^ m\ Oxj; and the line to be drawn is now OX, . 51 — Oxs 57 = 0 = w, . Join 17; let 1 Xj meet 7 X7 inp'; let Op' meet/r, in q^; let q^r^y parallel to 1 X5, meet 17 in r' ; 5 r* is the line required. The point r' is found by drawing 32 lines additional to the 67 already drawn before. The line v^ is (y.x^x—v.x)(y^ Jx^x + ]/x) (y^y . a;)(61) — (y. z^ — ya;) (y^. xH + yx) (y^*) (67)=o eiita ae 1346 61 81 4 Ciieiai aie|724a 67 874 296 BEV. T. p. KTRKMAN ON LINEAB CONSTBU0TION8. The drawing of 14 additional lines will reduce this to k\ OXs. OX^ 0X3 (61) = k^ Ox., Oxj Oxf, (67) ; twelve more lines give us the reduction, ^=» . OJTe • 0X9 . OXj = k^rn^ OT k* . Oxe . Oxj . Oxg = kW Ot. and the line or. (61)— 0^(67)rro = r, is given by drawing other four lines. We have thus found a fifth point of the conic (82349) by drawing 32 -}- 30 lines additional to the 67 lines ex- pended in finding a fifth point on (12349): in all, 129 lines. We can now proceed, after the elegant method of Mr. Weddle, to find the point 9 by five applications of Pascal's theorem, so that the mystic enneagram is completed by at most 150 applications of the ruler, after the drawing of the Cartesian co-ordinates ; a process which will be esteemed simplicity itself by those who have attempted to express x^ and ^9 in terms of ajiyi, &c., or even in terms of their nume- rical values. Note. — Since this paper was read, there has appeared in the Cambridge and Dublin Mathematical Journal of this year, 1851, a solution of thia problem of the ninth point, with the ruler only, by the Rev. A. S. Hart, F.T.C.D., which may be pronounced perfect, and which for elegaooe and simplicity leaves nothing to be desired. 29: XX. — On the Analysis of Gaseous Mixtures. By JoHN Leigh, Esq., M.R.C.S., F.C.S. Head January 7, 1851. The chief object of this paper is an examination of the methods hitherto employed for determining the constitu- ents of coal gas. Of late, considerable attention has been given by chemists to the products of the distillation of coal, as well as to those of its combustion, and of its spontaneous decomposition; and it is likely that their researches will erelong come to have a high value, and will throw much light on the great process of gas manufacture, on the eco- nomical employment of fuel as an agent in the production of steam, and in the great smelting operations of this coun- try ; and also in explanation of the production of the fire- damp, whose disastrous explosions so often occupy public attention. I shall confine myself to an examination of the methods employed, analytical or otherwise, for determining the com- position or the illuminating power of the gases resulting from the distillation of coal. There are few gaseous mix- tures that have offered more difficulties to the chemist than those which make up the composition of coal gas ; and as the time has now arrived, when a correct mode has become eminently desirable of ascertaining the proportions and constitution of the light-giving ingredients contained in the mixture, so that it may serve as a measure of the relative illuminating power of the gas, I will give a description of the method which I have been in the habit, for several years past, of employing in the examination of the gas produced at the Manchester gas-works. 2q 298 MR. JOHN LEIGH ON THE A brief statement of the constituents of coal gas, as de- termined by numerous analyses, without reference to their relative proportions, by affording an idea of the objects of the analytical enquiry, may best precede an enquiry into the nature and value of the methods employed. The gases eliminated from coal or cannel consist essen- tially of olefiant gas, volatile arid condensible hydro-carbons, light carburetted hydrogen, hydrogen, carbonic oxide, car- bonic acid, sulphuretted hydrogen, sulphuret of carbon, cyanogen, ammonia, and aqueous vapour. Of these the carbonic acid, sulphuretted hydrogen, and cyanogen, are positively injurious to health. The two latter, and ammo- nia, are injurious to the apparatus employed for the distri- bution of the gas, and these, with the sulphuret of carbon, to the furniture in rooms in which the gas is burnt. The nitrogen and oxygen generally found In gas, proceed from atmospheric air, which gains access by leakage of apparatus, and by opening the retorts when charging with fresh coal. In the process of purification now generally adopted at all well-regulated gas-works, the carbonic acid, sulphuretted hydrogen, cyanogen, and ammonia, are en- tirely removed ; so that there remain olefiant gas, illumina- ting hydro-carbons, light carburetted hydrogen, hydrogen, carbonic oxide, and a minute portion of sulphuret of car- bon, not recognizable by chemical tests in small quantities of the gas. Of these the hydrogen and carbonic oxide, though giving out much heat on combustion, yield scarcely any light, and bum with a very feeble blue flame ; they only dilute the gas, adding nothing to its illuminating power. The light carburetted hydrogen burns with a yellow flame ; the olefiant gas, and condensible hydro-carbons, with a very brilliant white flame, and give to the gas its chief illumina- ting power. The entire illuminating power of the gas, then, depends on the olefiant gas, hydro-carbons, and light carburetted hydrogen; the richness and value of a gas ANALYSIS OF GASEOUS MIXTURES. 299 may be determined by the proportions in which these exist in it, and a correct determination of their proportion and constitution will afford a correct and true measure of the quality and value of the gas, and a test of the excellence of the process by which the gas has been formed, as well as of the value of the coal, canuel, or other substance, used as a gas-producing material. Gas-makers have been much in the habit of relying on the weight or specific gravity of gas as an indication of its quality. The heavier the gas, and, it is said, the better is its quality. This notion has arisen from observing, that olefiant gas is one of the heaviest of the constituents of coal gas. But the specific gravity is not to be depended upon as a test of the excellence of the gas, and in any case could only give a very crude idea of the general quality of it, without giving any knowledge of the nature and propor- tion of its constituents. The specific gravity of carbonic oxide is 967*8, and 100 cubic inches weigh 29*83 grains, at 60*' Fahrenheit and 30 inches barometer. The specific gravity of olefiant gas is 985*2, and 100 cubic inches weigh 30*37 grains. The specific gravity of light carburetted hydrogen is 559*6, and 100 cubic inches weigh 17*25 grains. Now it is evident, that a gas containing much carbonic oxide, very little olefiant gas, and an inferior amount of light carburetted hydrogen, and consequently very poor in illuminating power, may weigh heavier, and seem better, than gas of far higher and better quality, so that the weight is only valuable as an adjunct to analysis. The measurement of the light by the eye, whilst, like the above, aflFording no index of the constitution of the gas, is open to many irregularities and fallacies. There is no con- stant means of comparison, and after a few trials the eye fails to appreciate any but large differences. The next method of determining the illuminating power of gas, by estimating the amount of its constituents condensible by chlorine, 300 MR. JOHN LEIGH ON THE originated with the late Dr. Henry, and has of late been much recommended by Dr. Fyfe. The method consists in mixing chlorine with the gas, allowing the mixture to re- main for some time in the dark, and then observing the diminution of bulk, and estimating half the diminution as olefiant gas. Chlorine has the property of combining with and condensing the olefiant gas and hydro-carbon contained in coal gas, whilst in the dark it exercises no action on the light carburetted hydrogen and other constituents. This is an attempt to estimate the value of a gas by the mere amount of olefiant gas that it contains ; but the method of accomplishing this is liable to constant errors. Formerly the whole of the diminution of volume was observed, and one-half of the entire amount estimated as olefiant gas. It was overlooked that a portion of the chlorine itself under- went absorption, and increased the apparent diminution, thus giving too large a volume of olefiant gas. Dr. Fyfe detected this, and proposed, therefore, to observe first, how much chlorine was absorbed alone by the confining water, and then to deduct this amount, from the total absorption, dividing the remainder as before. But this method is ex- tremely fallacious, and, though still practised and strongly recommended by Dr. Fyfe, is utterly untrustworthy. Dr. Fyfe estimates the absorption of chlorine alone at 1 or 2 per cent, within the time employed for examination. I have found it to vary from 2 to G per cent. The rate of absorption varies with the diameter of the tube employed. But what is of more consequence, and completely vitiates Dr. Fyfe's results, or any results obtained by this process, is the fact, that the rate of absorption of chlorine varies with the dilution of the latter by any other gas. The rate of ab- sorption, when mixed with atmospheric air or any other gas, is not the same as that with pure chlorine alone. Dr. Fyfe has either not tried this, or has entirely overlooked it; and it is as great an oversight as that which he has sought to cor- ANALYSIS OF GASEOUS MIXTURES. 301 rect. I have made a great number of experiments to deter-* mine this point. A quotation or two will serve to show how fallacious the method is. 1. Passed 2*9 cubic inches of nearly pure chlorine, col- lected over water that had been boiled to expel atmospheric air, into a graduated eudiometer standing over water at 50* Fah. The gas measured 1*9 cubic inch. After standing 15 minutes, the gas measured 1'68 cubic inch. This indi- cated an absorption amounting to 22 parts. 2. Passed equal measures of chlorine and atmospheric air into the same eudiometer. The mixed gases measured 1*9 cubic inches. After the lapse of 15 minutes, the mixed gases measured 1*8 cubic inch. The absorption, there- fore, in the same time and under the same circumstances as in No. 1 explanation, was only 10 parts. The rate of variation is not constant. It varies with the purity of the chlorine itself, and with the quality and pro- portion of the gas added to it. Besides, as I shall prove further on, the assumption by Dr. Fyfe, that the gases con- densed by chlorine consist entirely of defiant gas, is not correct; and therefore, even if the objections to the method just stated did not exist, it could only afford a crude approximation to the composition of the gas, and to its relative illuminating power. A better method of determining the amount of olefiant gas, and which was employed by Dr. Henry, and is recom- mended in some of the best analytical works, is to allow the chlorine and gas to re-act on each other, and then to remove the whole of the chlorine by an absorptive solution, determining the amount of olefiant gas by the diminution in bulk of the original quantity employed. This method I have subjected to a very rigid examination. Pure chlo- rine is recommended to be employed; but it has been over- looked that it is almost impossible to obtain pure chlorine over water, and mercury absorbs it so rapidly that experi- 302 MR. JOHN LEIGH ON THE ments cannot be performed over this metal with chlorine. I have prepared chlorine from perfectly pure material, in vessels completly filled v^ith fluid so as to exclude all air, and collected the gas over boiling water; and still, on acting upon it by an absorptive solution, there was always a re- siduum left. I collected the residuum from several opera- tions, and analysed it. I found it to be composed entirely of atmospheric air. Unless the exact proportion of this be accurately determined, it will vitiate the results, as in Dr. Fyfe's experiments, because it must be allowed for, which has not generally been done. But, unfortunately, the pro- portion of this unabsorbable residue varies with every por- tion collected; and, therefore, the examination of one bottle of chlorine does not give the true amount of impurity to be allowed for in that employed for the analysis. From the same quantity of gas (chlorine) I obtained in one instance, •625; in another, '637; in another, "03, as the amount of residuum. I was long puzzled as to the source of this impurity; but at length found it to proceed from the air, mechanically retained or absorbed by the water and other fluids employed in the processes. This was displaced by the more absorbable chlorine, and hence its constant pre- sence in the unabsorbed chlorine, and the constant varia- tion in its quality. Of late, Professor Bunsen of Marburg, finding the difficulty of truly determining the amount of olefiant gas by the means hitherto employed, has proposed and adopted the use of charcoal or coke balls, saturated with fuming sulphuric acid, for the removal of the olefiant gas. This process fully answered for the separation of the illu- minating gases; but the experiments, and the whole of the subsequent analysis of the residual gas, have to be performed over mercury, and require great care in the manipulations; still it is by much the best hitherto proposed. The best method of determining the general value of the gas is also due to the late Dr. Henry, and is the one which I have ANALYSIS OF GASEOUS MIXTURES. 