Cirgarclntesperte Ub hate Pesates tenn Ci! aoe vk Fret hae Sethe iss ve . y AeA RAAT, Saleem tat: pals Pas < ; ir tet ee 3. =e oo al ok fe a jo ~ wi eee rt fete Merete vee she 5 3) Seas ? ata 4) Sei hea Shee iat mt tat + Gig, Fei Stas ey ae wr ry: bas) “wt ace y = > re Bas Y ye ty Ts 4) Ohapatie . 3 ; Ae A ‘eae Oats aie ; Be hae « “3 Be \ (ie ) a ‘ < ies , if i f wh rhet > beats ents ne 9 son a . 7 ¥ Ung BB ‘ow. Smee e Digitized by the Internet Archive in 2009 with funding from University of Toronto http://www.archive.org/details/proceedingscambr01camb ot ip aS Pye ty Sep WENT ee nl - a + ROCEEDINGS oe OF THE : CAMBRIDGE PHILOSOPHICAL SOCIETY. | cd -VOL. (IT) (848 — 7é) Cambridge : PRINTED AT THE UNIVERSITY PRESS. 1876. a HY Se SAMS ST x ym Wipe! Mie OF ELAS : ba / Ak; pris hgkka 7 o€ ier vy 4P232 — / F/O PROCEEDINGS CAMBRIDGE PHILOSOPHICAL SOCIETY. November 27, 1843. On the Foundation of Algebra, No. III. By Augustus De Morgan, of Trinity College, Professor of Mathematics in University College, London, &c. . In the second paper of this series a general definition of the ope- ration A” was laid down, A and B being each of them any form of p+qv—1. The logarithm (or as Mr. De Morgan calls it, the logo- meter) of a line is thus described :—a line whose projection on the unit- axis is the logarithm of the length, and whose projection on the per- pendicular is the angle made with the unit-axis (or its arc to a radius unity). Thus a line r inclined at an angle @ has for its logometer a line /(log*r + 6?) inclined at an angle whose tangent is 4: log r. This being premised, the universal definition of A” is the line whose logometer is B x logom. A. ag _ The object of this third paper is-to-~show that the preceding defi- nition of the logometer is not the most general. Take any two lines whatsoever passing through the origin, and style them the bases of length and direction. Set off on the first a line representing the logarithm of the length in question, and on the second a line repre- senting the angle it makes with the unit-axis, both on any scale of representation. Then the diagonal of the parallelogram described on the lines just set off is a logometer to the length and direction from which it was derived; and if under this meaning of the word logometer the preceding definition of A” be employed, the equations A’ A°=A"'', A’ C'=(AC)", (A") =A” are universally true. There is no necessity for the introduction of this more general system, since all its results can be expressed in terms of those of the more simple definition in the second paper. This new definition of the logometer is really nothing more than the process answering to the extension of the theory of logarithms from the system constructed on the Napierian base, to that which is on any base whatsoever. No. I.—Procrepines or THe CamBringe Puit. Soc, 2 On the Measvre of the Force of Testimony in cases of Legal Evidence. By John Tozer, Esq., M.A., Barrister-at-Law, Fellow of Gonville and Caius College. ey The object of this paper is to show that the assumptions made by some English legal authorities on this subject, in opposition to the principles established by scientific processes, are not justified. The views more particularly dissented from, as extracted from a work of high legal authority, are thus enunciated :— ‘The notions of those who have supposed that mere moral pro- babilities or relations could ever be represented by numbers or space, and thus be subjected to arithmetical analysis, cannot but be regarded as visionary and chimerical. ‘‘ Whenever the probability is of a definite and limited nature (whether in the proportion of one hundred to one or of one thousand to one, is immaterial), it cannot be safely made the ground of con- viction; fer to act upon it in any case would be to decide, that for the sake of convicting many criminals the life of one innocent man might be sacrificed. «« The distinction between evidence of a conclusive tendency which is sufficient for the purpose, and that which is inconclusive, appears to be this: the latter is limited and concluded by some degree or other of finite probability beyond which it cannot go; the former, though not demonstrative, is attended with a degree of probability of an indefinite and unlimited nature.” The method pursued is that of investigating algebraic expressions for the probabilities that the allegations made in a case which ac- tually occurred, the trial of a female for murder, are true ; and thence deducing an expression for the probability of the truth of the charge, in passing from the symbolical to the numerical expression, the num- bers employed are not the actual values of the symbols but their limiting values; the resulting number is therefore a fraction which is not less than the value of the probability of the truth of the prin- cipal allegation, this being what in practice is required. The conclusion arrived at is, that the mode of estimating the force of evidence employed in a court is a process which algebraic inves- tigation analyses, and of which it explains the theory, and an ap- proximation to a result which is obtained with accuracy by assigning numerical values to the algebraic symbols: a clear conception of the nature of the practical process, it is conceived, must render its appli- cation more accurate, and to the extent of affording this the investi- gation is deemed to be of practical utility. December 11, 1843. On the Motion of Glaciers. By William Hopkins, M.A., F.R.S. Fellow of the Society (Second Memoir). In a previous memoir Mr. Hopkins had given the details of cer- tain experiments, by which it was proved that ice will descend with 3 a very slow unaccelerated motion down an inclined plane, presenting a surface like that of a common slab of paving-stone, at an angle scarcely exceeding half a degree (and probably also at still smaller angles), provided the lower surface of the ice in immediate contact with the inclined plane be in a state of constant but slow disintegra- tion. This experimental conclusion was brought forward in support of the sliding theory of De Saussure, and the author endeavoured to explain, according to that theory, different phenomena connected with the motion of glaciers. He there considered glacial ice as a solid substance, having a certain degree of plasticity and flezibiliiy, and the general mass of the glacier as a dislocated mass, the greater motion of the central portion of the glacier being much facilitated by these dislocations, though due partly, but in a comparatively small degree, to the plasticity of the general mass. In the present memoir Mr. Hopkins considers what would be the nature of the motion under other hypotheses respecting the constitution of glaciers. (1.) The lower part of a glacier may be conceived to be crushed, and conse- quently disintegrated, by the superincumbent weight, each compo- nent particle still retaining its solidity ; or (2.) the. whole mass may be conceived to be plastic, and to move by a change of form, pro- duced by gravity, in each component element. The author contends, if either of these hypotheses were true, that, ceteris paribus, the more superficial portion of the mass must tend to move the faster as the depth of the glacier should be greater; and that, consequently, the part of the glacier near the upper extremity must generally tend to move much faster than that near the lower extremity, assuming always the whole, or much the greater part of glacial motion, to be due to the plasticity of the mass, and to be independent of sliding over its bed. But in such case it is manifest that the general state of a glacier must be one of longitudinal compression, more particu- larly during the summer months, when the motion is greatest. Now the author contends that the general existence of transverse fissures (at least during summer) is a conclusive proof against the existence of general longitudinal compression; and he observes that no ob- server ventured to assert the fact of such compression to be dedu- cible from actual observation. He conceives this to be a serious ob- jection to the hypothesis here considered. In this memoir Mr. Hopkins has also investigated the directions in which transverse fissures must be formed when referrible to the internal tensions superinduced by the conditions to which glaciers in general are subjected, and more especially by the more rapid motion of their central portions. : Assuming the velocity of each particle of the glacier to be the same in any vertical line (which is at least true at points not remote from the surface), the glacier may, in this investigation, be consi- dered asa lamina. In this lamina take a rectangular element having two of its sides parallel to the axis of the glacier, and, therefore, the remaining sides perpendicular to it. Let X denote the intensity of the force acting normally to these latter sides of the element, Y that of the force acting normally to the two former sides. Also let f de- 4 note the intensity of the force acting tangentially on the sides on which X acts me hee It is proved that f will also be the ein. sity of the tangential force on the other two sides. Then, if : the angle which the line of maximum tension through the propose element makes with the axis of the glacier, it is proved that 2f tan 26= x_—y’ where X and Y are fensions. If either be a pressure, it must be made negative. i If the maximum tension become greater than the cohesion of the ice, a fissure will be formed in a direction perpendicular to that of the tension at each point, or at least approximately so. Conse- quently, the line whose direction is defined by the angle 6, will be a normal to the curve of fracture. Now, taking the case in which the glacial valley contracts in descending (which is the more frequent case), Y is doubtless most frequently a pressure, in which case ee tan 2 6 = X+Y > also f will be greatest at the sides (where the velocities of particles in a transverse line vary most rapidly), and will vanish at the centre. Hence 6 will vanish at the centre of the glacier, and will increase towards the sides, since the change in the value of the denominator cannot be great. Consequently, if a fissure were continued across the glacier it would form a curve, meeting the axis of the glacier at right angles; and its convexity will be turned towards the upper ex- tremity of the glacier, for the line defined by the angle 6, or the normal to the curve, meets the axis of the glacier when produced towards its Jower extremity. This is the well-known character of - transverse fissures, which the author conceives to be thus completely accounted for. In the previous memoir above referred to, this cu- rious character had been very imperfectly explained by referring it to the action of the longitudinal tension (X) alone. In conclusion the author has replied to the objections against the sliding theory urged by Prof. Forbes and others. February 5, 1844. On the Fundamental Antithesis of Philosophy. By W. Whewell, D.D. The fundamental antithesis here spoken of, is that which is vari- ously expressed by the opposition of thoughts and things, theory and fact, ideas and senses, necessary and experimental truth; also by the opposition of reflection and sensation, subject and object. It is remarked that we can have no knowledge without the union, no philosophy without the separation of these two elements. This fun- damental antithesis of philosophy is an antithesis of inseparable ele- 5 ments. It is also shown that the terms which denote the two ele- ments of this antithesis cannot in any case be applied absolutely and exclusively. We cannot say, this is a fact and not a theory, or this is a theory and not a fact; for a true theory is a fact; a fact is a fa- miliar theory. It was further observed, that the antithesis being inseparable, one element seems, and is asserted, in different systems of philosophy, to be derived from the other; ideas from experience, or experience from ideas. But we must always have both elements : thus in mechanics, and in our experience, we have necessary princi- ples, such as that every event must have a cause; and in chemistry also other necessary principles, as that the chemical composition of a body determines its kind and properties. March 4, 1844, On the Method of Least Squares. By R. L. Ellis, Esq. The aim of this paper is to give a succinct exposition of the differ- ent demonstrations by which it has been proposed to establish the validity of the rule known as the method of least squares. The first demonstration of this celebrated rule (which had been previously proposed by Legendre) is that given by Gauss in the Theoria Motds. The next appears to be that of Laplace, which has been followed, without variation of principle, by Poisson and other French writers. The demonstration of Gauss is based upon the assumption, that the arithmetical mean is the most probable result to be derived from a series of direct observations of an unknown magnitude. This as- sumption is alleged by Laplace to be altogether precarious ; and it appears that Gauss acquiesced in this remark, as he subsequently, in the Theoria Combinationis Observationum, produced another de- monstration, which