transformation would or would not lead to a new theorem, or merely, as is possible, to a repetition, with a new set of constants, of theone already obtained. It is hardly necessary to point out how strongly the analogies established in the preceding investigations point to the existence of some simple dynamical theory of the Cartesian ovals under the attraction of forces directed to their foci. The investigation of such theory cannot but tend materially to the elucidation of the essential properties of these most interesting and as yet little-understood curves, the natural parents of the conic sections viewed as focal curves. In conclusion it may be observed that, in the foregoing paper, it has been seen how a single orbital force passing through a fixed centre may be resolved into others of a more simple form. This suggests a more general subject of investigation, where the force to be so resolved, instead of passing through a fixed point, is tangential, or, better, normal to a fixed curve or surface. Such an inquiry by no means belongs to ideal mechanics; for it would correspond to the case of the motion under the earth's attraction of a body near the earth's surface, considered as a surface of fluid equilibrium. K, Woolwich Common. 20th December 1865. VIII. Notices respecting New Books. The Mathematical Writings of D. F. Gregory, M.A., late Fellow of Trinity College, Cambridge. Edited by W. WALTON, M.A. Cambridge: Deighton, Bell, and Co. 1865. IT T is very difficult for the outside world to do justice to the writings of a man removed by an early death. It is not that what he has produced is intrinsically unimportant (it may or may not be so), but that even when it is important it is still altogether disproportioned to the power to which it owes its origin, and consequently is inadequate to sustain the reputation in which the writer was held by the small circle of friends to whom he was intimately known. And this is true in the present case, although some of the articles which compose the Mathematical Writings of Mr. Gregory evince great originality, and at the time of their publication powerfully contributed to the advancement of mathematical science. Still it is very desirable that the original memoirs of a man like Mr. Gregory should be brought together and competently edited, both as a memorial of one who ought not soon to be forgotten, and as being in themselves interesting and suggestive articles, and in some cases of first-rate importance. Mr. Gregory was the youngest son of Dr. James Gregory, the distinguished professor of medicine in the University of Edinburgh. quite completed his thirty-first year. Previously to entering at Trinity College, Cambridge, he had attended classes at the University of Edinburgh, where he was a favourite pupil of Professor Wallace. He passed the examination for the degree of B.A. at Cambridge in 1837, and was elected Fellow of Trinity College in 1840. In consequence of illness he left Cambridge in the spring of 1843, and never returned. In the brief interval between the first and last of these dates he was actively employed in promoting the mathematical studies of the University of Cambridge. This he did in many ways-partly by lecturing and examining, but chiefly by his writings. He was mainly instrumental in establishing the Cambridge Mathematical Journal, and, excepting a short interval, was its editor from the time of its first appearance till a few months before his death. He contributed largely to the Journal; in fact nearly half the first volume came from his pen, as also did a considerable part of both the second and third volumes. These contributions, in addition to a paper read before the Royal Society of Edinburgh, compose the present volume. The most important are those in which he works out the relations existing between the symbols of operation and those of quantity, and applies his results to the solution of large classes of differential equations. These papers are undoubtedly of very great value, and will be read with pleasure by those who take an historical interest in mathematics. Another class of papers relates to the interpretation of results in symbolical algebra, such as that "On the Impossible Logarithms of Quantities,' 'On the existence of Branches of Curves in several Planes," &c. Other papers are on detached subjects— "Demonstrations of certain properties of Triangles," "Solutions of some Problems in Transversals," &c. The volume contains thirtyseven papers in all of these but three have any reference to physics; viz. one, "On the Sympathy of Pendulums," and another "On the Motion of a Pendulum whose point of Suspension is disturbed," and a "Note on a Problem in Dynamics." The two former Mr. Gregory wrote in conjunction with Mr. Archibald Smith. It is well known that, besides these articles, Mr. Gregory wrote a collection of "Examples of the Processes of the Differential and Integral Calculus." It may be added that in this work he embodies the results of his most important papers. At the time of his death he left an unfinished MS. Treatise on Geometry of Three dimensions, which was completed and published by Mr. Walton in 1845. The present volume is in some sort a companion volume to the 'Mathematical and other writings of Robert Leslie Ellis,' published about two years ago, and which we noticed at the time. As in the case of that volume, so of this, Mr. Walton has shown himself a faithful editor. The printing and the general appearance of the volume leave nothing to be desired. Mr. Ellis was an intimate friend of Mr. Gregory, and succeeded him as editor of the Mathematical Journal. On his death Mr. Ellis wrote a brief life of him, which appeared in the Mathematical Journal, and is reprinted here. Perhaps, as we are noticing a memorial volume, we may be allowed to make from it the following extract. Mr. Gregory had been aware for a short time before his death "that the end was at hand; and, with an unclouded mind, he prepared himself calmly and humbly for the great change, receiving and giving comfort and support from the thankful hope that the close of his suffering life here was to be the beginning of an endless existence of rest and happiness in another world. He retained to the last, when he knew that his own connexion with earthly things was soon to cease, the unselfish interest which he had ever felt in the pursuits and happiness of those he loved. A few words may be allowed about a character where rare and sterling qualities were combined. His upright, sincere, and honourable nature secured to him general respect. By his intimate friends he was admired for the extent and variety of his information, always communicated readily but without a thought of display, for his refinement and delicacy of taste and feeling,—for his conversational powers and playful wit; and he was beloved by them for his generous, amiable disposition, his active and disinterested kindness, and steady affection. And in this manner his high-toned