303 generally employed in the analysis of our own gas, and that of the neighbouring towns. In this process, the rela- tive value of the different gases is determined by the quan- tity of oxygen required to effect the complete combustion of the gas. This is done by firing the mixed gases by elec- tricity in graduated tubes, and calculating the oxygen con- sumed in the production of carbonic acid and water from the gas. By a combination of these two methods I have arranged a plan of analysis, by which the general value as well as the constitution of any coal gas can be determined with accu- racy, and the illuminating gases can receive an expression sufficiently high to indicate even small amounts of differ- ences in their proportions. I first determine, by an analysis with oxygen alone, the number of volumes of oxygen required for complete com- bustion, by 100 volumes of the gas to be examined, and the quantity of carbonic acid produced; I then, from another portion of the same gas, withdraw the olefiant and other illuminating gases by fuming sulphuric acid, and determine the exact amount of these ; I then subject a por- tion of the residual gas to examination with oxygen again, and determine the number of volumes required ; the differ- ence in the two examinations gives the amount of oxygen required by the illuminating gases removed by the sulphuric acid, and of carbonic acid produced. In the second portion of the residual gas I determine the quantity of each con- stituent gas, and estimate the amount of each in the whole mixture. This is accomplished in the way generally practised, viz., by exploding the residual gas with oxygen, determining the amount of oxygen consumed, and of carbonic acid produced; and from these data, calculating the proportions of light car- buretted hydrogen, hydrogen, carbonic oxide, and nitrogen. 304 MB. JOHN LEIGH ON THE I append an analysis made, after the manner described above, of gas from a cannel much used at the Manchester gas- works. 100 volumes of Ince Hall (Wigan) cannel gas required for complete combustion 146'5 volumes of oxygen; 100 vo- lumes of the same gas, being treated veith anhydrous sul- phuric acid, lost 8*5 parts, consisting of olefiant and other illuminating gases. The residual gas required 104 volumes of oxygen for complete combustion. So that the 8*5 parts which had been removed by the sulphuric acid, had required 42*5 parts of oxygen for combustion. This gives the number 5 as the expression in volumes of oxygen of each volume of the gas removed by sulphuric acid ; and to these 8*5 parts are due the chief portion of the illuminating power of the coal gas. This also proves what I before referred to, that the illumi- nating gas in coal gas does not consist of olefiant alone, as each volume of this gas requires only 3 volumes of oxygen for combustion ; it is a mixture of olefiant gas with what I would call trito and tetarto-carburetted hydrogen. It is evident that, with so large a multiple as 5 for each volume of illuminating gases, even small differences of the latter in any sample of gas can be correctly indicated and expressed. Below is the composition of the particular sample of gas taken in illustration. Ince Haix (Wigam) Caioiei. Gas. ^ Carbonic Acid 0*78 Olefiant Gas and Illuminating Hydro-carbons) > 8*60 represented by 42'5 vols, of Oxygen > Atmospheric Air 432 Nitrogen 0*19 Hydrogen 41*00 light Carburetted Hydrogen S3-83 Carbonic Oxide 1135 99-97 ANALYSIS OF GASEOUS MIXTURES. 305 I have made complete analyses of the gas from almost every considerable town in England and Scotland, and have examined analytically the gas from the greater number of coals andcannels; and shall be glad on a future occasion, if not objected to by the Manchester Gas Committee, to lay the results before the Society. 2b 306 XXI. — A Description of some supposed Meteorites found in Seams of Coal. By Mr. E. W. Binney. Read May 13, 1851. I . The component parts of sedimentary rocks afford the geologist most valuable data in assisting him to arrive at ■ an estimate of the forces which have been in operation on the earth's surface in very remote ages. Accordingly, we find that the earliest cultivators of geology paid consider- able attention to the conglomerates, sandstones, and slates of the older deposits, as well as to the gravels, sands, and clays of more recent formations. In a paper read by the author before this Society on the 1st day of December, 1846, and printed at p. 148 of vol. viii. (new series) of the Society's Transactions, the mechanical deposits of the coal- measures of Lancashire were investigated at some length, for the purpose of attempting to measure the intensity of the currents of water which brought them to the places where they are now found. At p. 166 is the following extract: ^'As before stated, rough gritstones, containing rounded pebbles of quartz, abound in the lower coal field; whilst the middle and upper measures, reaching to a thick- ness of 4,472 feet, as far as I know, have never yet afforded a piece of mineral matter, in their sedimentary deposits, of the size of a small pea. In two seams of coal, namely, the Four Feet Mine at Patricroft, and another seam under the same mine at Pendleton, I have obtained rounded stones of several pounds in weight ; but as both these speci- mens came from the neighbourhood of great faults, probably MR. E. W. BINNEY ON SOIHE SUPPOSED METEORITES, &c. 307 they may have been brought to the places where they were found by other causes than currents of water. They, how- ever, are interesting, and difficult to account for, being well rounded. Their composition is the same, though found in different seams and at different places, being of a hard crystalline quartz, more resembling gannister than any other stone in the carboniferous series. The outsides of both stones are well coated with a covering of coal, shewing that they must have lain long in the places where they were found." Ever since the reading of the above paper, I have devoted considerable time and trouble in attempting to obtain evi» dence of more stones having been found in coal seams — of course, by stones I don't mean any of those aggregations of iron pyrites and ironstone which are so frequently met with in coal seams, but foreign masses of stone, which must have been introduced into the coal when it was in a soft state, and not precipitations from water, or segregations from the substance of the coal itself, where they had pre- viously existed either in solution or admixture. All my enquiries, however, resulted in obtaining no proof of more specimens having been found in coal seams except the one next alluded to.* In the Mining Journal of the 9th day of November, 1850, appeared the following paragraph: "A large pebble of crystalline or primary limestone,! was found imbedded in the solid coal at the Rhydgaled Colliery, near Mold, on Monday the 4th instant. It is supposed to be * Since this paper was read, the author has had an opportunity of asking W. E. Logan, Esq., F.R.S., director of the geological survey of ^Canada, a gentleman of as great practical acquaintance with coal fields as any geologist of the day, and one who has investigated coal-measures in nearly all parts of the world, if he ever, in his great experience, had met with rounded pebbles of stone in the middle of coal seams, and that gen- tleman declared that he had not met with a single instance — E. W. B. f This stone, as will be seen by the analysis hereinafter given, is not a limestone, but nearly pure alica. 308 MR. E. W. BINNET ON SOME SUPPOSED the first instance known of such a pebble having been found in the coal strata." This convinced me more than ever that such stones were of rare occurrence, especially as none of the readers of that journal, which has an extensive circulation amongst the practical coal-miners of Great Britain, stated in its pages that any such pebbles had come under their observation. In answer to a letter addressed to Mr. Edward Jones, a gentleman who has the management of the Rhydgaled colliery, in March, 1851, the stone was liberally sent to me for examination, with a consent to analyse it, accompanied by the following letter : — " The stone was found by a per- son of the name of Edward Price, on the 4th November, 1850, whilst hewing the coal. It was imbedded in the upper part of the coal, within ten inches of the top of the seam, in a part of it called bone coal from its extreme hard- ness. The layers of coal that surrounded it were perfectly regular; so that, had the stone been immersed in a vessel containing metal in a fused state, and allowed to remain there until it was cooled, it could not have been more accu- rately fitted in its place. The seam in which it was found is called the main coal, and is the lowest that has been dis- covered in this neighbourhood. It is superior in quality to any other seam in the formation, and is the one most extensively worked here. Perhaps I should state, that the place where the stone was found was within twenty-five yards of a fault of considerable size." On comparing the Welsh stone with the two specimens of stones found in the Lancashire coal seams in my possession, their great resemblance in characters induced me to attri- bute them to a common origin, and to endeavour to find out what that origin was. My attention has therefore been directed to this enquiry ; and, although considerable time has been spent in hunting after more specimens of stones found in coal seams, no further information has been obtained of METEORITES FOUND IN SEAMS OF COAL. 309 the occurrence of any. Descriptions will now be given of the stones in my possession. THE PENDLETON SPECIMEN. This was found by Mr. Andrew Ray, the intelligent mana- ger of the colliery of the Pendleton Coal Company, in the year 1839, in sinking the new pit there. It was met with in the middle of the 6-feet seam of coal, at a depth of 245 yards from the surface. Mr. Ray, thinking it a great curiosity, brought it to me. At first I did not pay much attention to the specimen, thinking it was merely some boulder stone which had been squeezed into the coal from the great Irwell fault, which is not more than about 50 yards from the place where the specimen was found. On more careful examina- tion, the external characters as well as the composition of the stone, however, soon led me to consider it unlike any stone that ever previously came under my notice. This specimen is composed of a crystalline quartzose stone of a light gray, with mottled marks of a black colour, and con- taining small crystals of sulphuret of iron dispersed through the body of the stone. Its outside is moderately smooth, with traces of slickenside, as if it had been subjected to con- siderable pressure. The colour is dark black, with a slight polish on the stone, and some portions of a substance like the pulverulent carbonaceous matter, so commonly found in coals adhering to it. The black coating is a remarkably thin one on the outside of the stone, without penetrating scarcely at all into it. Fig. 1. Its form (see fig. 1) is that of g, an irregularly compressed oval, fe having one of its ends a little g" pointed, 5 inches in length by ^^^^ 3:^ inches in breadth. It has a specific gravity of 2*58, and weighed about 2f lbs. avoirdu- 310 MB. E. W. BISWET OS SOME SUPPOSED pois when whole. By the kindness of my friend, Dr. Robert Angus Smith, I am enabled to give an analysis of it, which is as follows : — Silica 96-463 Alumina 2578 Protoxide of Irou -644 Lime '161 99-846 THE PATRICROFT STONE. This was found by Mr. John Smith, in the colliery of Messrs. John Lancaster and Co., at Patricroft, near Man- chester, in the 4-feet seam, at a depth of 440 yards from the surface, about the year 1845. It is composed of a crystal- line quartzose stone of a darker gray than the specimen last described, and having very small black spots dispersed through its mass. The outside is partly smooth and partly irregular, of a shining black polish on the surface, but scarcely penetrating at all into the body of the stone. Marks of slickenside are seen on all parts of it, with several patches of sulphuret of iron. Fig. 2. Its form (see fig. 2) is that of an irregular pyramid, having ,,. mM ^ \ most of its angles rounded off. ^^ShII^HI JmiM "^^^^ greatest length is 7 inches, and the breadth 4 inches. It weighs 6^ lbs. avoirdupois, and has a specific gravity about 2*60. Dr. Smith's analysis of it is as follows : — Silica 96-050 Alumina 2-529 Protoxide of Iron "709 Lime '525 Magnesia '124 99-937 iiSiS' itETEOElTES FOUND IN SEAMS OP COAL. 311 THE RHYDGALED STONE. This, as previously mentioned, was found in November, 1850, in the main seam of coal at Rhydgaled, near Mold. It is composed of a crystalline stone of a grayish-white, with some small streaks and spots of a black colour dispersed throughout the mass. The outside is generally smooth, but contains a few little holes on its surface. It is coated with a shining black polish, just like a thin varnish, without pene- trating far into the body of the stone, as in the two last specimens. There are marks of slickenside on the outside, but not so strong as those on the specimen from Patricroft, Fig. 3. Its form (see fig. 3) is that of #''^^ an irregular oval, with three of its sides and one end compressed. The greatest length is 5^ inches V by 2 1 inches in breadth. It weighs 1| lb. avoirdupois, and has a specific gravity of 2r60." Dr. Smith's analysis of it is as follows : — Silica 99-182 Alumina "649 Protoxide of Iron '022 Lime *016 99-869 All the three stones were found in seams belonging to the middle division* of the coal field, the two first named in the higher, but the last named in the lower portion of it. Had they been found in the rough gritstones of the lower coal field, where most of the sandstone rocks prove that consider- able currents of water had been in action ; and that even some of the seams of coal, especially one known by the name of the Feather Edge Coal in Lancashire, are sometimes * For the definition of this part of the coal field, see Transactions of the British Association for the Advancement of Science, toI. xii. p. 46 ; and Sturgeon's Annals of Philosophical Discovery, and Monthly Reporter of the Progress of Practical Science, vol, i. p. 126. 