is independent of this assumption. As the first method of Gauss has been followed by later writers, of whom Encke is one, it seemed desirable to endeavour to ascertain if the objection of Laplace be well-founded; and this the writer has attempted to do in the first part of the present communication. His conclusion is, that although the practice of adopting the arithmetical mean as an approximation to the true value of the unknown magnitude ob- served, is founded on just principles, yet that we are not entitled to say that it leads to the most probable result; and consequently that the demonstration in question is invalid. The writer then proceeds to consider Laplace’s demonstration. This involves no precarious assumption, but the mathematical part of the investigation is of very considerable difficulty, and cannot be said to be altogether free from doubt. For Laplace’s analysis, an- other, founded on a theorem which was first made use of by Fourier in his researches on heat, is substituted; and by this change the mathematical difficulties of the subject are very much diminished. An attempt is also made to test the accuracy of Laplace’s methods by reference to a particular case. 6 The third part of the paper relates to Gauss’s second method. The relation in which it stands to that of Laplace is distinctly pointed out; the difference between them arising merely from this, that whereas Laplace considered the importance of a positive or ne- gative error (that is, of an error in excess or defect) to be propor- tional to its arithmetical magnitude, Gauss assumes the square of the magnitude of the error as the measure of its importance, alleging that this importance not being a magnitude, does not strictly admit of numerical evaluation ; that some assumption is therefore requisite, and that that which he proposes is not more arbitrary than Laplace’s, while, from the absence of discontinuity, it leads to far simpler and more satisfactory calculations. The writer then shows that neither Laplace’s investigations, nor that of Gauss in the Theoria Combinationis Observationum, tends to prove, that the results of the method of least squares are the most probable of all possible results. This point, with regard to which there has occasionally been some degree of confusion, seems to be essential to a just apprehension of the nature of the subject. It may be remarked, with reference to it, that Laplace uniformly speaks of the method of least squares as the most advantageous method of com- bining discordant observations, or as that which gives the most ad- vantageous results, and never as a method by which the most pro- bable results are to be obtained. Lastly, the writer proceeds to consider three demonstrations of the method of least squares, given by Mr. Ivory in the Philosophical Magazine. These demonstrations are independent of the theory of probabilities. The first is founded upon an assumed analogy between the equilibrium of weights on a lever and the combination of discordant observations ; the second upon another unsatisfactory analogy ; and the third upon the principle that the error committed at one observation is independent of that committed at any other. None of these demonstrations appear to the writer to be at all con- clusive, but they seemed to deserve consideration, not only from the high reputation of their author, but also from the terms in which they have been mentioned in a recent work on the theory of probabilities. On Divergent Series, and various Points of Analysis connected with them. By Augustus De Morgan, Esq. The author states that he does not pretend to have perfect confi- dence even in convergent series. It is the main object of his paper to show that the continental analysts are not justified in their rejec- tion of some classes of divergency, and retention of others, by any- thing but experience; that they have underrated the character of most which they reject, and overrated that of all they receive. Divergent series are either of infinite divergence, such as 1+2+3 +4+ &c., in which summation of terms may give any sum, however great; or of finite divergence, such as cos §+cos 26+ ..., in which no number of terms can give more than a certain quantity. The former are rejected by most modern continental writers, the latter are retained. 7 Section 1.—All divergent series, whether their divergence be finite or infinite, stand upon the same basis, and ought to be accepted or ce Ege together, as far as any grounds of confidence are concerned which are not directly derived from experience. The author exa- mines the reasons which Poisson gives for maintaining the contra- dictory of the preceding. That great analyst considers 1 —1+1 —1+... for instance, as 1—g+g*—g°+..., where g is less than unity by an infinitely small quantity. Mr. De Morgan maintains, that this method, if allowed in transformation of a finite diverging series into a convergent one, of which the convergency only begins after an infinite number of terms, must also be allowed, unless rea- son can be shown against it, in the destruction of the infinite cha- racter of an infinitely diverging series, by the tacit retention of the infinite remainder after an infinite number of terms. The author would not use any series, so as to place absolute de- io upon their results, unless the producing functions were own: and this because series themselves neither show disconti- nuity nor infinity, when it takes place ; and because it happens that divergent series, at least, and perhaps others, may represent one thing or another, according to the genera] form of which they are made ticular cases. _ Mr. De Morgan observes that a divergent series, which is not considered as arithmetically infinite, such as 1+2+4+ ... may be so in reality, in particular cases. This series being called §, satisfies the equation S =1+425, and this gives S = —1, the usual value of the series. But it is to be remembered that an equation may be a _ degenerate case of an equation of higher degree, in which case it has one or more roots infinite. An instance is produced in which ] + 2 +4+ ... certainly represents infinity. Finally, the author remarks that there is much more safety in se- ries with terms alternately positive and negative, whether their diver- gence be finite or infinite, than in series of finite divergence, as such. Section 2.—The operation of integration, as at present understood, is one of arithmetic, as distinguished from algebra, and must not be applied unreservedly to divergent series. The author supports the first part of this assertion upon the circumstance that the only defi- nition of integration which is generally applicable is the summatory one, in which / ¢ 2 dz does not mean the function whose differen- tial coefficient is ¢ z, but the limit of the summation expressed by =(¢zAz). He then goes on to show instances in which it is un- questionably net allowable to apply integration to infinitely diver- gent series: and he asserts throughout the paper generally, that all the instances in which error has been shown to arise from the use of infinitely divergent series, have been those in which integration has been applied, and those only. In this section warning is also given against the supposition that 0+0+0-+ ... must represent 0 in all cases. Section 3.—It generally happens that the real analytical equiva- lent of the different values of an indeterminate expression, is the mean 8 of these different values. This principle is the one which was adopted by Leibnitz in his well-known explanation of the meaning of 1—1 4+1—1+... Without assuming that anything like proof can be given, the author notes, as instances in which the thing asserted is true, algebraical series, trigonometrical series, Fourier’s integral, Poisson’s expression of a function between any limits by an infinite series of trigonometrical integrals, and also the sine and cosine of ! b - infinity. Assuming /” vd: (b—a) to represent the mean value a of x between a and 6, the author tries what ought to be the value of tan «©, if this principle be true, and finds + / —1, which on trial is found to satisfy the fundamental equations of trigonometry. _ Section 4.—Series of alternately positive and negative signs stand upon a much safer basis than those in which all the terms have the same signs, and that whether their divergence be finite or infinite. It has long been observed, that when the terms of an alternating series begin by diminishing, even though they afterwards increase, the converging portion may be made effective in approximating to the arithmetical equivalent of the series. The error committed by stopping at any term is not so great as the first of the rejected terms. In many alternating series this has been proved to be true, and it seems never to have been supposed that the theorem was anything but universal. In this section instances are produced in which the theorem is not true; and at the same time various proofs of it are given, each of which applies to very extensive cases, and the tendency of which is to show that it is only under definite and unusual condi- tions that the theorem can fail. Still, however, no positive criterion is established for ascertaining whether the theorem be true or not in any particular case. Section 5.—On double infinite series, in which the terms are in- finitely continued in both directions. It seems, in many different ways, that the series .»-+9(@—2)+6(#—-1)+6274+¢6(#@+1)+¢(e+2)+... can be resolved, by analytical transformation, into0-+0+0+0+4... When there is no discontinuity whatever in the relation between g2+o(e+1)+... the value of the preceding is 0. But when dis- continuity does exist, the value of the series may be some other so- lution of }(@+1)= ye. This assertion, derived from observation of instances, is here discussed in the case of 1 9? =TH (b+eap? the value of the double series is obtained, and some corresponding products of an infinite number of factors are deduced. April 20, 1844. On the Transport of Erratic Blocks. By W. Hopkins, M.A., F.R.S. &c. The principal object of this paper is to investigate the transport- ing power of currents of water in general, and to explain in parti- cular the nature of those which would arise from the instantaneous or paroxysmal elevation of any considerable extent of the earth’s sur- face lying beneath the surface of the sea. The author has termed them elevation currents. ‘The immediate effect of an elevation like that just supposed, would be the elevation to a nearly equal height, of the surface of the superincumbent water, whence a great wave would diverge in all directions. Such a wave would be attended by a current in the direction of the wave’s propagation, and has thence been called a wave of translation. When such a wave proceeds along a uniform canal, Mr. Russell has established experimentally the fol- lowing facts :— 1. Every particle in the same transverse section of the canal has the same motion. 2. The velocity with which the wave is propagated is equal to that due to half the height of the crest of the wave above the bottom of the canal. From these data the author has calculated the velocity of the cur- rents which would necessarily attend these waves of elevation. It depends principally on the height of the elevation and the depth of the sea, while the time during which the current will flow depends principally on the extent of the elevated area and the depth of the sea. Thus if the depth of the sea should be 300 feet, and the height of the crest of the wave above the even surface of the sea (which may be considered as approximately the same as the elevation of the sud- denly raised area) should be 50 feet, the wave would be propagated with a velocity of upwards of 70 miles an hour, and the attendant current would be upwards of 10 miles an hour. Also, if the elevated area were circular, the width of the wave would exceed the radius of the circle. The wave would have the essential character of a tidal wave termed a bore, except that it diverges in all directions, instead of proceeding along a confined channel. The author next proceeds to calculate the motive power of currents of water. Let v be the velocity of the current, g, the density of the water, and § the area of a plane surface on which the current acts, and so placed as to make an angle § with the direction of the cur- rent; then if R denote the whole normal action of the current on S, we have R = Fe S sin? 6, provided § do not deviate too much from 90°. When § = 90° the truth of this formula has been proved by numerous experiments, for all velocities up to 11 or 12 miles an hour, and may be assumed to 10 hold, at least approximately, for still greater velocities. It has also been proved experimentally to be approximately true for any value of § not differing by more than 45° from a right angle, as is the case in. the applications made of the formula. a The velocity of the current just sufficient to move a block will de- pend on the volume, the specific gravity, and the form of the block. If the block slide, much will depend on the nature of the surface over which it is transported, and thus a very uncertain element will be introduced into the calculations. This