character acquired a moral influence over his contemporaries and juniors, in a degree remarkable in one so early removed." The following communications were made to the Society : By Mr. A. R. Catton 1. On the Synthesis of Formic Acid." 2. "On the possibility of accounting for the double refraction of Light by the vibrations of a continuous elastic medium kept in a state of constraint by the action of the material molecules." By Professor Cayley 3. "A new Theorem on the Equilibrium of four Forces acting on a solid Body." Defining the "moment of two lines as the product of the shortest distance of the two lines into the sine of their inclination, then, if four forces acting along the lines 1, 2, 3, 4 respectively are in equilibrium, the lines must, as is known (Möbius), be four genera 1 and 2, and similarly 13 the moment of the lines 1 and 3, &c., the forces are as 23.34.42:34.41.13 : √/41.12.24:√12.23.31. Calling the four forces P1, P2, P3, P1, it follows as a corollary that we have P,P,.12=12.34/13.42./14.23=P,P1.34; viz. the product of any two of the forces into the moment of the lines along which they act is equal to the product of the other two forces into the moment of the lines along which they act,-which is equivalent to Chasles's theorem, that, representing a force by a finite line of proportional magnitude, then in whatever way a system of forces is resolved into two forces, the volume of the tetrahedron formed by joining the extremities of the two representative lines is constant. November 13.-By Professor Sedgwick, F.R.S., "A Sketch of the Geology of the Valley of Dent, with some account of a destructive Avalanche which fell in the year 1752." The valley of Dent lies in the north-west corner of Yorkshire, which is thrust in between Westmoreland and Lancashire, beyond the natural limits of the county. The upper part of the valley is excavated in the carboniferous groups which are continued southwards into Nottinghamshire, and northwards into Durham and Northumberland, and through the greatest part of their range form the watershed between the east and west coasts of England. All the valleys that drain down to the land are partly formed in rocks of the carboniferous age. In the upper part of Dent Dale, which is one of these tributaries, the great scar-limestone appears only near the bottom of the valley, while the sides are formed of soft shale alternating with harder bands of sandstone and limestone; and the whole series is capped by mill-stone grit. The rainfall in some portions of the Lake mountains is not less than 150 or 160 inches in the year. Among the neighbouring carboniferous mountains the rainfall is much less; but still it is at least three times the English average; and the winter fall of snow is in some years enormous. Hence the becks, or mountain-streams, are often greatly swollen, and the gills, or lateral branches, frequently descend in brawling torrents from the mountain-side into the lower valley through deep ravines and lateral valleys that have been excavated out of the shales and sandstones in the course of past ages. On rare occasions a great fall of snow, accompanied by a violent wind, will almost fill up the ravines and lateral valleys, and form a dam across the descending water; and should there be a sudden thaw afterwards, the descending gills may be held up for a while till the pressure of the water drives down the barrier, and an avalanche is formed of mingled snow and water (provincially called a brack), which rushes down with the roar of thunder, and bears all before it into the beck below. On the 6th of February, 1752, a very large one fell, destroying several houses and farm buildings, and killing seven people, besides several head of cattle. The following letter, written by an eye-witness, describes the catastrophe (the spelling and punctuation have been slightly modernized): "Dear Bro' and Sister, "Harbourgill, 6th of the 2nd month, 1752. "These few lines I hope will find excuse for it's not without a cause that I have written no sooner to you. I fully purposed to have seen you a considerable time since: but now, as things are at present, I have lost all hopes of coming. Yet through the good providence of Heaven we are all alive and pretty well in health: which is more than could be well expected, considering what dismal times it has been with us in Dent. I hope I shall never live to see the like again for we had the greatest storm of wind and snow that continued for above a week with very little intermission: so that all the watercourses, both in the mountains and elsewhere, were made level; the like never being remembered, for it excited the curiosity of several persons to view them with wonder and astonishment: yet little thinking that the consequences would have been so tragical to many. For at the breaking up of the storm (i. e. frost) it began to rain exceedingly in the evening, which continued all night and the next day to that degree that, by 11 o'clock, the dismal scene began. For the snow in the watercouses being no longer able to sustain the great quantities of water, all began to slide down the mountains together with incredible swiftness, driving great rocks, stones, and earth, all before it; roaring like claps of thunder; which made us run out of doors to see what was coming upon us. We ran to look at the Gill; and we directed our sights (by the noise that it made) the right way; and the frightfulness of the appearance at the very first sight, which was when about the middle of the pasture, made us run for our lives; and we got no further than from the yet (i. e. gate) to the sycamore trees, before the stable, peat-house, and all the calfparrack (i. e. paddock) and cow-parrack, was in a heap of the most shocking ruins that ever your eyes beheld. I believe from the first sight of it, when it was coming, till all was overturned, was less than the quarter of a minute's time. It has brought rocks down past the middle of the houme, which had gone through the peat-house and stable, that I think three or four yoke of oxen could not be able to move. The poor old horse was crushed to pieces in a moment. Nothing but the good providence of God has preserved us from perishing; for it's amazing to think how the barn stood the violence of the shock. The waters run round our dwelling house, broke down the garden wall, and continued running through it till next day in the morning; so that it's become a bed of sand. It was about 11 o'clock when this happened, and we went from place to place, not knowing where to be safe, expecting every moment more of the like nature; which accordingly happened; for I think in the space of two hours the face of things was so changed that one scarcely could have known them. For they came down almost every slack*, carrying all the walls before them; so that we were obliged to run from one place to another to escape their fury, which was with difficulty for it continued raining extremely, that we were wet to the naked skin, : |