312 ME. E. W. BINNEY ON SOME SUPPOSED found to have been wholly or partially removed by the effects of running water, it would not have been very remarkable; but when these stones are found in the midst of the most tranquil deposits of the whole series, with no trace of a portion of rolled mineral matter of the size of a pea for thousands of feet in vertical height, their occurrence, in the places where they were found, is difficult to account for. Sir Charles Lyell, at page 217 of the last edition of his Elements of Geology, in speaking of the pebbles in the chalk, states as follows : — " The general absence of sand and pebbles in the white chalk, has been already mentioned ; but the occurrence here and there, in the east of England, of a few isolated pebbles of quartz and green schist, some of them two or three inches in diameter, has justly excited much wonder. If these had been carried to the spots where we now find them, by waves or currents from the lands once bordering the cretaceous sea, how happened it that no sand or mud was transported thither at the same time ? We cannot conceive such rounded stones to have been drifted like erratic blocks by ice, for that would imply a cold climate in the cretaceous period, — a supposition incon- sistent with the luxuriant growth of large-chambered univalves, numerous corals and many fish, and other fossils of tropical forms. " Now, in Reeling's Island, one of those detached masses of coral which rise up in the wide Pacific, Captain Ross found a single fragment of greenstone, where every other particle of matter was calcareous; and Mr. Darwin con- cludes, that it must have come there entangled in the roots of a large tree. He reminds us that Chamisso, the dis- tinguished naturalist who accompanied Kozebue, affirms that the inhabitants of the Radack archipelago, a group of lagoon islands in the midst of the Pacific, obtained stones for sharpening their instruments, by searching the roots of trees which are cast up on the beach. METEORITES FOUND IN SEAMS OF COAL. 313 *' The only other mode of transport which suggests itself is seaweed. Dr. Beck informs me, that in Lym-Fiord in Jutland, the Fucus vesiculosus, often called Kelp, some- times grows to the height of ten feet ; and the branches rising from a single root form a cluster several feet in dia- meter. When the bladders are distended, the plant be- comes so buoyant as to float up loose stones several inches in diameter, and these are thrown by the waves high up on the beach. The Fucus giganteus of Solander, so common in Terra del Fuego, is said by Captain Cook to attain the length of 360 feet, although the stem is not much thicker than a man's thumb. It is often met with floating at sea, with shells attached, several hundred miles from the spots where it grew. * Some of these plants,' says Mr. Darwin, * were found adhering to large loose stones in the inland channels of Terra del Fuego, during the voyage of the Beagle in 1834 ; and that so firmly, that the stones were drawn up from the bottom into the boat, although so heavy that they could scarcely be lifted in by one person.'" No doubt there is a far greater abundance of fossil trees in the coal-measures than in the chalk ; but still there is little evidence to show that it is at all probable that the stones found in coal seams had been carried to the places where they are now met with in the roots of trees, any more than that the trees themselves were drifted. Doubtless a Sigillaria, having immense stigmarige roots, with radicles radiating from them in all directions to a great length, would be as likely a root as could be desired for the purpose of conveying a stone. But where and how is the Sigillaria to get loaded with its burden ? This is a difficult question to answer. The plant, of which this re- markable fossil is the root, must have grown beyond all question in soft mud, and not on a rocky bottom; and, even if it had grown in the latter position, coal seams, in Lan- cashire at least, as I have shown in a paper printed in the last 2s 314 &R. E. Ay. BtSrSEY on i^me strpposED volume of the Transactions of tMs Society, bear no evidence "of the vegetables composing them having been drifted, but, on the contrary, show that they were grown where they are now found, the seam of coal being simply a mass ■0f altered vegetable matter, lying upon a bed of tree roots, and having stems of similar trees resting upon it. A seam of coal like those in which the stones were found, bears no more evidence of a current of water than an ordinary peat bog does, and a rolled stoi^'e is Just as likely to be found iti the middle of the one as in the other, if we admit that the Vegetable matter now constituting coal, grew on the spot where it is found. In the bog, over which the Liverpool and Manchester Railway now passes, known by the name of Chat Moss, are some pits containing water called ringing holes. The people residing near the moss have a tradi- tion, that if any one can find a stone on the bog which has not been brought frorti a distance, and throws it into tliie holes, it will ring like a church bell. But this interesting^ experiment has not yet been tried, from the simple reason that no one, up to this time, has yet been able to discover 'imcih a stone ! If, therefore, it is difficult td accoiftit for the occurrence of the stones, previously described in this paper as found in seams of coal, being conveyed to the places where they M^e met with by the action of running water, we tffust look to some other cause for their origin. The shape of the stones is not such as we should expect to have been precipitated from solution in the water in which the vegetaT:>le matter was immersed, like the flints in chalk. Nor does their size allow of any probalbility of their being secreted from the sap of plants, like the crystals of silica, which are sometimes met with in the sugar-cane and some other plants. The trees of the cartwniferous series have, without doubt, been of a most extraordinary character when compared with those at present existing ; but still we cannot METEOKITES FOUND IN SEAMS OF COAL. 315, for a moment imagiae even tba^ they were capable of pro- ducing in tteir insides stones similar to those described in this paper. Now, in my humble opinion, there is another cause to ■yphich we can attribute the position of the stones in the seams of coal in whick they were, without doubt, founcl,^ by supposing that they are meteorites which fell from the atmosphere, and became imbedded in the coal when it waa in a soft state, and before it was covered by the overlying roof. Up to this time, few meteorites have been found in the strata composing the crust of the earth. In note 83 of Lieutenant-Colonel Sabine's translation of Baron Hum-- boldt's CosmoSy is the following passage : — " Olbers acutely observes, that it is a remarkable circumstance, not hitherto* noticed, that no fossil meteoric stones have as yet been, found, like fossil shells, in secondary and tertiary forma-< tions. Are we to infer that, previous to the last and pre- sent arrangement of the surface of our planet, no meteori© stones had fallen upon it; although, according to Schreibers, it is probable that 600 falls of aerolites now take place vOi each year?— (Olbers in Schum, JahrK, 1838, S. 329.) Problematical nickeliferous masses of native iron have beea found in Northern Asia, at a depth of 31 French feet, andi recently among the Carpathian mountains; both these masses are very like meteoric stones." Sir Charles Lyell, in the third edition of his Manual of Elementary Geology y at page 145, alludes to the first-named mass of native iron above mentioned, and as having been ibund in the alluvium at Petropawlowsker in the Mrasskei circle with more confidence. He, however, states that no sufiicient data are supplied to enable us to determine whether it be of post-pliocene or newer pliocene date. He further adds---" We ought not, I think, to feel surprise that we have not hitherto succeeded in detecting signs of 316 MR. E. W. BINNEY ON SOME SUPPOSED such aerolites in older rocks ; for, besides their rarity in our own days, those which fell into the sea (and it is with marine strata that geologists have usually to deal), being chiefly composed of native iron, would rapidly enter into new chemical combinations, the water and mud being charged with chloride of sodium and other acids. We find that anchors, cannon, and other cast-iron implements, which have been buried for a few hundred years off our English coast, have decomposed in part or entirely, turning the sand and gravel which enclosed them into a conglo- merate, cemented together by oxide of iron. In like man- ner meteoric iron, although its rusting would be somewhat checked by the alloy of nickel, could scarcely ever fail to decompose in the course of thousands of years, becoming oxide, sulphuret, or carbonate of iron, and its origin being then no longer distinguishable. The greater the antiquity of the rocks — the oftener they have been heated and cooled, permeated by gases or by the waters of the sea, the atmo- sphere or mineral springs — the smaller must be the chance of meeting with a mass of native iron unaltered ; but the presei-vation of the ancient meteorite of the Altai, and the presence of nickel in these curious bodies, renders the recognition of them in deposits of remote periods less hope- less than we might have anticipated." In the translation of Humboldt's Cosmos, before cited, at page 11 8 of Vol. I. is the foUojving passage, which, as it contains valuable information on the subject before us, will be given at length : — " The solid masses which reach the earth— whether they have been seen to fall at night from balls of fire, or in the daytime from a small dark cloud, usually in a clear sky, and with a loud noise — though con- siderably heated, are not incandescent. They exhibit, on the whole, a general unmistakeable resemblance to one another in their external form, in the nature of their crust, and in the chemical composition of their principal constitu- METEOEITES FOUND IN SEAMS OF COAL. 317 ents; and this resemblance is traceable, when and wherever they have been collected, at all periods of time, and in all parts of the earth. But this remarkable and early recog- nised similarity of general character in solid meteoric masses, suffers many exceptions in detail. How different are the very malleable masses of iron from Hradschina, in the district of Agram ; or those from the banks of Sisim, in the Jeniseisk government, mentioned by Pallas; or those which I brought from Mexico — allofwhich contain 96 percent, of iron — from the aerolite of Sienna, which hardly contains 2 per cent, of iron ; from the earthy meteoric stone of Alais, in the De- partment du Gard, which falls to pieces when immersed in water ; and from those of Jonzac and Juvenas, which are without any metallic iron, and are composed of various crystalline ingredients? These diversities have led to the division of the cosmical masses under consideration into two classes — nickeliferous meteoric iron, and fine or close- grained meteoric stones. The crust of these masses, which is only a few tenths of a line in thickness, is very charac- teristic; it has often a pitchy lustre,* and is sometimes veined.