uncertainty, however, will be in a great degree removed if we calculate the force sufficient to make the block rol/. Each block would present a separate problem if it were required to find accurately the current necessary to move it, but as great accuracy is not necessary in the cases here contem- plated, it is sufficient to make the calculations for a few determinate and simple forms as those to which more irregular forms may be re- ferred with a sufficient approximation to accuracy. Thus the author has considered the cases of blocks whose’sections perpendicular to their length are squares, pentagons, hexagons, &c., and has calcu- lated their dimensions, that a current of about 10 miles an hour might just be sufficient to make them move by rolling. Assuming the specific gravity of the blocks to be 2°5, we have the following results :— 1. A parallelopiped. . Side of the square section perpendicular to its length = 2°73 feet. 2. A pentagonal prism. Side of the pentagonal section perpendicular to its length = 2°27 feet. 3. A hexagonal prism. Side of hexagonal section perpendicular to its length = 2°3 feet. When the motion takes place, as here supposed, in a direction perpendicular to the length of the block, the efficiency of the cur- rent to move it will evidently be independent of the length of the block. If we suppose the length of the parallelopiped to be equal to the side of a section of it taken as above, it becomes a cube; and if we take the lengths of the blocks in the other two cases to be equal to twice the length of the sides of their sections respectively, their lengths will not much exceed their heights. Then the weights of the blocks will be 14 ton in the first, nearly 3 tons in the second, and upwards of 4 tons in the third case. Again, if the block be an oblate spheroid resting with its axis vertical, and the polar axis = $ths of the equatorial diameter, the current of about 10 miles an hour will just make it roll if its height be about 2 feet, and its weight about 4 tons. If the polar axis = 4ths of the equatorial diameter, the block will be just moved, provided its height be 34 feet and its weight 14 or 15 tons. In this part of the investigation it is shown that the power of rapid currents to transport blocks of enormous magnitude is per- fectly consistent with the almost inappreciable power of currents of which the velocity does not exceed, for instance, 2 miles an hour ; for it is shown that the weight of a block of given form and specific 11 - gravity, which may thus be moved, varies as the 6th power of the velo- city of the current. Thus if a current of 10 miles an hour will just move a block of a certain form, whose weight is 5 tons, a current of 15 miles an hour would move a block of similar form of upwards of 55 tons. A current of 20 miles an hour would, according to the same law, move a block of 320 tons, while a current of 2 miles an hour would scarcely move a small pebble. In the previous calculations the relation between the magnitude of the block and the velocity of the current has been determined on the supposition that the current, at the instant it acquires its greatest velocity, shall just be able to move the block, which would again be left at rest without being moved through any sensible space. If the velocity be greater or the mass smaller, the block will be transported to a distance which the author has calculated. Let v, be the velocity of a current just sufficient to move an assigned block ; u, the velocity of the transporting current acting on the above block, uv, being greater than vu, ; 1 the breadth of the great wave of translation producing the current ; h the height of the highest point of the wave above the level of the ocean ; Hi the depth of the ocean ; s the space through which the block is transported by the wave. The following Table gives corresponding values of these quanti- | H. | A. | V. | o | $. So: feet. | feet. | miles. /miles.|miles. 5 i nearly! | ; 200} 50| 62/12 30 5p Dearly av ter 1 372 °° 300| 50| 73\lo4, sf & | & 52 14 i Rg co 300| 100! 77/|19°4 14 oa 10) 2 8 : 34 l Fos 400'100 86/17 20 4. 19) 10 60 450 150) 95/23°5; 10, — 600 150; 106) 20°5, 10| — 800) 200 | 140 | 28 10 i) © 12 ties. The last column gives the corresponding value of the space (so) through which a particle of the water, or any body floating in the water, will be carried by the wave. The expressions for s and s, are mio i (v, — v,)* L 2 Vu, be ity aby Ee er Ya V being much greater than v,. [See preceding page. ] In estimating the magnitude of a block which may be moved bya given current, the transport is supposed to take place over a hori- zontal surface sufficiently hard and even for the block to roll upon it without impediment. In other states of the surface the transport might be more or less impeded. The constant action of denuding causes would be highly favourable to the transport by the successive removal of local impediments. ‘The author conceives that the ob- jection to this mode of transport, founded on inequalities of surface which now exist between the original site of a block and its present position, have been far too much insisted on by some geologists, for, he contends, such inequalities could not generally exist under the continued action of denuding causes, among the most powerful of which may be reckoned the transporting currents themselves. It should be remarked, that it appears from the values of s given in the preceding table, that the space through which any consider- able block could be moved by a single wave of elevation, is only equal to a small fraction of the breadth of the wave. Consequently, if such a block has been moved by this agency to a considerable di- stance from its original site, the transport must have been effected by a repetition of transporting waves; and, therefore, since a wave of considerable height can only be produced by a sudden elevation, this theory of transport is ultimately associated with the theory which attributes the more marked phenomena of geological elevation to a repetition of paroxysmal movements. The author concludes with some general observations on the evi- dence by which we may hope to distinguish between the effects of the three different agencies to which the transport of blocks may be attributed—glaciers, floating ice, and currents of water. Large an- gular blocks in the immediate neighbourhood of glacial mountains (such as the alpine blocks) may doubtless, in many cases, be referred to glaciers, while the transport of similar blocks to great distances may be referred to floating ice. Smooth rounded blocks of smaller dimensions, especially when spread out with other detrital matter in layers of considerable horizontal extent, the author would refer to the action of aqueous currents. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. October 28, 1844. On the Foundation of Algebra, No. IV.—On Triple Algebra. By Augustus De Morgan, Esq., of Trinity College. The extensions which have successively been made in algebraical ~ interpretation have been consequences of efforts to interpret symbols which presented themselves as necessary parts of the algebraical lan- guage which is suggested by arithmetic. The now well-known sig- nification of a+6,/—1 did not yield any new imaginary or unex- plained quantities: and accordingly no effort (within the author’s knowledge) was made to produce an algebra which should require three dimensions of space for its interpretation, until Sir William Rowan Hamilton wrote a paper (the first part of which appeared in the Philosophical Magazine* before the present one was begun) on a System of Quaternions. This system, as the name imports, involves four distinct species of units, one of which may by analogy be called real, the three others being imaginaries, as distinct from one another as the imaginary of ordinary algebra is from the real. These ima- ginaries are not deductions, but inventions; their laws of action on each other are assigned: this idea Mr. De Morgan desires to acknow- ledge as entirely borrowed from Sir William Hamilton. Sir William Hamilton has rejected the idea of producing a triple algebra, apparently on account of the impossibility of forming one in which such a symbol as a£ + by +c represents a line of the length V/(a2+62+c*%). Mr. De Morgan does not admit the necessity of having a symmetrical function of a, 6, c, and, throwing away this stipulation, points out a variety of triple systems, partially or wholly interpreted. Sir William Hamilton’s quaternion algebra is not entirely the same in its symbolical rules as the ordinary algebra: differing in that the equation AB=BA is discarded and AB=—BA supplies its place. Those of Mr. De Morgan’s system, which are imperfect, all give AB=BaA, but none of them give A(BC)=(AB)C, except in particu- lar cases. * Vol. xxv. pp. 10, 241. No. II.—Procrrepines or THE CamBRIDGE Puit. Soc. 14 Mr. De Morgan gives systems of triple algebra, which he distin- guishes into quadratic, cubic, and biquadratic, according as the in- vented imaginary units represent square roots, cube roots, or square and fourth roots, of the negative real unit. It would not be easy in _an abstract to give any account of these, but among them are found,— 1. An imperfect quadratic system, strongly resembling the common double algebra, and which would, but for its imperfect character, be at once recognised as the proper and natural extension of the inter- pretation of imaginary quantities to three dimensions of space: the ultimate symbol for a line is /(cos 6+sin§ /”,,—1). 2. An imperfect quadratic system, very like the former one, except in having a peculiar inversion in the operation of multiplication, and a somewhat remarkable mode of representing what would by analogy be called arithmetical multipliers. 3. A perfect quadratic system, the interpretation of which has con- siderable resemblance to that of the first-mentioned system, and is completely attainable, though not of great interest. 4. Three perfect cubic systems, each irreconcileable with the others, though closely connected with them. Each system presents a triple trigonometry, the cosine and two sines of which are each a function of two angles; but these can be easily expressed as func- tions of common circular and hyperbolic sines and cosines. The in- terpretations of these systems are very imperfect, and appear to pre- sent great difficulty, but their symbolical character is unimpeachable. 5. A perfect biquadratic system, which is of a redundant charac- ter, that is, its fundamental form represents a line drawn in space from a given origin, with a symbol to spare, which may represent the time of drawing it, its density, its tendency to a given position, &c. at pleasure. Many interpretations are attainable, but Mr. De Morgan does not pretend to say that he knows the one which ought to be adopted. It is singular that every attempt to reduce this algebra, by assigning a condition among the subsidiary symbols of its fundamental form, leads to an imperfect algebra. The system first mentioned in this abstract is one such result, and fails in its rules of multiplication, as before mentioned. Another is obtained, which is perfect as to its rules of multiplication, &c., but fails in its rules of addition. Mr. De Morgan concludes by giving some formule which may be useful to those who would try to interpret algebra of three dimen- sions by the use of solid angles in the place of plane ones. December 9, 1844. On the Values of the Sine and Cosine of an Infinite Angle. By the Rev. 8. Earnshaw, of St. John’s College. It has been usual with mathematicians to write zero as an equiva- lent for both sinz and cosz when ¢ becomes infinite. The object 15 of this paper is to examine into the propriety of this usage. The inquiry derives importance from its bearing on the general correct- ness of Fourier’s theorem for the transformation of functions, and from its affecting the truth of many remarkable results in definite in- tegrals. Certain principles also which have been assumed and acted on by Poisson, Fourier, Cauchy and others, in treating of periodic infinite series, are examined, and shown to be untenable: for exam- ple, it is shown, that as 1—z approaches zero, 1—x + z*— 25+... ad inf. does not approach 1—1+1—1+... ad inf. as its limit; that - this last series has not a unique value, and that its value is not > as has generally been argued. It is also remarked that every series of the form a, a” +a, +...4,2" +... is discontinuous in those terms which are at an infinite distance from the first, unless the co- efficients tend to zero as m andy tend to o. ‘The truth of this de- pends on a circumstance which does not seem to have been remarked before, viz. that however small 1—.z may be, a value of y can always be found so large that (1—) y may be finite, and therefore z’, which is equal to (1—1—z)’, is not equal to 1 in the limit, but to elim. of (1-2) - It is lastly proved that sino and cos are not equivalent to zero, whether we regard them as the results of integration between limits, or as the limiting cases of more general forms. February 10, 1845. On the Connexion between the Sciences of Mechanics and Geo- metry. By the Rev. H. Goodwin, of Caius College. This paper contains an attempt to determine the ground of the truth of the elementary propositions of mechanics. The remarkable analogy between mechanics and geometry suggests the thought, that perhaps there may be something more than analogy, that in fact the basis of the two may be the same. The author endeavours to show that this is really the case; the ground of the reasoning is, that force is a physical expression of the two ideas of magnitude and direction, of which a straight line is the geometrical expression, and therefore that propositions which are true for one event are true for the other. Hence it is argued, that inasmuch as the giving two sides of a tri- angle gives the third, so that the third may be considered as the re- sultant of the two already given, so if the two sides represent forces, the third will still represent the resultant of the two forces already ven. = Reasoning of this kind does not, of course, admit of a very de- monstrative character primd facie; it is the author’s design rather to point out a path to the truth, than to assert that he has cleared away every difficulty. . 16 The subject is further elucidated by the application of the remark- able symbol ef V=1,a symbol which in geometry serves to indicate the direction in which a line is drawn with respect to a given fixed line; the same symbol is perfectly applicable as a sign of affection for forces, and hence the conclusion is strengthened that the ground of truth in the two sciences is the same. The reasoning of this paper extends not only to forces, but also to velocities and moments, and to all expressions of whatever kind of the pure ideas of magnitude and direction. If the author’s reasoning be sound, the elementary propositions of mechanics are necessary truths in as strict, in fact, in exactly the same, sense as the elementary propositions of geometry; and to a mind which dwells upon them, the truths of the one science ought to appear in as axiomatic a light as those of the other. April 14, 1845. On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids. By G. G. Stokes, M.A., Fellow of Pembroke College. The theory of the equilibrium of fluids depends on the funda- mental principle, that the mutual action of two contiguous portions of a fluid is normal to the surface which separates them. This prin- ciple is assumed to be true in the common theory of fluid motion. But although the theory of hydrostatics is fully borne out by expe- riment, there are many instances of fluid motion, the laws of which entirely depend on a certain tangential force called into play by the sliding of one portion of fluid over another, or over the surface of a solid. The object of the first part of this paper is to form the equa- tions of motion of a fluid when account is taken of this tangential force, and consequently the pressure not supposed normal to the surface on which it acts, nor alike in all directions. Since the pressure in a fluid, or medium of any sort, arises di- rectly from molecular action, being in fact merely a quantity by the introduction of which we may dispense with the more immediate consideration of the molecular forces, and since the molecular forces are sensible at only insensible distances, it follows that the pressure at any point depends only on the state of the fluid in the immediate neighbourhood of that point. Let the system of pressures which exists about any point P of a fluid in motion be decomposed into a normal pressure p, alike in all directions, due to the degree of com- pression of the fluid about P, anda system S of pressures due to the motion. The author assumes that the pressures belonging to the system S depend only on the relative velocities of the parts of the fluid immediately about P, as expressed. by the nine differential co- efficients of u, v and w with respect to z,y and z. [The common no- tation is here employed.] He assumes, further, that the -relative velocities due to any arbitrary motion of rotation may be eliminated 17 without affecting the pressures of the system S. Choosing for the motion of rotation that for which the angular velocities are : ae a ae about the axis of z, with similar expressions for the ~ of y and z, the residual relative motion depends on only six independent quantities. Considering only this residual relative motion, the author shows that apatbes are always three directions, which he names aves of extension, at right angles to one another, such that if they be made the axes of 2, y,, z,, the resolved parts of the relative velocity of the point P’, whose relative co-ordinates are 2;, ¥;, 2;, Will be e' x, e!'y,, ez, along the three axes of exten- sion respectively, the point P’ being supposed indefinitely near to P. Thus the system: of pressures S is made to depend on the three quantities e’, e’, e'", which i in the case of an incompressible fluid are connected by die equation e'+e''+e"'=0. Moreover, on account of the symmetry of the motion, the pressures on planes perpendi- cular to the axes of extension will be normal to those planes. They will here be denoted by p’, p’’, p'". By what precedes, any one of these pressures, as p’, will be ex- pressed by ¢(e’, e’, e'’), the function ¢ being symmetrical with re- spect to the second and third variables. For reasons stated in the paper itself, the author was led to take, as the form of the function g, Ge’ + C'(e’ + e'"). The general expressions for the pressures would thus contain two arbitrary constants (or rather functions of the pressure and temperature), which in the case of an incompres- sible fluid would unite into one. But it is shown by the author, that in all probability p '=0 when e/= e’ =e"'; and he accordingly makes this assumption, which reduces the two constants to one, even in the case of a gas. The expression for p’ finally adopted is 2 aCe" =e 2e'). The pressures on three planes passing through P being known, the pressure on any other plane passing through that point may be found by the consideration of the motion of an indefinitely small tetrahedron of the fluid. Thus expressions are obtained for the pressures on planes parallel to the co-ordinate planes. These ex- pressions, however, contain quantities which refer to the axes of extension; and it is necessary to transform them into others con- taining quantities which refer to the axes of co-ordinates. This transformation is easily effected by means of an artifice, and then no difficulty remains in forming the equations of motion. When mu is supposed to be constant, a supposition which it is shown may in many cases be made, the equations thus obtained are those which would be yor from the common equations by subtracting oe du -dv dw e asi at ga) + Fae aoe dy ‘dz d from - in the first, and making similar changes in the other two. The particular conditions which must be satisfied at the boundaries 18 of the fluid are then considered, and the general equations applied to a few simple cases. On considering these equations the author was led to observe, that both Lagrange’s and Poisson’s proofs of the theorem that uda + vdy + wdz is always an exact differential when it is so at any instant (the pressure being supposed equal in all directions), would still apply, whereas the theorem is manifestly untrue when the tan- gential force is taken into account. This led him to perceive that one objection to these proofs is of essential importance. He has given a new proof of the theorem, which however was not necessary — to establish it, as it has been proved by M. Cauchy in a manner perfectly satisfactory. 1 . The methods employed in this paper in the case of fluids apply with equal facility to the determination of the equations of equili- brium and motion of homogeneous, uncrystallized, elastic solids, the only difference being that we have to deal with relative velocities in the former case, and with relative displacements in the latter. The only assumption which it is necessary to make, is that the pressures are linear functions of the displacements, or rather relative displace- ments, the displacements being throughout supposed extremely small. The equations thus arrived at contain two arbitrary constants, and agree with those obtained in a different manner by M. Cauchy. If Wwe suppose a certain relation to hold good between these constants, the equations reduce themselves to Poisson’s, which contain but one arbitrary constant. The equations of fluid motion which would have been arrived at by the method of this paper if the two constants €, €’ had been re- tained, have been already obtained by Poisson in a very different manner. The author has shown, that according to Poisson’s own principles, a relation may be obtained between his two constants, which reduces his equations to those finally adopted in this paper. There is one hypothesis made by Poisson in his theory of elastic solids, by virtue of which his equations contain but one arbitrary constant, which the author has pointed out reasons for regarding as improbable. He has also shown that there is ground to believe that the cubical compressibility of solids, as deduced by means of Pois- son’s theory from their extensibility when formed into rods or wires, is much too great, a conclusion which he afterwards found had been previously established by the experiments of Prof. Oersted. The equations of motion of elastic solids with two arbitrary con- stants, are the same as those which have been obtained by different authors as the equations of motion of the luminiferous ether in va- cuum. In the concluding part of ‘his paper the author has endea- voured to show that it is probable, or at least quite conceivable, that the same equations should apply to the motion of a solid, and to those very small motions of a fluid, such as the zether, which accord- ing to the undulatory theory constitute light. 19 May 12, 1845. On the Aberration of Light. By G. G. Stokes, M.A., Fellow of Pembroke College. In the common explanation of aberration, it is supposed that light comes in a straight line from a heavenly body to the surface of the earth, except in so far as it is bent by refraction. This, of course, would follow at once from the theory of emissions; but it appears at first sight difficult to reconcile with the theory of undulations, unless we make the startling supposition that the ether passes freely through the earth as the earth moves round the sun. The object of this paper is to show that if we make the following suppo- sitions, that the earth in its motion pushes the ether out of its way, that the ether close to the surface of the earth is at rest relatively to the earth, and that light is propagated through the disturbed zther as we suppose sound to be propagated air in motion, the ob- served law of aberration will still result, provided the motion of the zther be such that udx + vdy+wdz is an exact differential, where u, v, w are the resolved parts of the velocity of any particle of the z her along the rectangular axes of z, y, z. On the Pure Science of Magnitude and Direction. By the Rev. H. Goodwin, Fellow of Caius College, and of the Cambridge Philo- sophical Society. . This memoir may be considered in some degree supplementary to the preceding one by the same author, “On the Connexion of the Sciences of Mechanics and Geometry.” In that memoir it was argued, that if the views there advanced were sound, there must be such a science as that of pure direction, or rather a pure science of magnitude and direction which should include within itself the sciences of geometry, of kinematics, and of mechanics; in this the attempt is made to establish mathematically the fundamental proposition of such a science. By making use of De Moivre’s formula, the author conceives him- self to have established this proposition, that if P represents the magnitude of any cause which varies uniformly and continuously into its exact opposite, 7. e. into — P while its direction varies uni- formly from a given direction to the exactly opposite direction; and if § be the angle which the direction of P makes with a given direc- tion, then P is equivalent to two causes, P cos @ in that given direc- tion, and P sin @ in the direction perpendicular to it. The author is aware of the tmprobability which may appear to exist, that so general a proposition should be susceptible of proof without reference to particular instances, and has therefore endea- voured to obviate some objections, which will be more or less strongly felt, according to the nature of the philosophy of knowledge adopted by the mind which makes them, and which in some cases will probably be invincible. The memoir concludes with some remarks on the general question 20 of the transition of a quantity from the + to the — affection, which the author conceives to be illustrative of his general design. December 8, 1845. On the Heights of the Aurora Borealis of September 17 and October 12, 1833. By Professor Potter, A.M., of Queen’s College. The data for the calculations are almost entirely taken from the conspectuses of the observations printed and distributed in 1833 by the British Association; and although so long time has elapsed, no calculations of the heights of the phenomena, which are the