- The only instance which I know of the absence of this crust, is in the meteoric stone of Chantonnay in La Vendee, which is marked by another circumstance equally rare, viz., the presence of pores and vesicular cavities, like the meteoric stone of Juvenas. The separation of the black crust from the light grey mass beneath, is always as sharply defined as in that of the dark leaden-coloured crust of the white granite blocks which I brought from the cataracts of Orinoco, and which are also found by the side of many cataracts in other parts of the world, as those of the Nile and the Congo. The greatest heat of our porcelain fur- naces can produce nothing similar to the crust of the aero- * Pliny has remarked the peculiar colour of the crust of aerolites " colore adusto " (11, 66 and 58). The expression "lateribus pluisse " also refers to the burnt appearance of the exterior. ^18 MR. B. W. BINJSfEY ON SOME SUPPOSED lites, so distinctly and sharply separated from the unaltered mass beneath. Appearances which might seem to indicate a softening of the fragments, have been occasionally recog- nised; but, in general, the condition of the greater part of the mass — the absence of any flattening from the effects of the fall — and the moderate degree of heat perceived on, touching the newly-fallen aerolite— are far from indicating a state of internal fusion during its rapid passage from the limits of the atmosphere of the earth." The chemical composition of the three stones previously described in this communication, undoubtedly shows less iron than exists in the majority of meteoric stones hitherto examined. The composition of the Waterloo stone, found in Seneca county, New York, and described by Professor Shepard in his Report on Meteorites, at page 40, No. 3.1, Vol. XI. of SilUman's Journal, has some analogy in its composition and specific gravity to the stones now under consideration. The analysis of this stone gave. Silica 78-80 Peroxide of iron 8-72 Alumina 6'28 Moisture 4-76 Lime and magnesia, and loss 1 '45 10000 Specific gravity 2-30. — The Waterville and Concord stones, also described by Professor Shepard at pp. 414 and 416 of No. 18 of Vol. VI. of SilUman's Journal, contain no iron, but^ a large quantity of magnesia. According to Dr. Shepard, the first five chemical elements thus far known to exist in meteoric masses, in the supposed order of their prevalence,^ are as follows: — Iron, nickel, magnesium, oxygen, and sili- con, p. 386, Vol. II. No. 6 (second series), of Sillimar^s, Journal. For a long time scarcely any meteorites were recognised as such, which did not contain a large amount of metallic iron; but, now attention has been directed to METEORITES FOUIiTD IN SEAMS OF COAL. 319 these -bodies, many without irom will doubtless be met with. So tliere is nothing in the composition of the stones de- scribed in this paper to prevent them from being considered as meteoric, even supposing that such bodies which fell to the earth in so remote an age as the carboniferous strata, ^ete exactly similar in their nature to those which visit the earth at the present time— a supposition not very probable. Scarcely any of the strata of the earth, in England at least, have been so thoroughly explored as the valuable seams of coal, which have contributed so largely to our national re- Boiirces; and therefore it is in those strata that we should certainly look with the greatest probability for finding ancient meteoric stones. Also, as Sir Charles Lyell has observed in the preceding quotation, in his remarks on fossil meteorites, it is not the masses of iron which fell in the waters of the ancient globe that we should expect to find preserved in the strata, but rather sttch stones as we have previously described, consisting nearly altogether of silica, and, therefore, capable of resisting the decomposing 'agents, which metallic iron, and many other bodies, would be nearly incapable of enduring. The rare occurrence of «uch bodies as the stones before described in seams of coal, may, with propriety, be adduced in attributing them to ^ome extraordinaiy cause rather than the eft'ects of currents ■of water, and their transport by the roots and branches o£ trees. The shape of the three specimens, the specific gravities and chemical composition, do not prove much either for ot against their being considered as meteorites ; for if the seams of coal in which they were found had been formed of drifted karees, as was formerly the favourite hypothesis for accouirt- iBg for the origin of coal, the stones might be taken for travelled pieces of quartzose stone, equally with the drifted vegetables; but each of the seams in which they occurred (and they were found in the middle of the beds), is placed on 320 ME. E. W. BINNEY ON SOME SUPPOSED METEORITES, &c. a floor full of stigmariae roots, thus affording conclusive proof that the plants of which the seams were formed grew upon the spots where they are now found. The crust of the specimens, consisting of only a few tenths of a line in thickness, of a shining black lustre, and so sharply defined from the light gray mass of the stone beneath, is the strongest evidence of their being meteorites. This is the peculiar character of meteorites, so forcibly alluded to by Humboldt in the quotation previously given. The black colour of the stones might certainly have been derived from the decomposing vegetable matter in which they have been so long enveloped; but it is very remarkable that the colouring matter should have penetrated so slightly into the body of the stone. An ordinary pebble of quartz, after only a few years' immersion in black mud, composed of decaying vegetable matter, would be much more dis- coloured in depth than the specimens in question, which have lain for countless ages; and the present slight coating of the latter can only be attributed to their outsides having been subjected to the action of great heat, and thus so vitrified as to prevent the colouring matter from entering into the body of the stones before they came amongst the vegetable matter now forming coal. Nothing is more difficult than to establish, without question, the meteoric character of a stone. For ages doubts were thrown even on those specimens which were seen to fall from the heavens, and were actually found in a hot state. With fossil meteorites still greater difficulties occur; and the specimens described in this paper can only be considered as such bodies, by their resemblance in their characters to recent meteorites, and by their being found in a position where it is more probable to suppose they came through the atmosphere, than by currents of water, or any other ordinary cause. 321 XXn. — On the Volvox Globator, J% William Cra\vford Williamson, Professor of Natural History in Owens College^ Manchester, Eead May 27, 1851. From the long period that has elapsed since observers first became interested in the Volvox globator as a microscopic object, it is a matter of some surprise that its structure should still be so imperfectly understood. Such, however, is really the fact. The idea of its unity as an individual animal having been rejected by Professor Ehrenberg, many naturalists adopted the more novel interpretation of its his- tory, enunciated by the illustrious Prussian, who was almost universally considered to have dispelled the obscurity with which this elegant organism was previously invested. Whilst all honour is acknowledged to be due to this great observer for the brilliant results of many of his labours, it has become manifest, that in points relating to the internal structure and physiology of some of the objects which he has investigated, his conclusions require to be received with a degree of caution. It is now known that he has included in his great work on infusorial animals, a large number of plants, to which he has assigned organs and functions which are really confined to animal life. The consequence has been, that doubts have gradually suggested themselves as to how far his conclusions are to be relied upon, in reference to many of the other objects delineated in his magnificent volume. The Volvox globator has come in for a share of this scepticism, and not without reason. But whilst many have disputed the accuracy of Ehrenberg's interpretation of its structure and history, I 2 T 322 PROFESSOR W. C. WILLIAMSON ON VOL VOX GLOBATOR. am not aware that any other observer has grappled with the subject in order to supply the deficient knowledge. Havmg recently met with large numbers of this beautiful object in a pond near Rusholme, I have subjected it to a careful and protracted examination, in the hope of clearing up some of the points considered to remain unsettled. As is well known, Professor Ehrenberg regards each of the small green specks with which this organism is studded, as a distinct polygastric animalcule, having an oral orifice leading to several stomachs, an eye, organs of generation, and divergent canals, which are supposed to maintain a communication between it and the individuals by which it is surrounded. According to this view, the Volvos^ con- sists of an association of similar individual animals, uniting to form a hollow sphere, a structure somewhat analogous to that of the zoophytic polyparies. Though this interpretation has recently received the sanction of Professor R. Jones (Encyclopcedia of Anatomy and Physiology^ Article Polygasirica, Nov. 1847j^ and is also adopted by some other leading naturalists, I confess I cannot reconcile it with the appearances which the careful use of a good microscope reveals. The details of its struc- ture and development appear to exhibit less affinity with the phenomena of animal than of vegetable life. Regard- ing it as a plant, we have comparatively little difficulty in understanding its history ; but it is no easy task to bring it into accordance with our ideas of what is essential to animal life, modified though these have been within late years. Having little doubt in my own mind, that it is a true Con- fervoid plant, I will proceed to explain the details of its structure, in accordance with this general conclusion. Commencing with the examination of a very young in- dividual, we find that it is a hollow vesicle, the walls of which consist of numerous angular cells {Fig. 1 a), filled with green endochrome, mixed with minute granules, the PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOR. 323 intercellular spaces being more or less transparent. After a while, the green colouring matter changes its form, losing its regularly angular aspect, and assuming the appearance of Fig, 2 a. In this stage we cease to perceive the original outlines of the cell ; but on the application of some re-agents they can readily be brought into view. This irregularity of form appears to arise from the existence of a ductile cell- membrane which primarily lines the interior of each cell, whilst it surrounds the cell-contents. This inner mem- brane becomes separated from the outer cell wall, excepting at a few points {Fig. 2 h), where it is retained in contact. It now undergoes some curious changes, which alike aifect its form and the character of its contents. After a time the angular projections of the inner mem- branes of diflPerent cells appear to become still more inti- mately united, as represented in Fig. 3 a. This appearance seems to be caused by the further diminution of the con- nection between the outer cell-wall and its internal cell or lining membrane, from the shrinking of the latter tissue. Whilst these changes are in progress, an increase in the diameter of the entire organism is taking place, unaccom- panied by any corresponding addition to the number of the cells, or to the amount of endochrome which they contain. Hence the older organisms appear much more transparent than the younger ones. Along with this general enlarge- ment, there appears to be an increase in the diameter of the individual cells ; but whether this is owing to a real disten- sion of the outer cell-membrane, or merely to the flattening of cells which were previously somewhat spherical, is doubt- ful. I suspect that the former may be the case. It is obvious that the superficial area of each cell enlarges ; and as the projecting radii of the inner cell and its endochrome retain their attachment to the outer cell-wall, the expansion of the latter leads to an elongation of the former ; this is, of course, accomplished at the expense of the central 324 PEOFESSOB W. C. WILLIAMSON ON VOLVOX GLOBATOE. mass of lining membrane and endochrome, which diminishes as the radiating processes become elongated. Owing to these changes, the green cell-contents assume a stellate form {Fig. 4 a), and the intermediate transpa- rent spaces become proportionately enlarged. This ex- pansion of the cell- wall, and elongation of the thread-like processes of the lining membrane, go on until we have the appearance presented by Fig, 5, in which state the central green cell {Fig. 5 a) is seen to be much reduced in size, as well as altered in shape — having become more triangu- lar ; whilst the processes {Fig. 5 b) attaching it to its outer cell-walls, are elongated, and often branched. In this stage it is very rarely possible to trace the outlines of the original hexagonal cells. I have, however, been able to identify them sufficiently often to establish their existence. By rupturing a Volvox under water containing a slight trace of tincture of iodine, and at the same time paying great at- tention to the management of the light, I have seen them with great distinctness, as represented by the faint lines, Fig. 5 c ; each one containing its own endochrome. The projecting threads which maintain the inner cell in its posi- tion {Fig. 5 h), and which are usually attached to the cell-walls