first steps to be taken in finding the nature of the meteor, have, to the author’s knowledge, been hitherto published; the only imperfect discussion being given in the Philosophical Magazine for December 1833. The observers of the display of September 17, were Mr. J. Phil- lips, at York; Mr. Clare, Mr. Hadfield and the author, at or near Manchester; Professor Airy, at Cambridge; and the Hon. C. Harris, near Gosport. ; The observers of that on October 12, were Professor Sedgwick, at Dent; Mr. W. L. Wharton, near Guisborough; Mr. J. Phillips, at York; Mr. Clare, Mr. Hadfield and the author, at or near Man- chester; Dr. Robinson, at Armagh; Professor Airy, at Cambridge ; and the Hon. C. Harris, at Heron Court. The observations of the aurora of September 17 at Cambridge at 8» 25™ Greenwich time, taken with those at Manchester at 8" 24™, give the height of the lower edge of the arch 56 English miles, and ofthe upper edge 71 miles. The observations of another arch, seen from 10 49™ to 1]5 19™ at York, and from 10" 492™ to 11" 42™ near Gosport, give the height of the lower edge 389 miles. The observations on October 12, at 7 56™ at York and at 75 54™ at Cambridge, give the height of the upper edge of an arch 72°2 miles. The observations at Guisborough at 88 20™, and at Heron Court at 8 22™, give the height of the under edge of the arch seen at that time 70°9 miles, and of the upper edge 85:5 miles. The observations at Dent at 85 55™, taken with others at Man- chester at 8" 54™, give the height of the upper edge of that arch 84°97 miles. . The last arch remained stationary about a quarter of an hour, and therefore the observations are the more valuable; but combining an observation at Armagh with those at Manchester, the height comes out only 64°47 miles; and even with the utmost allowable latitude €.she deductions from the observations, the height comes out 66°5 miles. . The last arch having been noticed to have risen to a higher alti- 21 tude at the same places, a calculation with the corresponding data gives the height 65°4 miles. These last three results are remarkably in accordance with each other, but considerably different from those for other places at nearly the same time; so that probably the method which was used, of ob- taining a base line by projecting the places of observation upon an intermediate magnetic meridian, is only approximately correct, from the course of the arch over the earth’s surface, rather than geome- trical reasons. Another arch was noted by most of the observers from 105 34™ to 108 45™. The observations at Dent at 10° 40™, and Heron Court at 10" 37™, give the height of the upper edge 59-4 miles. An observation made by the author on the extent of the arch, on September 17, upon the horizon at 85 40}, and its altitude, for ap- plication to the formula he has given in the Edinburgh Journal of Science, before it was joined with the Philosophical Magazine, for de- termining the height from observations at one place by the help of an hypothesis, gave the height 53-9 miles, which is a near approxi- mateo to the height found by the trigonometrical method for 8 25™. The author concludes that the meteor occurs immediately beyond the ordinary limits assigned to the earth’s atmosphere, and from that to very much greater altitudes, as shown by many other calcula- tions; and states his conviction that the meteor will be some time observed with much more accurate means than hitherto, from its connexion with the earth’s magnetism. February 23, 1846. Analytical Investigation of the Disease prevalent in the Potato during the year 1845. By Geo. Kemp., M.D., Pet. Coll. This communication may be resolved into two parts; the analysis of the diseased portion of the potato, as compared with Boussin- gault’s analysis of the healthy tuber, and certain deductions derived from the empirical formule proposed as representing their respective compositions. The author recognises three stages of the disease: the first ap- pearing as dark brown patches under the skin; the second as strie of the same colour proceeding towards the centre; and the third as a soft, pultaceous, blackish, and offensive mass, in which all traces — of organization are lost. From the impossibility of isolating the portions affected by the disease, in the first two stages, from the surrounding sound parts, the examination was principally directed to the third stage. A potato having been selected in which the above characters were well-developed, a sufficiently large portion for comparison still remaining perfectly sound, gave the following results as indicative 22 of the relative proportions of organic and inorganic matter of the sound and unsound part. Of the sound portion, — I. 247 milligrammes gave 10° 5 milligrammes of ash. II. 205°5 Of the unsound portion,— I. 311 milligrammes gave 18 milligrammes of ash. II. 234 hae pee 13 III. 236 4. st 13°5 Or, reducing to 100 parts, the sound portion doniaiits of— ii II. Organic matter.... 95°75 95°86 Inorganic matter... 4°25 4°14 100°00 100-00 whilst the unsound portion gives,— I. II. ee Organic matter .... 94°22 94°45 94°29 Inorganic matter.... 5°78 — 5°55 5°71 100-00 100°00 ~—:100°00 The mean of the former is 4°19 per cent. of ash, that of the latter 5°68, making an excess of 1°49 per cent. arising from loss of organic matter. The ultimate analysis of the unsound portion afforded the follow-. ing results :— I. 155 milligrammes gave carbonic acid 238°5, water 98. rH 3139 ti ta iced 202 oad ee Ill. 163 er ee, a 251, water not esti- mated. Mean of two analyses for nitrogen, after the method of MM. Varrentrapp and Will, 1:23 per cent. These data furnish the following summary :— I, II. Ill. Carbon .... 42°09 41°73 41:99 Hydrogen .. 7°02 6°56 7°02 Nitrogen.... 1°23 1°23 1°23 Oxygen .... 43°98 44°80 44°08 LS 5°68 5°68 5°68 100-00 100-00 100:00 The analysis of the sound potato, by Boussingault, is as fol- OWS :— Carbon :'). n7aerse 44°1 Hydrogen ........ 5°8 Nitrogen 1:2 Oxygen’. 3s. Gapelt 43°9 As 0 0 23 The object of the second part of the communication is to show, that, while Boussingault’s analysis of the sound potato may be ex- pressed by an empirical formula, representing the elements of pro- teine, starch, and cellulose, the analysis of the tuber, after under- going the action of the late prevalent disease, admits of no such so- lution; but may be expressed by an empirical formula, representing proteine, starch, and butyric acid, with a very large excess of the elements of water. Butyric acid has been found in the diseased potato by Mr. Tilley; but the author’s principal object is to connect the changes developed by his analyses with the researches of Erdmann, Mar- chand, and Scharling, on the germination of seeds and tubers. These researches are totally independent and irrespective of the dis- ease in question, whilst it is clear that the same changes occur in both cases. After reviewing the physical circumstances with respect to soil and culture, which have proved remarkably favourable to the development of the morbid changes, the author arrives at the gene- ral conclusion, that the disease in question essentially consists in an unnatural tendency to premature germination. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. February 1846, The Mathematical Theory of the two great Solitary Waves of the first Order. By S. Earnshaw, M.A. The nomenclature of this paper is adopted from a Report on Waves by Mr. J. S. Russell, printed in the Proceedings of the British Association. From the extreme comprehensiveness of the equations of fluid motion, the author infers a necessity of appealing to experiments for the suggestion of data which may be used in mo- difying the generality of those equations so as to suit actual cases of known fluid motion. With this view he has made use of the expe- riments recorded in Mr. Russell’s report, and thence selected the two following properties :—1st. ‘The velocity of transmission of a wave in a uniform canal is constant. 2nd. The horizontal velocity is the same for all particles situated in a vertical plane, cutting the axis of the canal at right angles. By reference to Mr. Russell’s report, it will be seen that these two properties, selected on account of their sim- plicity and ready experimental examination, are distinguishing cha- racteristics of what he has denominated the two great solitary waves of the first order. By the aid of them the equations of motion take such modified forms as to admit of exact integration; so that without employing any analytical approximations the author is en- abled to obtain theoretical expressions for all the circumstances of the two solitary waves. ‘The results are tested by a comparison of the velocities of transmission of various waves given by theory and by experiment. The greatest difference of these in the case of the positive wave is not found to exceed ,},th part of the whole velocity ; but in the case of the negative wave it is found to be much greater, and to amount in one instance to as much as }th of the whole velo- city. The reason of this discrepancy is conjectured ; and the agree- ment in the case of the positive wave is considered to be exact. It is found in the course of the investigation that one of the ne- cessary conditions of fluid motion is not satisfied; and it is shown that it cannot be satisfied as long as the two principles, adopted from Mr. Russell’s report, are supposed to coexist. They are proved in fact to be incompatible with each other. But as the second prin- No. III].—Procerpines or roe CamBripGe Pui. Soc. 26 ciple was found by Mr. Russell to be so nearly exact that he could not detect any deviation from it in his experiments, it 1s shown by theory that from this circumstance there will be a rapid degradation of the summit of the wave, and a consequent loss of the velocity of its transmission, both which results of theory were observed to be true experimentally. ‘The memoir concludes with pointing out the agreement of theory with some minor phenomena noticed by Russell. May 11, 1846. A theory of Luminous Rays on the Hypothesis of Undulations. By the Rev. J. Challis, M.A., Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge. In this communication, the ether, which is supposed to be the medium of the transmission of light, is regarded as a continuous fluid substance, such that small increments of its pressure are proportional to small increments of density, and is treated mathematically accord- ing to hydrodynamical principles. The author shows, by means of the usual hydrodynamical equations, and by an additional equation of continuity, the existence and necessity of which he has considered in the Cambridge Philosophical ‘Transactions (vol. vii. part ill. pp. 885 and 386), that a given slender cylindrical portion of the fluid may continue in motion without tendency to lateral spreading, while all other parts remain at rest. It is shown,—1l, that the motion in this filament of fluid may be propagated with a uniform velocity ; 2, that in one straight line, which may be called its axis, the motion is entirely longitudinal; 3, that at all other points the motion is partly longitudinal and partly transversal; 4, that the motion is vibratory, the vibrations both longitudinal and transversal following the law of sines; 5, that the condensation (s) in any transverse plane, at a point whose co-ordinates in that plane reckoned from the axis are w and y, is given by the equation d2s ds . det dy® +gs=0, g being a certain constant. It follows that the condensation in any transverse plane, being determined by a partial differential equation, is arbitrary, and by consequence that the transverse velocity varies at a given time from point to point of any transverse plane in an arbitrary manner. To obtain the foregoing equation, it is assumed » that the condensation at any point of a transverse plane, has to the condensation at the intersection of the plane with the axis, a ratio not variable with the time. Each fluid filament in vibration is supposed in this theory to cor- respond to a ray of light. The vibrations in different fluid filaments may co-exist, and consequently rays be propagated in the same direc- 27 tion independently of each other. A ray of common light has the condensation symmetrically arranged about the axis. May 25, 1846. A Theory of the Polarization of Light on the Hypothesis of Un- dulations. By the same Author. _ This paper is a continuation of the foregoing. A ray in which the condensation is not arranged symmetrically about the axis is considered to be polarized. Polarization in this theory corresponds to difference of condensation in different directions transverse to the axis of the ray. ‘The sensation of light is due to the transverse vi- brations. By assuming that the bifurcation of a ray takes place so that the transverse velocity at each point is resolved into two velocities at right angles to each other, and that these are respectively the velocities at the corresponding points of the two parts into which the ray is divided, Professor Challis finds,—1, that if the original ray be one of common light, the two parts are symmetrical about planes at right angles to each other passing through the axis, and are each of half the intensity of the original ray; 2, ‘that if the original ray has been once polarized, the ratio of the two parts is equal to the square of the tangent of the angle which the plane of the second polariza- tion makes with that of the first; 3, that whether the original ray be one of common light or a polarized ray, the two parts, on pursuing the same path, form a compound ray the intensity of which is in- dependent of the difference of phase. According to this theory, elliptically or circularly polarized light is produced whenever a ray of first polarization is divided into two parts which subsequently pursue the same path in different phases. If the parts be made to meet in the same phase, they constitute the original polarized ray. Hence is explained the necessity of the analysing plate for the pro- duction of colours by polarized rays transmitted through thin pieces of uniaxal or biaxal crystals. The compound rays, if received directly by the eye on leaving the crystal, would be of the same intensity whatever be the difference of phase. But when they fall on the plate, those incident in the same phase, being equivalent to rays of first polarization, are incapable of reflexion, while the remainder, which are incident in the form of elliptically or circularly polarized light, are reflected in different degrees of intensity according to the difference of phase. The author states that he has extended this theory to the phenomena of double refraction. On a Change in the State of Vision of an Eye affected with a mal-formation. By G. B. Airy, Esg., Astronomer Royal. Twenty years ago, the author communicated to the Society a state- ment of the effects of a mal-formation in his left eye. The rays of light coming from a luminous point, and falling on the whole surface 28 of the pupil, do not converge to a point at any position within the eye, but converge so as to pass through two lines at right angles to each other, and, in the ordinary position of the head, inclined to the vertical, as formerly described (Transactions of the Society, vol. ii). As the luminous point is moved further from or nearer to the eye, the image of the point becomes a straight line in one or other of the positions above-mentioned. Since 1825 the inclinations of the two focal lines to the vertical, their length, and their sharpness do not appear to have undergone any sensible change, but the distances at which the luminous point must be placed to bring the focal lines re- spectively exactly upon the retina are increased, having been formerly" 8°5 and 6 inches, and being now 4°7 and 8°9 inches. Thus while the shortsightedness of the eye is diminished the astigmatism remains the same, On the Geometrical Representation of the Roots of Algebraic Equations, By the Rev. H. Goodwin, late Fellow of Caius Col- lege, and Fellow of the Cambridge Philosophical Society. The changes of value of any function of a, f (#), may be very clearly, and for some purposes very usefully represented, by tracing the curve defined by the equation z=/f (a); and the positive and negative roots of the equation f (#)=0 will be the distances from the origin at which the curve cuts the axis of 2. In this memoir a similar method is applied to the representation . of the changes of value of a function of (a), corresponding not only to real values of x, but also to values of the forma+y +/ —1. Ifwe make z=f (x+y —1), and restrict ourselves to real values of z, the equation separates itself into two, which, it is shown, may be represented symbolically by z=Ccos (5) 7@ and O= fey.) Some Xs are Ys .. : >> ia | Some Xs are not Ys 4s 'O and names which are contraries are denoted by large and small let- ters. Aristotle having excluded the contrary of a name from formal logic, and having thereby reduced the forms of proposition to four, these forms (universal affirmative, universal negative, particular affir- mative, particular negative) the writers on logic in the middle ages represented by the letters A, E, I, O. Thus X)Y and Y)X are equally represented by A. When contraries are expressly intro- duced, all the forms of assertion or denial which can obtain between two terms and their contraries, are eight in number; and the most convenient mode of representing them is as follows :—Let the letters A, E, I, O have the above meaning, but only when the order of sub- ject and predicate is XY. Then let a, e, i, o stand for the same propositions, after 2 and y, the contraries, are written for X and Y. The complete system then is— A=X)Y a=x)y=Y)X O=X:Y o=wv:y=Y:X E=X.Y e=ry and every form in which subject and predicate are in any manner chosen out of the four X, Y, z, y, so that one shall be either X or z, and the other either Y or y, is reducible to one or other of the pre- ceding. The propositions e and i,-which are thus newly introduced, are only expressible as follows, with reference to X and Y. (i.) There are things which are neither X nor Y. (e.) There is nothing but is either X or Y or both. The connexion of these eight forms is fully considered, and the 31 various syllogisms to which they lead. Rejecting every form of syl- logism in which as strong a conclusion can be deduced from a weaker premise ; rejecting, for instance, Y)X+Y)Z=XZ because XZ equally follows from Y)X+YZ, in which YZ is weaker than Y)Z—ali the forms of inference are reduced to three sets. 1. A set of two, called single because the interchange of the terms of the conclusion does not alter the syllogism. Neither of these forms are in the Aristotelian list. One of them is X)Y+Z)Y=2z; or if every X be a Y, and also every Z, then there are things which are neither X nor Z; namely, all which are not Ys. 2. A set of six, in which the interchange produces really different syllogisms of the same form, and in which both premises and con- clusion can be expressed in terms of three names, without the con- trary of either. This set includes the whole Aristotelian list, except those in which a weaker premise will give as strong a conclusion, or the one in which the same premises will give a stronger conclusion. 3. A set of six resembling the Jast in everything but this, that no one of them is expressible without the new forms e and 7; that is, requiring three names and the contraries of one or more of them. Those of the third set are not reducible to Aristotelian syllogisms, as long as the eight standard forms of assertion are adhered to. The second theory of the syllogism has its principles Jaid down in the memoir before us; but those principles are only applied to the evolution of the cases which are not admitted into the Aristotelian system. ‘The formal statement of the manner in which the ordinary cases of syllogism are connected with those peculiar to this second system is contained in an Addition. In providing that premises shall certainly furnish a conclusion, the common system requires that one at least of the premises shall speak universally of the middle term; that is, shall make its asser- tion or denial of every object of thought which is named by the middle term. Mr. De Morgan points out that this is not necessary: m being the fraction of all the cases of the middle term mentioned in one premise, and z in the other, all that is necessary is that m+n should be greater than unity. In such case, the real middle term, being the collection of all the cases by comparison of which with other things inference arises, is the fraction m+n—1 of all the possible cases of the middle term. ‘Thus, from the premises ‘most Ys are Xs’ and ‘most Ys are Zs,’ it can be inferred that some Xs are Zs, since m and n are both greater than one-half. The assignment of definite quantity to the middle term in both premises, gives a canon of inference, of which the Aristotelian rule is only a particular case. In the addition above alluded to, this same canon, namely ‘that more Ys in number than there exist separate Ys shall be spoken of in both premises together,’ is made to take the following form :—If in an affirmation or negation, in ‘As are Bs’ and ‘ As are not Bs,’ definite numerical quantity be given to both subject and predicate, if 32 it oe stated how many As are spoken of and how many Bs—the number of effective cases of the middle term is seen to be the num- ber of subjects in an affirmative proposition, whether the middle term be subject or predicate. Hence, defining the effective number of a premise to be the number of subjects if the proposition be affirmative, and the number of cases of the middle term if it be negative, all that is necessary for inference (over and above the usual condition that both premises must not be negative) is that the sum of the effective numbers of the two premises shall exceed the number of existing cases of the middle term; and the excess (being the fraction denoted by m+n—1 in the Memoir) gives the number of cases in which inference can be made. To attempt to combine these two systems of form and of quantity is rendered useless by language not possessing the forms of mixed assertion and denial, which the syllogisms deduced from the combi- nation would require. As far as the combination can, in Mr. De Morgan’s opinion, be made, nothing is required but a distinct con- ception of, and nomenclature for, the usual modes of expressing a logical form, and implying one or the other of the alternations which the mere expression leaves unsettled. Mr. De Morgan proposes the following language. , Two names are identical when each contains all that the other contains: but when all the first (and more) is contained in the second, then the first is called a subidentical of the second, and the second a superidentical of the first. ‘Two names are contrary when every- thing (or everything intended to be spoken of) is in one or the other and nothing in both. But when the two names have nothing in common, and do not between them contain everything, they are called subcontraries of one another. And again, if everything be in one or the other, and some things in both, they are called supercon- traries of one another. Lastly, if the two names have each seme- thing in common and something not in common, and moreover do not between them contain everything, each is called a complete par- ticular of the other. A table is then given, which contains every form of complex syllogism. If X and Z be the terms of the conclusion, and both be described in terms of Y, the middle term: it can be seen from this table what can be affirmed and what denied, of X with respect to Z. For in- stance, if X be supercontrary of Y, and Z subcontrary, then X must be a superidentical of Z: but if X and Z be both subidenticals of Y, nothing can be affirmed; only it may be denied that X is either contrary or supercontrary of Z. The remaining part of this paper relates to the application of the theory of probabilities above-mentioned. Mr. De Morgan asserts that no conclusion of a definite amount of probability can be formed from argument alone; but that all the results of argument must be modified by the testimony to the conclusion which exists in the mind, whether derived from the authority of others, or from the previous state of the mind itself. The foundation of this assertion is the circumstance that the insufficiency of the argument is no index of 33 the falsehood of the conclusion. Various cases are examined; but it must here be sufficient to cite one or two results. If w be the probability which the mind attaches to a certain con- clusion, @ the probability that a certain argument is valid, and 6 the probability that a certain argument for the contradiction is valid : then the probability of the truth of the conclusion is (1—d)p (1—6)u+ (1—a)(1—p)’ If 6=0, or if there be no argument against, and if the mind be unbiassed, or if p= ss this becomes 1 orat+ U sa ie 2—a For this writers on logic generally substitute a, confounding the absolute truth of the conclusion with the validity of the argument, and neglecting the possible case of the argument being invalid, and yet the conclusion true. November 23, 1846. On a New Notation for expressing various Conditions and Equa- tions in Geometry, Mechanics and Astronomy. By the Rev. M, O’Brien. If A, P, P’ be any three points in space, whether in the same straight line or not, and if the lines AP and AP’ be represented in magnitude and direction by the symbols uw and uw’, then, according to principles now well-known and universally admitted, the line PP’ is represented in magnitude and direction by the symbol u'—u. Now if AP and AP’ be equal in magnitude, and make an indefinitely small angle with each other, PP’ is an indefinitely small line at right angles to AP, and u'—u becomes du. Hence it follows, that, if « be the symbol of a line of invariable magnitude, du is the symbol of an in- definitely small line at right angles to it; and therefore, if \ be any arbitrary coefficient, Adu is the general expression for a right line perpendicular to wu. The sign dd therefore indicates perpendicularity, when put before the symbol of a line of invariable length. The object of the author is to develope this idea, and to show that it not only leads to a simple method of expressing perpendicularity, but also furnishes a notation of considerable use in expressing various conditions and equations in geometry, mechanics, astronomy, and other sciences involving the consideration of direction and magnitude. The author first reduces the sign Ad to a more convenient form, _ which not only secures the condition that u is invariable in length, but also defines the magnitude and direction of the perpendicular which Adu denotes. This he does in the following manner. He assumes u=r2a+yB+2y, 34 (where « 6 y represent three lines, each a unit in length, drawn at right angles to each other, and # y z are any arbitrary numerical coefficients,) and supposes that the differentiation denoted by d affects a By, but not xyz. This secures the condition that u is invariable in length, and leads to the following expression for Adu, viz. Adu= (zy!