at points exactly opposite the corresponding processes of adjoining cells, give the whole the appearance of continuous canals, connecting together the separate masses of endochrome. It is in this light that they were r^arded by Ehrenberg, who appears never to have seen the hexagonal cells within which they are enclosed, and the thin cell-walls of which intervene between the opposite extremities of apparently continuous threads. During the progress of these transitions from the angular to the stellate form, corresponding changes are affecting the character of the other cell-contents. In the early stages of growth, each cell, as already stated, contains an abundance of dark green endochrome, along with numerous minute PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOE. 325 granules of an uncertain nature, but apparently analogous to those seen in so many of the fresh-water Algaj. After a while, a single large green granule {Fig. 3 and 4 V) makes its appearance in each cell, whilst many of the minute ones, previously observed, disappear. One or two of the latter, however, not only remain unabsorbed, but continue to enlarge; and from their pale colour, and high refracting power, become very brilliant (Fig. 5 a and 12 a). The large green granule now disappears in its turn, whilst a new feature becomes increasingly conspicuous : this is the celebrated red spot which Ehrenberg regards as the eye of his animalcule. For some time I thought that the latter object was the direct result of a transmutation of the former granule, so fre- quently did the appearance of the one mark the disappear- ance of the other; but this is not the case. They are perfectly distinct, though the red speck is rarely to be seen when the green granule is fully developed, or even beginning to be absorbed. On the disappearance of the latter object, we find that the remaining green endochrome has assumed a paler hue; but the one or two brilliant granules {Fig. 5 f, 12 a) just referred to, have materially increased in size and con- spicuity, occupying a large proportion of each green central area. In fact the only contents of the latter, in the present stage, are the granules, the red spot, and a small quantity of pale green homogeneous endochrome. The brilliant points, which are the generative organs of Ehrenberg, are perfect and well-defined spheres; but I have not been able, by any adjustment of the instrument, to obtain so definite an outline to the eye spot {Fig. 5 d and 12 6) of the Prussian naturalist. It appears to be the result of an alteration in the condition both of the lining mem- brane, and of some of the granular cell-contents which ad- here to it. The degree of its distinctness is very variable. In some specimens its presence is barely to be detected, even in the most advanced condition of the cells. 326 PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOR. Implanted above each of the green areas, we have two cilise, or " proboscides" of other writers {Fig. 5 e, and 12 c), to which further reference will be made. On rupturing a Volvox between two glasses moistened with water, we find ample confirmation of the accuracy of the above views. By the bursting of the organism, a num- ber of cells along the torn margins are laid open. The cell-contents of the cells whose outer walls are so ruptured, instantly lose their stellate form and become spherical {Fig. 4 d and 12) ; a result produced by the liberation of the elongated processes from their points of attachment to the cell-walls, and which is probably owing to the existence of some elasticity in the inner cell-membrane. But whilst many of the cell-contents, thus altered in their contour, escape from the cavities of their respective outer cells, individuals are sometimes retained in connection, by means of a long thread, as at Fig. 4 d and e. It is evident that the whole of the cell-contents, with the exception of the brilliant spherical granules, are remarkably ductile and cohesive. They may be drawn out into long threads to the extent of several times their own diameter {Fig. 4 e). This singular viscid character readily accounts for the ex- tension of the stellate processes, on the enlargement of the areas of the individual cells. It is evidently the property which has caused the elongation of the threads at the expense of the central mass, and is possibly owing to the existence of a large quantity of gummy matter in their sub- stance. In a little time after water has obtained access to the interior of the Volvox, we find that an analogous change gradually steals over the contents of all the other cells. The processes become successively detached from the cell- wall, and are drawn in towards their respective centres, which also become globular {Fig. 12). For a time the cell-contents, though thus altered in form, do not escape from the cells in which they are contained, — as we found PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOR. 327 to be the case when the cell-walls had been torn across ; but after a while many of the cell-walls appear to become bo far softened as to allow the cell-contents to float out. When this is the case, the latter objects invariably pass into the interior of the Volvox. They never appear to break through the outer wall. On examining the cell-contents, after they have thus made their escape, we find that they have under- gone no change beyond that of external form: their com- position is the same as when seen in situ, according to the degree of development which the individual under examina- tion has imdergone. When thus liberated, they exhibit no traces of the two ciliae or **proboscides" of Ehrenberg, and which he describes as belonging to the individual ** animaU cule," and being merely projected through the investing membrane. Neither have we any thing resembling the oral aperture, or sacculated stomachs, delineated in his figure. Indeed, the whole appears exactly like the ordinary cell of an Ulva, deprived of its external cell-wall. On turning our attention to the external membrane, from the under surface of which these objects have escaped, we find that the cilias have been left behind, and are im- planted in pairs upon its surface, in a very regular order {Fig. 6). We observe a series of small specks arranged in pairs, and which preserve a degree of relative parallelism (6 a) ; from each of these points a long cilia is projected. It has already been remarked, that when the cell-contents have escaped from a ruptured cell, the vesicle is frequently retained in connection with the latter, by means of a deli- cate ductile thread (Fig. 4 e). In this case the thread always terminates at one or both of these small specks, in- dicating a more intimate union between the cell and its contents at these points than at any other. It is not easy to say what these specks are. They do not appear to be merely minute apertures, since they exhibit a power of condensing transmitted light. 328 PROFESSOR W. C. WILLIAMSON ON VOL VOX GLOBATOK. For a short time after the rupture of the Volvoxy the fila- mentous ciliae continue their active whip-like movements ; but these gradually cease, and soon afterwards the filaments detach themselves from the membrane in great numbers, floating away into the surrounding medium. I was long puzzled by the appearances which the liberated filaments assume. Soon after they make their escape from these attachments, they appear to become thickened at one end, and assume the appearance of large spermatozoa {Fig. 7 a). I was for some time in doubt whether this was a real bulbous thickening, or whether, as some specimens seemed to indicate, it was merely the result of a flexure of that portion of the filament, which, when in situ, was implanted in the outer membrane; but I am now satisfied that the latter of these is the true explanation. The base of the filament curls up (Fig. 7 b c), and produces the bulbous appearance in question. I am indebted to my friend, Mr. Dancer, for his assistance in this matter ; and, after a very careful examination of the filaments, he has arrived at the same conclusion. The filament is evidently a distinct appendage, and not a mere protrusion of a part of the cell-contents through the small apertures already referred to. This is shown, both by the readiness with which they drop off, and also by the little change which they subsequently undergo, whether retained in the fluid, or dried upon the glass. Thus far I have confined my attention to the changes which have alike modified the great bulk of the individual cells ; but there are other phenomena which only affect a few of them. Ehrenberg observed, that some <' animalcules" were selected in each VolvoXt to be the seat of changes of a dif- ferent character, and that, by a continued process of divi- sion and subdivision, every one of these became converted into a young Volvox. PBOFESSOR W. G WILLIAMSON ON VOLVOX GLOBATOE. 329 On examining a number of individuals, we shall find, that whilst many of them contain young fac- similes of the parent object, others exhibit no obvious traces of such young organisms. The latter remark especially applies to those small specimens which are the least developed. But even in these a careful examination reveals a slight enlargement of eight or nine cells, dispersed through different parts of the structure (Fig. 1 d). This enlargement goes on until each of the cells referred to, attains to a diameter four or five times greater than those by which it is surrounded {Fig. 2 c). The inner membrane also continues in close union with the cell-wall, and never assumes the stellated contour seen in the ordinary cells. In fact, the process begins whilst the germs are contained within the parent; and before the contents of any of the cells have become detached from the cell-walls {Fig. 3 h). Two new cells are soo.i seen to have been developed within the old one; and, by a repetition of a similar process, each of these becomes the parent of two more {Fig. 4 f). I have never yet observed one of these germs within which eight cells could be seen at once. I have no doubt that this arises from the fact, that the subsequent division has taken place in the plane of the external surface, and at right angles to the axis of vision. The next obvious development always increases the number to sixteen {Fig. 8), at which stage an internal cavity appears to have been formed within the germ. From this point, the multiplica- tion of cells, by the ordinary process of cell-development, progresses {Figs. 9 and 10), until at length a condition is attained, beyond which no farther increase takes place in their number. It is at this stage, apparently, that each cell is furnished with its pair of filaments. These appendages are added before the young germ is set free by the rupture of the parent, and occasionally the young ones may be seen revolving within the old organism. This, however, is a rare occurrence, since, though their ciliae move freely, the germs are usually stationary. 2u 330 PROFESSOK W. C. WILLIAMSON ON tOLtOX GLOBATOR. Up to this Stage, each germ is still retained within a large transparent vesicle {Figs. 8 a, 9 a, and 10 a), apparently the relique of the cell-wall of the primary cell. Its tenuity is extreme, and not the slightest trace of structure can be detected in it; but it is invariably present. I apprehend that this is the sole bond of union with the parent Volvox; and that, even though the latter may be torn, unless these special vesicles be also ruptured, the young Volvox will not make its escape — a phenomenon which we may constantly wit- ness. Usually, however, this membrane gives way along with, if not prior to, the laceration of the older individuals, and the ciliae being already in action, the young ones float away, and commence their independent life. At this time, the cells constituting the substance of the young Volvox are angular, fitting closely together, and hav- ing the lining membrane and endochrome in close apposi- tion with their walls ; they are in fact in the state with which we commenced our sketch. The cell-contents soon shrink away from the cell-wall, excepting at the points of contact, where the radiating threads still retain the contracting mem- brane in a central position, and all the other changes already described are again gone through. In this development of germs by a process of cell division, we have merely an ordinary example of the production of a bud — a process common alike to the ani- mal and vegetable worlds. It presents nothing like the development of an ovum, or a seed — though it is very similar in its results to the production and growth of the embryo, as it is developed from the membrane lining the embryo sac, only wanting the pollen tube and its influences, the sac being represented in the Volvox by the entire sphere. But though the germs produced are not true ova or seeds, may they not be endowed with a potentiality which will enable them to develop something analogous to seeds? It is consistent with what we know of other forms of or- PBOFESSOR W. C. WILLIAMSON ON VOL VOX GLOBAXOR. 