—z'y)a+ (w2'—a'2)B + (ya!'—y'a)y, z' y' 2' being arbitrary coefficients. : j Assuming u'=2'a+y'B+z2'y, it appears from this expression for Adu, that du=0O when u=w', and therefore that d denotes a differen- tial taken on the supposition that uw! is constant. On this account the author substitutes the symbol D, in place of Ad; he then shows that the operation D, is distributive with respect to w! (i. e. that Dy+.”==Dy+D,”), and to indicate this he elevates the subscript index u!, and writes Du!.u instead of Du. Thus he obtains the expression Du! .u=(zy!—2'y)a+ (wz! —2'z) B+ (ya'—y'z)y. From this it follows that Dw’.u is a line perpendicular both to w! and wu, and that the numerical magnitude of Du'.u is rr’ sin §, where r and 7 are the numerical magnitudes of u and u', and § the angle made by w and wu’. Having investigated the principal properties of the operation Du’, the author, -by a similar method, obtains another notation, Aw!.u, which represents the expression xa’ +yy'+22', or rr'cos§. He then gives some instances of the application of these two notations to mechanics, which may be briefly stated as follows :— Ist. If U, U', U’, &c. be the symbols* of any forces acting upon a rigid body, and u, uw’, u”, &c. the symbols of their respective points of application, then the six equations of equilibrium are included in the two equations LU=0 and TDu.U=0. 2nd. That these two equations are the necessary and sufficient conditions of equilibrium, may be proved very simply from first prin- ciples by the use of the notation Du. 3rd. The theory of couples is included in the equation 2Du.U =0. In fact the symbol Du.U expresses, in magnitude and direction, the axis of the couple by which the force U is transferred from its point of application U to the origin. 4th. Supposing that the forces U, U’, U”, &c. do not balance each other, and putting SU=V, =Du.U=W, we may show immediately, by the use of the notation Aw, that the condition of there being a single resultant is AV.W=0; and when there is not a single resultant, the axis of the couple of minimum moment is * By the symbol of a forceis meant the expression X2+Y6+Zy, where X Y Z are the three components of the citi, Rae t By the symbol of a point is meant the expression xa-} yb+z2y, where + y x are the coordinates of the point. 35 AV.W AV.V. 5th. The three equations of motion of a rigid body about its centre of gravity are included in the equation d dt u being the symbol of the position of any particle dm of the body, and U the symbol of the accelerating force acting on dm. 6th. If w be assumed to represent the expression w,a@+ w+ wyy, where w,, w,, w, are the angular velocities of the planes of yz, zz, zy about the axes of «x, y, z respectively, then the symbol of the velocity of dm is Dw.u; from which follow immediately the three well-known equations, Wi =Du. im) =2DuUam . sit Ste ao ctts GES rad =w.z—wy ¥ =wyt— wz, S =w,y—wn. dt Wat ge dt The symbol w represents in direction the axis of instantaneous rotation, and in magnitude the angular velocity about that axis. 7th. The equation (1.) may be reduced to the form d << qi (Awe Bw23 + Cw sy} ==Du.Udm, which includes ee s three equations of motion about a fixed point. 8th. If the forces U, U’, U’, &c. arise from the attraction of a distant body, the symbol of whose position is wu’, this equation may be further reduced to the form ! eee a (422 +Bu26+ Cay) = = Du'.(Az'a + By'B+ Cz'y). 9th. In the case of the earth AES by the sun or moon, this equation becomes = = aa A(Au!.y)(Du'-y) ; y being the polar axis, and A= ——. 10th. The mean daily eee of y is given by the equation OY = 0" x(ul (Duly) which equation gives “a all the well-known expressions for solar and lunar precession and nutation, for a is the symbol of the velocity of the north pole, representing that velocity both in magnitude and direction. Supplement to a Memoir on some cases of Fluid Motion. By G. G. Stokes, M.A., Fellow of Pembroke College, Cambridge. In a former paper the author had given the mathematical calcula- 36 tion of an instance of fluid motion, which seemed to offer an accurate means of comparing theory and observation in a class of motions, in which, so far as the author is aware, they had not been hitherto com- pared. The instance referred to is that in which a vessel or box of the form of a rectangular parallelepiped is filled with fluid, closed, and made to perform small oscillations. It appears from theory that the effect. of the inertia of the fluid is the same as that of a solid having the same mass, centre of gravity and principal axes, as the solidified fluid, but different principal moments of inertia. In this supplement the author gave a series for the calculation of the prin- cipal moments, which is more rapidly convergent than one which he had previously given. It is remarkable that these series, though numerically equal, appear under very different forms, the aca of the latter containing exponentials of the forms e”** and ¢ 2, while the nth term of the furmer contains exponentials of the second form only. In conclusion, the author referred to some experiments which he had performed with a box, such as that described, filled with water, employing the method of bifilar oscillations. The moment of inertia of the fluid about an axis passing through its centre of gra- vity (¢. e. the moment of inertia of the imaginary solid which may be substituted for the fluid), was a little greater as determined by experiment than as determined by theory, as might have been ex- pected, since the friction of the fluid was not considered in the cal- culation. The difference between theory and experiment varied in different cases from the ;!;th to the s;st part of the whole quantity. December 7, 1846. On the Principle of Continuity in reference to certain results of Analysis. By Professor Young of Belfast College. The object of this paper is to inquire into the influence of the law of continuity, as it affects the extreme or ultimate values of variable functions, more especially those involving infinite series and definite integrals. : The author considers that this influence has hitherto been impro- perly overlooked; and that to this circumstance is to be attributed the errors and perplexities with which the different theories of those functions are found to be embarrassed. He shows that every parti- cular case of a general analytical form—even the ultimate or limiting case—must come under the control of the law implied in that form ; this law being equally efficient throughout the entire range of indi- vidual values. Except in the limiting cases, the law in question is palpably impressed on the several particular forms; but at the limits it has been suffered to escape recognition, because indications of its presence have not been actually preseryed in the notation. It is in this way that the series 1—1+1—1+4 &c. has been con- founded with the limits of the series 1—xr+a2?—a3+ &c.; these 37 limits being arrived at by the continuous variation of zx from some inferior value up to z=1, and from some superior value down to #=1. It is shown however that the series 1—1+ &c. has no equi- valent. among the individual cases of 1—a+a?— &c., with which latter, indeed, it has no connexion whatever. By properly distinguishing between the real limits, and what is generally confounded with them, the author arrives at several con- clusions respecting the limiting values of infinite series directly op- posed to those of Cauchy, Poisson, and others. And to prevent a recurrence of errors arising from a neglect of the distinction here noticed, he proposes to call such an isolated series as 1—1+1—A&c. independent or neutral; and the extreme cases of 1—a+a*—&c., dependent series: the difference between a dependent and a neutral series becomes sufficiently marked, as respects notation, by introdu- cing into the former what the author calls the symbol of continuity, which indeed is no other than the factor, whose ascending powers Poisson introduces—and, as here shown, unwarrantably—into the successive terms of strictly neutral series; thus bringing such series under the control of a law to which in reality they owe no obedi- ence. An error somewhat analogous to this is shown to be committed in the treatment of certain definite integrals, which are here submit- ted to examination and correction, and some disputed and hitherto unsettled points in their theory fully considered. The author is thus led to what he considers an interesting fact in analysis; viz. that the differentials of certain forms require indeterminate corrections, in a manner similar to that by which determinate corrections are intro- duced into integrals; and he attributes to the neglect of these the many erroneous summations assigned to certain trigonometrical series. This is illustrated by a reference to the processes of Poisson. The paper concludes with some observations on what has been called discontinuity ; a term which the author thinks is sometimes injudiciously employed in analysis, and prefers to treat discontinuous functions as implying distinct continuities ; and by considering these in accordance with the principles established in the former part of the paper, he arrives at results for definite integrals of the form +n Kk x~P dz totally different from those obtained by Poisson. Two m notes are appended to the paper; one explaining what the author denominates insensible convergency and insensible divergency, and the other discussing some conclusions of Abel in reference to certain trigonometrical developments. March 1, 1847. On the Theory of Oscillatory Waves. By G. G. Stokes, M.A., Fellow of Pembroke College. The waves which form the subject of this paper are characterized 38 by the property of being propagated with a constant velocity, and without degradation, or change of form of any kind. The principal object of the paper is to investigate the form of these waves, and their velocity of propagation, toa second approximation; the height of the waves being supposed small, but finite. It is shown that the elevated and depressed portions of the fluid are not similar, as is the case to a first approximation ; but the hollows are broad and shallow, the elevations comparatively narrow and high. The velocity of pro- pagation is the same as to a first approximation, and is therefore independent of the height of the waves. It is remarkable that the for- ward motion of the particles near the surface is not exactly compen- sated by their backward motion, as is the case to a first approxima- tion; so that the fluid near the surface, in addition to its motion of oscillation, is flowing with a small velocity in the direction in which the waves are propagated; and this velocity admits of expression in terms of the length and height of the waves. The knowledge of this circumstance may be of some use in leading to a more correct estimate of the allowance to be made for leeway in the case of a ship at sea. The author has proceeded to a third approximation in the case in which the depth of the fluid is very great, and finds that the velocity of propagation is increased by a small quantity, which bears to the whole a ratio depending on the square of the ratio of the height ef the waves to their length. In the concluding part of the paper is given the velocity of pro- pagation of a series of waves propagated along the common surface of two fluids, of which the upper is bounded by a horizontal rigid plane. There is also given the velocity of propagation of the aboye series, as well as that of the series propagated along the upper sur- face of the upper fluid, in the case in which the upper surface is free. In these investigations the squares of small quantities are omitted. March 15, 1847. Contributions towards a System of Symbolical Geometry and Mechanics. By the Rev. M. O’Brien. The distinction which has been made by an eminent authority in mathematics between arithmetical and symbolical algebra, may be extended to most of the sciences which call in the aid of algebra. Thus we may distinguish between symbolical geometry and arithme- tical geometry, symbolical mechanics and arithmetical mechanics. ‘This distinction does not imply that in one division numbers only are used, and in the other symbols, for symbols are equally used in both ; but it relates to the degree of generality of the symbolization. In the arithmetical science, the symbols have a purely numerical signi- fication ; but in the symbolical they represent, not only abstract quantity, but also all the circumstances which, as it is expressed, affect quantity. The arithmetical science is in fact the first step of generalization, the symbolical is the complete generalization. 