331 ganic life, to presume that the process of gemmiparous generation is not the only one through which the species is perpetuated ; but that, in one form or another, germs of a diflferent kind, capable of existing through the winter, are produced. We naturally turn to the cell-contents in the advanced stages of their development, in search of these objects, and we are immediately struck with the existence of the large granules {Fig, 12 «), of which one or two are developed in each matured cell. The production of these, which invariably exist, would appear to have been one of the objects of all the antecedent changes ; and it becomes very possible that they may be the true germs, or reproduc- tive spores. This is, of course, a fact that it would be very difficult to establish by any process of direct observation, owing to the exceeding minuteness of the objects. When one of the cells, containing these granules, has been immersed in water for some time after the rupture of the parent, it assumes the appearance represented in Fig. 12 ; a are the granules in question ; b is the brown or pinkish spot already referred to ; c are the two filaments, which are always implanted over the contracted globular cell-contents ; and d is the outline of t^he primary cell- wall. I have already rem:.rked, that these cell-walls are very difficult to trace in the fully developed specimens, whilst in their ordinary state ; but on mounting a number of the objects for my cabinet, the fluid used being merely distilled water, I found that in a few days these cells came beautifully into view (Fig. 11). I have scarcely one specimen, in which a careful management of the light does not make them very conspicuous. When they are in close contact they are angular, the angles being sharp and well defined ; but when the cells are apart, which is often the case, they appear more circular. The inner cell-membrane, and other cell-contents, ehrink up into an irregular central mass, as in Fig. 11, which represents a portion of one of these specimens. 532 PROFESSOR W. C WILLIAMSON ON VOLVOX GLOBATOK. There is evidently an intercellular substance of some kind, in addition to the outer common investing membrane, which supports the filaments externally, and encloses the cells within it. The investing membrane is most probably the result of an alteration and condensation of the outer walls of some of the primary cells of the young germ. Since all the existing cells were originally developed within others, it is evident that the walls of the latter have either been absorbed, or they still exist in the form of thin layers, investing the cells to which they gave birth. The latter is the more probable conclusion of the two ; by their de- velopment and consolidation they may have produced both the intercellular substance and the common investing in- tegument. The existence of the former of these tissues appears to be established by specimens that have been acted upon in the way represented by Fig. 11. When these objects have been mounted a few days in a solution of iodine, which, by the way, renders their cilias beautifully distinct, the cell-contents separate into two dis- tinct portions. One of these is homogeneous, and of a pale green colour ; the other, which comprehends the lining membrane and the granular substances, assumes a dark- brown hue. The existence of a true internal cell or membrane lining each cell, distinct from the outer cell-wall, is obvious from an examination of young half-developed individuals, in which the cells are of comparatively large size. On subjecting one of these to gentle pressure under water, so as slightly to rupture it, the green cell-contents soon flow out, as al- ready described. As they do so, we see that their form is capable of modification, enabling them to glide through a very narrow fissure, or to be packed together in a small space ; but each one retains its pristine integrity ; and, as soon as the pressure is removed, resumes its spherical .form. If, on the other hand, the pressure is increased. PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOR. 833 each of the individuals becomes ruptured, when the fluid and granular cell-contents flow out ; they mingle freely with the water and with each other, but never regain their primary spherical contour. From the foregoing outline of the principal facts pre- sented by the Volvoiv ghhator, we may now proceed to consider its probable position in the kingdom of nature. I am aware, that prior to arriving at a conclusion as to whether it is an animal or a vegetable, it will be expected that I should define what I comprehend in each of these terms. I confess myself unable to do this satisfactorily. The attempt has frequently been made by others ; but none of their definitions are free from objections, or embrace all the numerous deviations from the typical forms of each kingdom. The most plausible distinction between plants and animals, is that which assigns to the former the power of assimilating inorganic mineral food; whilst the latter can only take into their systems that which is already organized. It is probable that this distinction is a valid one ; but it is scarcely one of practical application as a test of special moot cases. If the digestive process involved the necessity for an internal digestive cavity with an external oral orifice, the case would be different ; but there is every reason to believe, that some examples of animal organisms receive no food into internal cavities, but are endowed with a power of embracing the object on which they are about to feed, and thus absorb nutriment from the body with which they are merely in contact. The AriKsba is still a case in point — even though we should concede the vegetability of marine and fresh-water sponges, which I am not prepared to do. M. Agassiz and Dr. Gould, in their recently pubhshed Principles of Zoology, speak unhesitatingly on this point. They enumerate " distinctly limited cavities, destined for the lodgement of certain organs," as existing "in all ani- mals without exception." They also say, that " the well- 334 PROFESSOR W. C. WILLIAMSON ON VOLVOX GLOBATOR. defined and compact form of the organs lodged in these cavi- ties, is also another peculiarity of animals. In plants, the organs for special purposes are not embodied in one mass, but are distributed over various parts of the individual." That all this is strictly true, when merely applied to the higher forms of each kingdom, cannot be denied. But surely the vegetable nature of the sponges and Amcebce, closely allied as these groups are to the Foraminifera and jther Protozoa, cannot be regarded as so indisputably set- tled as to admit of the recognition of the above generaliza- tion. If the latter objects are animals, they present all the features just quoted which these distinguished writers con- sider characteristic of animal life. It is only proper to add, that they entertain no doubt of the vegetable nature of sponges. They also consider " voluntary motion and sen- sation" as characteristic of animal life. The existence of the latter function would be difficult to prove in many un- doubted animals ; and on comparing the motions of the Volvox, of many Confervoid spores, and other vegetable organisms, with those of the ciliated germs of numerous Acrite animals, as well as those of the infusorial Animalcules, we at once perceive that the one class exhibits just as many evidences of volition as the other. In such examples as that now under consideration, it appears to me, that we cannot safely do morie than ascer- tain to which of the two kingdoms the object presents the greatest amount of affinity on the one hand, and the fewest discrepancies on the other. By thus weighing the various positive and negative arguments, we may arrive at an accu- rate conclusion, without the necessity of attempting to succeed where so many able men have previously failed. On subjecting the Volvoon to what is apparently the only kind of test that the present state of knowledge renders practical, we are brought to the conclusion, that its true place is amongst the vegetable Algae, rather than amongst the animal polygastric Infusoria. PEOFESSOE W. C. WILLIAMSON ON VOLVOX GLOBATOE. 335 On comparing one of the cells of a young Volvoa^, prior to the shrinking of its cell-contents, with those of many of the Alg9B, we find the very closest resemblance existing between them. On advancing a stage further, when the cells have become ciliated, and the organism capable of locomotion, we have a condition which is common among the zoospores of the Confervse. In the development and temporary existence of the large green granule {Fig. 4 b), we have another point of resemblance to the Confervse. It exists under precisely similar conditions in many of the Algae. If we watch the development of the various species of Coccochloris in their earliest stages, we shall invariably detect the appearance of a single corresponding granule in the interior of each cell. It exists in the young states of most of the DesmidiacecB, and is especially obvious in the species of Cosmarium, Euastrum, and Arihrodesmus. It is also a curious fact, that when two new segments are pro- duced between two others of older growth (a common phenomenon amongst the Desmidiace(s\ each of the new portions exhibits the characteristic granule. In some stages of its development amongst the Desmi- diacecB, this granule appears to contain an abundance of starch. Hence we may regard it as the analogue of the similar granules, of which a few exist in the very young cells of the Zygenemata, and into the composition of which starch enters largely. The number of these granules in each cell varies considerably. In Coccochloris, and other fresh-water Ulvaceous plants, I have never seen more than one. In the young states of the Arihrodesmus ^ Euastrum, and Cosma- rium, we have invariably one, and occasionally two, in each segment. Mr. Ralfs considers each individual of these genera as consisting of but one cell, with symmetrical con- strictions ; and consequently we have as many granules for each cell as there are segments. I have just examined a young example of Cosmxirium margaritiferum, which con- 336 PROFESSOR W. C. WnxIAMSON ON VOLVOX GLOBATOR. tained four of these large granules, and which assumed the characteristic purple colour under the influence of iodine. In this example they were surrounded by minute dark- coloured granules in an active state of molecular motion, though themselves perfectly quiescent. In many Desmi- diaceae, however, I have failed to effect any change in the colour of the granule by the addition of iodine, beyond that produced upon the Volvox when similarly acted upon, viz., the conversion of the pale green hue into a varying tint of brown. From these circumstances, I have little doubt that the granule possesses the same nature and functions in all these known vegetable forms, and in the Volvoa;. It does not ultimately become converted into a mass of starch in many true plants, though it is in a number of instances ; consequently, the absence of the violet hue in the cells of Volvoja, when they are acted upon by iodine, neither militates against their vegetable nature, nor against my conclusion, that the large green granule is identical with that seen in the cells of a young Zygonema. What may be its use I know not ; but in all these cases it assumes the same form as in the Volvoa; ; not existing in the first instance, but being gradually developed ; and after fulfilling its office, whatever that may be, it is re-absorbed before the plant arrives at maturity. The red speck to which Ehrenberg has assigned the functions of an organ of vision, is also known to exist in the ciliated moving zoospores of Conferva glomerata and C. ciliaris. Of the vegetable character of these spores there can be no doubt ; consequently we must not only reject the physiological conclusion of the Prussian professor, but cease to regard the red speck as an indication of animal life. The peculiar appearances presented by the shrinking of the endochrorae and inner cell-membrane, as seen in Mg. 11, are identical with those exhibited by Pediastrum, Cocco- chlorisy and numerous other Confervae, when subjected to the .(^ V/.C WilliamBon.Del V Smi.th, LitB.. Printeci ty W MunklioiiSo, Yovi Pig. 8 TiiS. Tii.ll TT C "Wmiamson.Del. TV. Smith , LiUx Prmtei ""oy TV MbiiJtlio\*se ,. ?or> PROFESSOB W. a WILLIAMSON ON VOLVOX GLOBATOB. 337 same influences. Though we have something approaching to it in the case of the white cartilage cells of animals, I have seen nothing like it amongst the undoubted infusorial forms of animal life. In its globular form, the object approximates somewhat to the well-known Ulva ghhosa. This little parasitic plant has also a spherical contour, and consists of a saccated membrane, on the under surface of which the numerous cells are developed in a gelatinous intercellular substance. This membrane appears to be nothing more than the ex- panded and condensed tissues of the primary cells. I have frequently found, that when it has been ruptured, the in- ternal cells have floated out, when its cavity has become filled with Naviculae and other minute Diatomaceae. The Volvox exhibits a still closer affinity to the Botridina vulgaris of Brebisson, both in its structure and mode of growth. This latter plant, like the Volvox^ is spherical, being primarily developed from a single independent cell ; only this cell if. solitary in the first instance, and not aggre- gate. It develops in its interior a number of other cells, of which those occupying its centre are subsequently absorbed. *'The whole frond is then constituted of vesicles, closely heaped together, and inclosing in the centre, granules. The primitive membranej inclosing in its midst the interwoven or cellular structure, is so closely united with the peripheral stratum, of vesicles, that it can in no way he separated from, it. The last development having been accomplished, the peripheral stratum of vesicles altogether loses its granules. Whether these disappear by absorption, or escape out- wardly, 1 have never been able to perceive." Such is the description of Botridina given by Meneghini, as quoted by Mr. Hassal {British Fresh-water Algce, vol. i. p. 320). In every point it exhibits so close an approximation to what occurs in the Volvox, as to leave no room to doubt that a close affinity exists between them. 2x 338 PROFESSOE W. C. WILLIAMSON ON VOL VOX GLOBATOK; In the development of the young germs of Volvox, we have a process closely resembling that by which the em- bryos of all phaneroganic plants are formed ; for vrhilst it is seen amongst the ova of animals, it is also one of the ordinary phenomena of vegetable life^ The cells of Volvox, in their varying conditions, throw some light upon an interesting problem in physiological botany, since they may probably be regarded as the proto- types of a structure found in some of the higher plants. The cells entering into the composition of the hard endo- carps of such fruits as the plum, and even the pear, are, as is well known, lined by successive layers of sclerogen. These layers are penetrated by tubular extensions of the central cavity, the extremities of these tubes being usually in con- tact with the corresponding ones of adjoining cells, as in the case of the radiating prolongations of the inner cell- membrane of Volvox. Now it appears probable, that we have here the same phenomena under different conditions. Dr. Carpenter inclines to the opinion, that as the inner cell-membranes become detached from the outer cell-walls in the plum-stone, they throw off successive layers of sclero- gen, which occupy what would otherwise have been an intervening cavity. In the Volvox we have the same reces- sion of the inner from the outer cell- wall ; but we have not the cognate development of sclerogen ; the intervening space being merely filled with colourless fluid. The struc- ture of the Volvox cell thus appears to give support to Dr. Carpenter's explanation of those of hard endocarps, an ex- planation which equally applies to the development of most forms of pleurenchyma. We thus find, that a vegetable analogue is to be found for every portion of the structure of Volvox, as well as of every function which those structures fulfil, so far as we can com- prehend them. But on comparing it with known and un- doubted animal organisms, we find that it is wanting in PEOFESSOE W. C. WILLIAMSON ON VOL VOX GLOBATOE. 339 many points. We have no trace of an oral orifice, or an internal digestive cavity ; neither has it the compensating power of investing the food from which it obtains its nutri- ment, as we see to be the case with the Am'jt^ .Vhlliu,mson,l)elt "W Smith, Lilh Tit y. M . 10. ni 9. Y\k. 8. i ,^.: a. h "b a /\'^) W C Williamson. neU. 1'rirn.ea by W Moullio u ao.Tork. "W Smith .r.it.li HITHEETO KJTOWN AS STEENBERGIJ3. 357 separates the yellow from the white jasmine, or some species of Carya in which it exists, from the others in which it does not. It has never been regarded as constituting a generic distinction. Subsequently to the penning of the preceding observa- tions, I have been favoured by G. W. Ormerod with per- mission to examine a specimen of Sternbergia obtained from the celebrated quarry at Peel, in Lancashire. I find in this interesting fragment demonstrative evidence of the accuracy of my previous determination. The specimen is partly covered with the usual thin film of carbonaceous matter, in which the cellular structure is beautifully preserved; the cells, which exhibit a very strong disposition to be arranged in vertical lines, have also left a definite impression upon the exterior xii the Sternbergia, which consists of clay ironstone. Horizontal laminae of brown carbonaceous matter are prolonged inwards from the smooth investing layer, and Separate the contiguous disks. In these laminae, also, the cellular structure is beautifully defined. In the disks inter- vening between the cellular laminae, there is no trace of structure whatever. They wholly consist of inorganic clay ironstone. This specimen appears also to support my con- clusion, that the centre of each horizontal lamina has usually been absorbed; but I cannot decide positively whether this has actually been the case, or whether the clay has been forced in at one extremity by an external pressure, which has been sufficient to break through the delicate layers of piths, and thus connect the disk — like portions of the cast at their centres. One part of the specimen ex- hibits a very different external aspect from the remainder, showing how very much influence mere pressure has had in modifying the external surfaces of the so-called Stem- bergias. 358 PKOFESSOE W. C. WILLI^iMSON ON STERNBERG I-E. INDEX TO THE PLATES OF THE ABOVE PAPER. Fig. 1. Fragment of wood from Coalbrookdale, fractured longitudinally ; a and b, bark: c, woody zone; g, medulla; h, cast of hollow pith, or Sternbergia. 2. Transverse section of the same fragment. The same letters are employed as in fig. 1. 3. Transverse section of the bark ; a, Epiphloeum and Mesophloeum ; b, Endophloeum ; c, Pleurenchyma, or woody fibre. 4. Vertical section of the bark ; a, Epiphloeum and Mesophlojum ; 6, Endophloeum ; c, Pleurenchyma. 5. Transverse section of the woody zone and pith ; c, Pleurenchyma ; d, medullary rays ; ff, medulla. 6. Vertical section of the same portions as fig. 6 ; c, Pleurenchyma ; d, medullary rays ; ff, medulla. 7. Fibres of Pleurenchyma from fig. 6, still more highly magnified, and exhibiting the hexagonal disks. 8. Vertical section of the Pleurenchyma, parallel with the bark, and at right angles to the medullary rays; «, transverse bars marking the positions of the disks ; d, intersected medullary rays. 9. Small portion of fig. 8 still more highly magnified. 10. Specimen of Sternbergia from the Lancashire coal field ; c, woody zone ; g, pulverulent carbonaceous matter, representing the medulla ; /*, cast of medulla, or Sternbergia approximata. 11. Vertical section of restored medulla of Dadoxylon (Sternbergia) approximatum ; c, woody zone ; g, medulla ; A, cast of medullary cavity. 12 Vertical section of a young branch of Gary a angustif olia, or hickory j a, bark ; b, wood ; c, pith ; d, hollow cavities. 359 INDEX TO VOL. IX. Page. Bateman, John Fked., Report on the Fall of Rain 1 BiNNET, E. W., Description of a Mineral Vein in the Lancashire Coal Field near Skelmersdale.... 116 A description of some Supposed Meteorites found in seams of Coal 306 Remarks on a VeiD of Lead found in the Carboniferous Strata in Derbyshire 125 Bridges, Tubular Girder, on the Security and limit of Strength of, constructed of Wrought Iron 179 Calvebt, F. Crace, Memoir on the Oxides and Nitrates of Lead...... 130 Clare, Peter, Report on the Fall of Rain 1 Coal, Chemical Changes attending the Formation of. 260 Coal Field, Mineral Vein in the Lancashire, near Skelmersdale 116 Dew, on the Formation of 46 Engine, Locomotive, Experimental Enquiries into the relative powers of, and the resistance of Railway Gradients 149 Equations, on Impossible and certain other Surd 207 on Impossible 236 Fairbaikn, William, F.R.S., Experimental Enquiry into the Relative Powers of the Locomotive Engine, and the resistance of Railway Gradients 149 On the Security and limit of Strength of Tubular Girder Bridges constructed of Wrought Iron 179 Farming, Faults in 93 Finlay, Robert, Professor, en Impossible Equations 236 Gas, on the Composition of, produced by the joint Distillation of Tar and Water at a high temperature 243 360 INI^EX. Page Gas-making, on the Cbemical Changes attending the Formation of Coal, and the relation of these changes to the philosophy of Gas- making 250 Gaseous Mixtures, on the Analysis of 297 Goodman, John, Researches into the Identity of the Existences or Forces of Heat, Light, &c 80 Gradients, Railway, Experimental Enquiry into the Resistance of..... 149 Harlet, Robert, on Impossible and certain other Surd Equations... 207 Heat, some Remarks on, and the Constitution of Elastic Fluids 107 SoriuNS, Thomas, on the Formation of Dew 46 — — — Cause of Unequal Falls of Rain in Cumberland 196 JocjLE, J, P., F.R.S., SQme Remarks on Heat, and the Constitution of Elastic Fluids 107 Just, John, Faults in Farming 93 KiRKMAN, Rev. Thos. Pentngton, Mnemonic Aids in the Study of Analysis 29 . On Linear Constructions 279 Lead, Remarks on a Vein of, found in the Carboniferous Strata in Derbyshire 125 Memoir on the Oxides and Nitrates of 130 Leigh, John, M.R.C.S., on the Composition of the Gas produced by the joint Distillation of Tar and Water at a high temperature 243 on the Chemical Changes attending the Formation of Coal, and their relation to the Philosophy of Gas-making 250 ., on the Analysis of Gaseous Mixtures 297 Light, Heat, &c., Researches intc the Identity of, by John Goodman, M.D 80 Lightning, and Lightning Conductors 66 Linear Constructions 279 Meteorite, Description of a, which fell at AUport 146 Meteorites, a description of some supposed, found in seams of Coal... 306 Mnemonic Aids in the Study of Analysis 29 Rain, Report on the Fall of 1 ., Cause of Unequal Falls of, in Cumberland 196 INDEX. 361 PA0£. Smith, Robt. Angus, Description of a Meteorite which fell at Allport 146 Sternbergiae, on the Structure and Affinities of the Plants hitherto known as 840 Stobgeon, WiixiaHj on Lightning and Lightning Conductors 56 Volvox globator, on the.~ 321 WiLUAMsoN, Professor W. C, on the Volvox globator 321 Professor W. C, on the Structure and Affinities of the Plants hitherto known as Sternbergiae , 340 END OF VOLUME NINTH. BXADSHAW AND BLAOKI.OCK, FRINTEKS, MANCHSSTEB AND LONDON. 3 A 363 LIST OF BOOKS PRESENTED TO THIS SOCIETY, SINCE APRIL IIth, 184^. S. C.HOMEBSHAM, C.T!. FsoKBSsoB^ Yoim, Esd. The Council of the Society of Anti- quaries, London. Iltles of Books. Report ott the Supply o£ Water to Manches- ter, Salford, and Stockport. By S. C. HOMERSHAM, CE. Pamphlet, " Oii an Extension of a Theorem by Euler." By J. R. Young, Professor of Mathematics, Belfast College. Report on R^n, for 1847. By Mr. J. F. MiUiBR of Whitehaven. Successful Case of Caesarian Operation, an 1848, 1849. Transactions of the American Philosophical Society, Vol. X. Part 1. Pk)ceedings of the American Philosophical Society. Vol. I., Nos. 40—46. Littell's Living Age. Transactions of the Royal Irish Academy. Vol. XXI., Part 2. The Eighth and Ninth Annual Reports of the Registrar-General of Births, Deaths, and Marriages iu England. jOn the Sanitary Condition of Oxford; By- W. P. Ormerod, Esq. Archaeologia. Vol. XXXII. 864 LIST 6t BOOKS Doaoft, The Council of the British Association. Ecoi/B DEB Mines op Pabis. The Council op the LlTEBAET AND PHI- LOSOPHICAL Society CI- LiVEBPOOt. Pbopessob Kupppbb. The Council of the Leeds Philosoph. & LiTEBABT SOCIETT. Mb. F. W. Dteb. The Council of the ■ RoTAL Scottish So- ciety OF Abts. Rev. H. H. Jones. The Council of the Geneva Society fok Physics and Natu- BAL Histobt. Geo. Wabeing Obhe- BOD, M.A. Wm. Lassel, F.R.S., Titles of Book». Lalande and Lacaille's Catalogue of Staw, Annales des Mines. 12th Vol. to the 19th Vol. inclusive. Proceedings of the Literary and Philosophi-- cal Society of Liverpool, No. 14, Vol. V^ Annuaire Magnetique et Met^orologique des Corps des Ingenieurs des Mines. Nos. 1 & 2 for 1848 ; Nos. 1 & 2 for 1846. Resumes des Observations M^teorologiques for 1846. By Professob Kupffeb of St. Petersburg. Annual Report for 1847, 1848, 1849, 1850, of the Leeds Philosophical and Literary Society. The Slave Girl : a Tale of the Nineteenth Century. By F. W. Dyek, Esq. Report of the Directors of the Dublin Uni- versity Museum. Transactions of the Royal Scottish Society of Arts. Vol. III., Parts 2 & 3. Description of a Machine for Polishing Spe- cula, with Directions for its Use. By H. Lassel, F.R.S. Me'moires de la Societ6 de Physique et d'His- toire Naturelle de Geneve. Vol. II., p. 2. Outlines of the Principal Geological Features of the Salt Field of Cheshire. By G. W. Ohmebod, M.A. Description of a Machine for Polishing Spe- cula. By Wm. Lassel, F.R.S., &c. PRESENTED TO THIS SOCIETY. 365 Donors. The Board of Regents OF THE Smithsonian Institution. Mb. Samuel Sai.t. J. Fletcher Mellon, F.R.A.S., &c. Mb. H. W. Fbebland. Pbof. Hodgkinson. The Cambridge Phi- i-osoPHicAL Society. The Council op the RoTAL Society of Edinburgh. Mb. Richard Buxton. Mb. T. H. Paslet. The Council op the RoTAL Society. Titles of Books. No. 1 of the Smithsonian Contributions to Knowledge — Ancient Monuments of the Mississipi Valley. Third Annual Report of the Smithsonian Institution. Statistics and Calculations essentially neces- sary to Persons connected with Rail- ways or Canals, &c. Calculated and Arranged by Samuel Salt. Facts and Figures, principally relating to Railways and Commerce. By S. Salt. Railway and Commercial Information. By S. Salt. On the Meteorology of the Lake District. By J. Fletcher Mellon. Poems. By H. W, Freeland. Report of the Commissioners appointed to inquire into the Application of Iron to Railway Structures. By Eaton Hodg- kinson, F.R.S., &c. Transactions of the Cambridge Philosophi- cal Society. Vol. VIII., Parts 3, 4, & 5 ; Vol. IX.,'Part 1. Quarterly Journal of the Chemical Society. Transactions of the Royal Society of Edin- burgh, Vol. XVIII.; Vol. XVI., Part 4 and 5 ; with No. 31 to 41 of Proceedings. Botanical Guide to the Flowering Plants, Ferns, Mosses, and Algae, found indi- genous within sixteen miles of Man- chester. By Richard Buxton. The Philosophy which shows the Physiology of Mesmerism. By T. H. Pasley. Philosophical Transactions of the Royal So- ciety. Parts 1 and 2, 1848; Parts 1 and 2, 1849, with the address of the Pre- sident ; Parts 1 & 2, 1850 ; Part 1, 1851. w LIST OF BOOKS Donors. E. J. Lqwb, F. B. a. S. The Council of the Philosophic Ai. So- ciKTr or Glasgow. The Council op the CuviBKiAN Society of Cobk. By tub Author, Mr. A. H. Palmer. From Mr. Wakefield. William Fatbbaibn, F. R. S. The Physical Society of Berlin. Db. E. du Bois Ret- MOND, The Council of the Zoological Society. Mji, J. R. Dyer. TiU«BOfBook«. A Treatise on Atmospheric Phenomena. By Edmund Joseph Lowe. Prognostications of the Weather, and Signs of Atmospheric Changes. By the same Author. Samson's Thermometer Stand. Proceedings of the Philosophical Society o^ Glasgow, Contributions towards a Fauna and Flora of the County of Cork. By the Cuviehian Society of Cork. Memoirs, Geographical and Commercial, on Liberia, &c. By A. H. Palmer. Remarks on the Smoke Nuisance. By John Wakefield. An account of the Construction of the Bri- tannia and Conway Tubular Bridges. By William Fairbairn, C.E., F.R.S. A New System of Architecture. By Wm. Rose Peckitt. Ah Experimental Enquiry into the Strength of Wrought-Iron Plates. By Wm. Fair- bairn, F.R.S., &c. Two Lectures on the Construction of Boilers, and on Boiler Explosions ; also on the Combustion of Smoke. By Wm. Fair- bairn, F.R.S. Die Fortschritte der Physik im Jahre, 1846, dargestellt von der Physikalischen Ge- sellschaft zu Berlin, 1 1 Jahrgang. Redi- girt von Dr. G. Karsten. Die Fortschritte der Physik, for 1847. 31 Nos. and 3 Reports of the Council of the Zoological Society of London. Remarks on Education. PEESEKTED TO THIS SOCIETY. Donors. ThB HiOTOBIC SodETT OF Lancashibe and Cheshie. Thb Conarciii of thb Bbitish Association. Fbooi the French Aca- SBKY. Fbom Samuel Cbomp- ton, M.R.C.S. Fbom the Council of THE Statistical So- CIETT, Pbofessob Stokes. J; e.ADAMs, F.R.S. E. ScHCNCKi F.R.S. Fbom Mb. Hedley. W. F. Stevenson, F.R.S. John Allan Beown, lately Dibectoe of the Obsebvatoby. Bebiah Bottield, Esq. The Royal Obsebva- TOBY OF EdINBUBGH. Tiuetof Boolu. Historic Society of Lancashire and Cheshire — 1st, 2nd, and 3rd Vols, of Proceedings and Papers. The British Association's Catalo]^ of Stars. Memoires de I'Academie des Sciences 4# I'Institut de France, t. 20, 21, & 22. Also, Memoires Pres6nt6s par divers Savants a I'Academie, t. 10 & 11. The 4th, 6tli, 7th, and 8th Edition of the Elements of Chemistry, by the late Db. Wm. Henby, F.R.S., &c., contain- ing annotations in his haadwriting. A complete Series of the Journals of the Statistical Society of London. 14 Vols. Sixteen Papers on various Scientific Sub- jects. By Pbofessob Stojkjes. An Explanation of the observed Irregularity of the Motions of Uranus. By J. !p. Adams, F.R.S. On Rubian and its products of Decompositio'fl.' By Edwabd Schunck, F.R.S, Letter to Dr. Lardner. By Mb. RetdlBy, Shield Row, Newcastle. On Hydrogen Gas and Water. By W. F. Stevenson, F.R.S. Report on the Completion of the publication of the Makerstown Observations; also, Makerstown Magnetical and Meteorolo- gical Observations for 1845 and 1846. By J. A. Bbown, Director. Makerstown Magnetical and Meteorological Observations in 1845 and 1846. Notes on the Cathedral Libraries of England. By Bebiah Botfield. Professor Henderson's Astronomical Obser- vations, made at the Royal Observatory of Edinburgh. Reduced and edited by Chables Piazzi Smyth. 368 LIST OF BOOKS PRESENTED TO THIS SOCIETY. Donors. John Jambs Wild, Esq. The Churchwardens OP Makchesteh. PbtbbClaeb, F.R.A.S., Vice-Pebs. J. F. Miller, F.R.S. Pbofbssob Habe. Chas. Daubent, F.R.S. M. Emmanuel Liais. Royal Cornwall Po- lttechnic Society, COMTB DE WbONT- chenko, l'etat Ma- job Du Corps des Ingenieurs. MM. Rayen, Soubei- BAN, AND BoCILLABD. J. Hastings, Esq., of the Liverpool Obser- vatory. Rev. Henry Halfobd Jones. The Society or Civil Enoineebs. Wm. Johnson, Esq. Titles of Bookn. Letter to Lord Brougham. By John Jambs Wild. Report of Evidence taken before the House of Commons on the Manchester Rectorv Division Bill. ^ An Account of some Thunder Storms and extraordinary Electrical Phenomena at Manchester. By Peter Clare, F.R. A.S. Meteorological Observations in Cumberland. By J. F. Miller, F.R.S. of Whitehaven. Memoir on the Explosion of Nitre. By Robert Hare, M.D., Professor of Chemistry in the University of Pennsyl- vania. Introduction to the Atomic Theory. By Charles Daubeny, F.R.S., &c. Thfeorie Math^matique des Oscillations du Barometre. Par M. Emmanuel Liais. Royal Cornwall Polytechnic Society's Eight- eenth Annual Report. Annales de I'Observatoire Physique Central de Russie. Par A. T. Kupffeb. "Digitaline" (2 copies). Meteorological Results, &c., during 1850. By John Hastings, Esq. Philosophy of Education. By H. H. Jones, F.R.A.S. Proceedings and Transactions of the Society of Civil Engineers, from the Commence- ment. Climate of Sidmouth,by W. H. Cullen, M.D« THE COUNCIL OF OF MANCHESTER. APRIL 16TH. 1851. ^rBsikirt: PROF. EATON HODGKINSON, F.R.S., M.R.I.A., F.G.S., &c. JOHN MOORE, F.L.S. PETER CLARE, F.R.A.S. JOSEPH ATKINSON RANSOME, F.R.C.S. WILLIAM FAIRBAIRN, F.R.S., M. INST. C.E. ijmtamH : JAMES P. JOULE, F.R.S., &c. EDWARD WILLIAM BINNEY. ^xmmx : SIR BENJAMIN HEYWOOD, BART., F.R.S. E. W. MAKINSON, A.M. M % Cnunril : THOMAS HOPKINS REV. H. H. JONES, F.R.A.S. JOHN GRAHAM. WILLIAM FLEMING, M.D. LAURENCE BUCHAN. ROBERT ANGUS SMITH, PH. D. 3b 371 AN ALPHABETICAL LIST OF THE MEMBERS OF THE LITERARY AiJD PHILOSOPHICAL SOCIETY OF MANCHESTER. APRIL 16Tn, 1851. DATE OF ELECTION. James Ainsworth January 25th, 1805 Ralph F. Ainsworth, M.D April 30th, 1839 Thomas Ashton, Mosley Street October 29th, 1824 Thomas Ash ton, Hyde August 11th, 1837 John Atkinson January 27th, 1846 W. H. Ash April 17th, 1849 Richard Parr Bamber, F.L.S October 19th, 1821 Robert Barbour January 23rd, 1824 Joseph Barratt April 19th, 1842 John Frederick Bateman, M. Inst. C.E January 21st, 1840 Thomas Bazley January 26th, 1847 Charles Bell, M.D January 25th, 1848 William Bell January 26th, 1847 James Bevan January 23rd, 1844 Edward William Binney January 25th, 1842 Alfred Binyon January 26th, 1838 Richard Birley April 18th, 1834 James Black, M.D., F.G.8 April 30th, 1839 372 ALPHABETICAL LIST OF MEMBEES. DATE OF ELECnOK. John Blackwall, F.L.S January 26th, 1821 Henry Bowman October 29th, 1839 Edward Brooke April 30th, 1824 W. C. Brooks, M.A January 23rd, 1844 Henry Browne, M.B January 27th, 1846 Laurence Buchan November 1st, 1810 John Burd January 27th, 1846 Rev, R. Bassnett, A.M April 17th, 1849 Henry Cadogan Campbell January 23rd, 1835 Frederick Grace Calvert, M.R.A.T January 26th, 1847 John Young Caw ...April 15th, 1841 Henry Charlewood January 24th, 1832 David Christie October 19th, 1847 Peter Clare, F.R.A.S April 27th, 1810 Charles Clay, M.D April 15th, 1841 Rev. John Colston October 29th, 1850 Thomas Cooke, jun April 12th, 1838 Samuel Elsworth Cottam, F.R.A.S October 20th, 1837 James Crossley Jaimary 22nd, 1839 Joseph S. Crowther January 25th, 1848 Charles Cumber November 1st, 1833 Matthew Curtis AprU 18th, 1843 John Benjamin Dancer April 19th, 1842 Samuel Dukinfield Darbishire January 25th, 1822 Rev. John Davies, M.A January 21st, 1851 James Joseph Dean November 15th, 1842 Joseph Cheeseborough Dyer April 24th, 1818 Frederick Nathaniel Dyer April 30th, 1850 The Right Hon. the Earl of Ellesmere, F.G.S April 15th, 1841 Thomas Fairbairn April 30th, 1850 William Fairbairn, F.R.S., M.Inst. C.E October 29th, 1824 W. A. Fairbairn October 30th, 1849 Octavius Allen Ferris January 26th, 1847 David Gibson Fleming January 25th, 1842 William Fleming, M.D AprU 18tb, 1828 ALPHABETICAL LIST OF MEMBEBS. 878 DATE OF ELECTION. Richard Flint October 31st, 1818 James William Fraser « January 22nd, 1839 Rev.William Gaskell, M.A '. January 21st, 1840 Samuel Giles , April 20th, 1836 Thomas Glover January 21st, 1831 John Goodman, M.D January 25th, 1842 John Gould April 20th, 1847 John Graham August 11th, 1837 Robert Hyde Greg, F.G.S January 24th, 1817 William Rathbone Greg April 26tb, 1833 Robert Philips Greg October 30th, 1849 John Edgar Gregan : January 25th, 1848 John Clowes Grundy January 25 th, 1848 Rev. Robert Halley, D.D... April 29th, 1845 Richard Hampson January 23rd, 1844 John Hawkshaw, F.G.S. and M. Inst. C.E January 22nd, 1839 William Charles Henry, M.D., F.R.S October 31st, 1828 Sir Benjamin Heywood, Bart., F.R.S January 27th, 1815 James Heywood, M.P., F.R.S. and G.S April 26th, 1833 James Higgins April 29th, 1845 Peter Higson October 31st, 1848 John Hobson January 22nd, 1839 Eaton Hodgkinson, F.R.S., M.R.I.A., F.G.S., &c. January 21st, 1820 James Piatt Holden January 27th, 1846 Thomas Hopkins January 18th, 1823 Henry Houldsworth January 23rd, 1824 Paul Moon James January 27th, 1837 John Jesse, F.R.S., R.A.S., and L.S January 24th, 1823 Rev. Henry Halford Jones, F.R.A.S April 21st, 1846 Joseph Jordan October 19th, 1821 James Prescott Joule, F.R.S., &c January 25th, 1842 Benjamin Joule, jun April 18th, 1848 William Joynson January 27th, 1846 Richard Johnson April 30th, 1850 374 ALPHABETICAL LIST OP MEMBERS. DATE OF ELECTION. Alexander Kay October 30th, 1818 Samuel Kay January 24th, 1843 John Kennedy April 29th, 1803 Richard Lane April 26th, 1822 William Langton April 30th, 1830 John Leigh April 17th, 1849 John Rowson Lingard January 26th, 1847 Thomas Littler January 27th, 1825 John Lockett January 25th, 1842 Joseph Lockett October 29th, 1839 Benjamin Love April 19 th, 1842 Joseph Leese, jun April 30th, 1 850 Edward Lund April 30th, 1850 James M'Connel October 30th, 1829 William M'Connel April 17th, 1838 Alexander Macdougal April 30th, 1844 John Macfarlane January 24th, 1823 Edward William Makinson, B.A October 20th, 1846 The Right Rev. t^e Lord Bishop of Manchester, D.D., F.R.S., F.G.S April 17th, 1849 Robert Manners Mann January 27th, 1846 James Meadows April 30th, 1830 Thomas Mellor January 25th, 1842 William Mellor January 27th, 1837 John Moore, F.L.S January 27th, 1815 L. A. J. Mordacque October 29th, 1830 David Morris January 23rd, 1849 George Murray January 27th, 1815 Alfred Neild January 25th, 1848 William Neild April 26th, 1822 John Ashton NichoUs, F.R.A.S January 21st, 1845 William Nicholson January 26th, 1827 George Wareing Ormerod, M.A., F.G.S ..January 26th, 1841 Henry Mere Ormerod April 30th, 1844 ALPHABETICAL LIST OF MEMBERS. 875 DATE OF ELECTION. John Owen April 30th, 1839 Joseph Owen February 5th, 1850 George Parr April 30th, 1844 John Parry April 26th, 1833 George Clark Pauling January 25th, 1842 George Peel, M. Inst. C.E April 15th, 1841 Peter Pmcoffs, M.D January 25th, 1848 Archibald Prentice January 22nd, 1819 Joseph Atkinson Ransome, F.R.C.S April 29th, 1836 Thomas Ransome January 26th, 1847 Rev. William Read, A.M January 23rd, 1824 Rev. John Gooch Robberds... April 26th, 1811 Richard Roberts, M. Inst. C.E January 181h, 1823 Samuel Robinson January 25th, 1822 Alan Royle January 25th, 1842 Samuel Salt April 18th, 1848 Michael Satterthwaite, M.D January 26th, 1847 Edward Schunck, Ph. D., F.R.S January 25th, 1842 Sails Schwabe April 20th, 1847 John Sharp October 28th, 1824 John Shuttlcworth October 30th, 1835 George S. Fereday Smith, M.A., F.G.S January 26th, 1838 Robert Angus Smith, Ph. D April 29th, 1845 Edward Stephens, M.D January 24th, 1834 James Stephens April 20th, 1847 Daniel Stone, jun January 23rd, 1849 Robert Stuart January 21st, 1814 Rev. John James Tayler, B.A January 26th, 1821 John Thorn January 27th, 1846 James Aspinal Turner April 29th, 1836 Thomas Turner, F.R.C.S April 19th, 1821 Absolom Watkin January 24th, 1823 Joseph Whitworth January 22nd, 1832 Matthew A. Eason Wilkinson, M.D January 26th, 1841 William James Wilson, F.R.C.S April 29th, 1814 ^p 376 ALPHABETICAL LIST OF MEMBERS. DATE OF ELECTIOW. Gilbert Winter November 2nd, 1810 George Bancroft Withington January 21st, 1851 William Rayner Wood January 22nd, 1839 George Woodhead April 21st, 1846 Edward Woods ....April 30th, 1839 Robert Worthington April 28th, 1840 James WooUey November 15th, 1842 Joseph St. John Yates January 26th, 1841 James Young October 19th, 1847 CONSERVED & BOUND 30 AUG 1988