39 In this view of the case, the author has entitled his paper Contri- butions towards a System of Symbolical Geometry and Mechanics. The proposed geometrical system consists, first, in representing curves and surfaces, not by equations, as in the Cartesian method, but by single symbols; and secondly, in using the differential notation proposed in a former paper* to denote perpendicularity, and to ex- press various equations and conditions. The proposed mechanical system is analogous in many respects. Examples of it have already been given in the paper just quoted. The author uses the term direction unit to denote a line of a unity of length drawn in any particular direction; and he employs the symbols « ( y to denote any three direction units at right angles to each other. He defines the position of any point P in space by the symbol re- presenting the line OP (O being the origin) in magnitude and direc- tion. If x y z be the numerical values of the coordinates of P, and a 6 vy the direction units of the coordinate axes, the expression watyp+2y represents the line OP in magnitude and direction, and therefore defines the position of P. This expression he calls the symbol of the point P. If r be the numerical magnitude, and ¢ the direction unit, of OP, we have re=axat+yPBt+zy: re is therefore another form for the symbol of the point P. The following is the method by which the author represents curves and surfaces. If the symbol of a point involves an arbitrary quantity, or, as it is called, a variable parameter, the position of the point becomes inde- terminate, but so far restricted that it will be always found on some line or curve. Hence the symbol of a point becomes the symbol of a line or curve when it involves a variable parameter. In like manner, when the symbol of a point involves two variable parameters, it becomes the symbol of a surface. The parameters here spoken of are supposed to be numerical quantities. An arbitrary direction unit is clearly equivalent to two such parameters; and therefore, when the symbol of a point involves an arbitrary direction unit, it becomes the symbol of a surface. The following are examples of this method :— 1. If u be the symbol of any particular point of a right line whose direction unit is ¢, then the symbol of that right line is U-+-Tré, r being arbitrary. 2. If u be the symbol of the centre of a sphere, and r its radius, the.symbol of the surface of a sphere is ut+re, s being an arbitrary direction unit. * Read Noy. 23, 1846. 40 3. If u be the symbol of any particular point of a plane, ¢ and e! the direction units of any two lines in the plane, the symbol of the plane is utretr'e', r and r’ being arbitrary. ; 4. If « be the direction unit and r the numerical magnitude of the perpendicular from the origin on a plane, the symbol of the plane is re+Dv.e, v being an arbitrary line symbol, é. e. denoting in magnitude and direction any arbitrary line. 5. If u and w! be the symbols of two points, the symbol of the right line drawn through them is u+m(u'—u), m being arbitrary. 6. If u be the symbol of any curve in space, the symbol of the tangent at the point w is ‘ u+mdu, m being arbitrary. 7. The symbol of the osculating plane at the point u is u+mdu+m'du, m and m! being arbitrary. 8. If s denotes the length of the arc of the curve, and ¢ the direc- tion unit of the tangent, then du s= —. ds de 1 du ‘ ; 9. ay ot a d ( 9) represents a line equal to the reciprocal of the radius of curvature drawn from the point wu towards the centre of curvature, 7. e. it represents what may be called the index of curva- ture in magnitude and direction. Hence, since u=2a + y( + zy, the numerical magnitude of z d ) 1 dx\ 2 dy \2 dz a — d— d= — . ds ( a) tk ) +(4=) which is the general expression for the reciprocal of the radius of curvature. 10. The symbol of the normal which lies in the osculating plane is du u+md 3, : m being arbitrary. 11. The symbol of any normal at the point u, i. e. the symbol of the normal plane, is is u+ Dv.du, v being an arbitrary line symbol. 41 12. The symbol of the normal perpendicular to the osculating plane is u+mDd2u.du, m being arbitrary. 13. If w be the symbol of a surface, involving therefore two vari- able parameters, A and w% suppose, then the symbol of the normal at the point w is - du du u+mD Fe i m being arbitrary. 14, The symbol of the tangent plane at the point u is du du u+mdu, or e+ S Saat _ mand n being arbitrary. - 15. The symbol of the plane which contains the three points uuu! is u+m(u'—u)+n(ul!—u). 16. If u be the symbol of a right line, the symbol of the plane containing it and the point w’ is u+m(u'—u). The following are examples of the proposed mechanical system in addition to those given in the paper already quoted. 1. If r be the radius vector of a planet, and @ 3 y be chosen so that a is the direction unit of the radius vector, and y perpendicular to the plane of the orbit, it may be shown immediately by the sym- bolical method, that the symbol of the force acting on the planet is 2 1 d(r2w) or naar 6 where w is the angular velocity of the planet, and w' that of the plane of the orbit about the radius vector. The expressions for the three component forces along r, perpendicular to r, and perpendicular to the plane of the orbit, are the coefficients of a 3 y in this expres- sion. 2. The equation of motion of the planet, when the force is the attraction of a fixed centre varying as the inverse square of the di- stance, is B+ruw'y, It is curious that this equation is immediately integrable, the in- tegral being the two equations rew=h; The latter equation is the symbolical equation of a conic section, 42 the origin being focus, 4 c and «¢ being the arbitrary constants intro- duced by integration. 3. The application of this method to the case of a planet acted on by a disturbing force is worthy of particular notice, as it expresses the variations of the elements of the orbit with great facility, in the following manner :— If U be the symbol of the disturbing force, we have d(hy) _ | A =DuU 2... ss ss Cl) d(ee) _ pr aca BAB.U+U." Wat. Sea These two equations determine with great facility all the elements of the orbit. For y is a direction unit perpendicular to the plane of the orbit (é. e. it is the symbol of the pole of the orbit), and there- fore it defines completely the position of the plane of the orbit. Also e is a direction unit in the plane of the orbit at right angles to the axis major, and therefore it determines the position of the axis major ; in fact the direction unit of the axis major is Dy.e. The letters h and e have their usual signification. To find h and y separately from (1.), suppose that we obtain by integration of (1.) hy=W ; then 42 =AW.W ; and / being thus found, we have y= ul The same observation applies to (2.). 4. The expression for the parallax of the planet is e a ap. fa F~BAB.U + v). These instances suffice to show the nature of the proposed sym- bolical method. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. 2 March 1, 1847. On the Partitions of Numbers, on Combinations, and on Permu- tations. By Henry Warburton, M.P., F.R.S., F.G.S., Member of the Senate of the University of London; formerly of Trinity Col- lege, A.M. The use made by Waring of the Partitions of numbers in develo- ping the power of a polynome, induced the author to seek for some _ general and ready method of determining in how many different ways a given number can be resolved into a given number of parts. On his communicating the method described in article 5 of Section I. of this abstract, to Professor De Morgan, in the autumn of 1846, that gentleman intimated a wish that the author would turn his attention also to Combinations ; and such was the origin of the re- searches which form the subject of the 2nd and 3rd sections. I. On the Partitions of Numbers. 1. Let [N, px] denote how many different ways there are of re- solving the integer N into p integral’parts, none lessthan y. Then CN, poJ=CNt pips]. - - - - - CO) 2. Such of the p-partitions of N as contain y as a part, and no part less than y, are obtained by resolving N—y into p—1 parts not less than 7, and by adding y, as a pth part, to every such (p—1)- partition. That is, LN, pn] — [N, p I= LN» pra » «+ (IL) 3. In (II.), substitute ger, 4+2, &c. sebebatvely for 7. The sum of the results is CN, po] —CN. pr te41]=S,0N—a—sp, p—1]. - (IIL) n In this expression, when §=I* q —y, the term [N, p,4¢41] vanishes, and the formula then becomes analogous to one published N * (5 is employed to avoid the long phrase, “the integer nearest to and not exceeding —.” No. [V.—Procrxrpines of THE CamMBRIDGE Puit. Soc, 44 anonymously by Professor De Morgan in a paper printed in the fourth volume, p. 87, of the Cambridge Mathematical Journal. 4. In(II.), for [N, p,41] substitute [N—py, p,], and transpose the terms. Then (N. pa] —(N-9.p— 11 =(N—py pds ene and this leads to [N—y, p—1]—[N—2, p—2]=(N—pr, p—1] and that leads to the summation CN, pn ]=SoCN—py, 21]. . «fee The lower limit of z in (V.) is made 0, in order that the formula may comprehend the extreme case (0, U, ]=1, analogous to the ex- treme case in Combinations. 5. After substituting 1 for y, the author applies formula (IV.) to determining in how many different ways N can be resolved into p parts not less than 1. Let [N, p,] be the term in a table of double entry corresponding to column N, line p, in the table. From the head, in line 0, of each of the columns 0, 1, 2, 3, &c., draw a diagonal, advancing one column and one line atatime. Take these diagonals one after another, and in each of them compute by formula (1V.) the terms situate on lines 0, 1, 2, 3, &c., one by one in succession. IfN be the number at the head of the column from which any diagonal takes its departure, there will be only N terms to compute on that diagonal, the further terms being only repetitions of the term on the line N. For the diagonal in question intersects line N in column 2N; and, by formula V, [2N,N,J=SOLN, 2] Zz = the sum of all the terms in column N. But, moreover, [2N+y,Ni+y]=SoLN, 4] = the same constant. The leading property of the table, indicated by the formula [N, p.]=So[N—p, 4], is, that the term [N, p, ]= the sum of all the terms in column N—p, from line 0 to line p inclusive. After the publication of the anony- mous paper before referred to, Professor De Morgan discovered this theorem also, but he did not announce it*. II. On Combinations. 1. In ordinary Combinations, the combining elements are of differ- ent kinds, and there is but one element of a kind: in the case here considered, there are different kinds of elements, and there may be many elements of a kind; and more than one element of a kind may enter into the same combination. * The author has recently discovered an equivalent formula in p. 264 of Euler’s Int. in An. Infinitorum ; but investigated by a totally different method, and not applied as the author has applied it. 45 2. If u elements enter at a time into each combination, and the kinds are determinate in number, and their number is s, let { c \ denote how many different combinations can then be formed: if the elements are determinate in number, and their number is ¢, let the number of the combinations which can then be constructed, be denoted by {u,o}. If (x) be any function of x, let Dv g(~) denote the co- efficient of 2” in that function developed according to the powers of x. 3. The same things as before being assumed, let a given set of elements consist of a elements of the kind A, + elements of the kind B, + &c. Take the product, K, of the s geometrical pregres- sions, [1+Az+A2e®+ ....4A%*], [14+ Br+Bex?+....+BSr*],&. Then K will be of the form, 1+S[A]a+S[A?+AB]a2+ S[A*+A*B+ ABC ]a° + &c., and D«[K] will be of the form S[A?BICr &c.], the last expression being an aggregate of terms of the form A?B/C’... , each containing a different combination of u of the given elements, and their sum comprehending all the possible combinations of those elements taken u at a time. Now, if A, B,C, &c. be each made equal to 1, K will become k=[1l+e+a°+.,+2a*][l+a24+2°+.. +2°}. &e. ; each of the terms A?.B7. C”. &c. will become 1, and the number of all the terms of the form A?B7C"... which D¥[K] or S [A?B7C’... ] contains, that is to say, {u, 7} will be represented by Dv [k]; which latter coefficient the author next proceeds to determine. Now pe rattt 1oatt! l—x - 1-2 =[1-a*t'yp1—at7..8% pee] 1ejl . &e. =(i—a"thypi—aPt}..[1—2]~"] ie pitty pi—-#t'}... 4 8% [ cut 1a | (VIL.) For brevity, write u,, a, A, &c. respectively, foru+1,a+1,8+1, &c.; andalso write [1] for [1—2]—*; [2]for[1—a][1—a2]~*? that is, for [1—2%1].[1]; [3] for [1—a] (1—a] [1-2] ~*3 that is, for [1—a,].[2], and soon. Then D“[2]=D*[1]—D*—™[1]; and D*“[3]=D"[2]—D*—“ 2}; * According to the factorial notation, here used by the author, suit! represents s [s+1][s+2]...[s+(u—1)]. 46 © D“[4]=D"[3]—D“%— [3]; and so on; (VIII.) and the developed product of the binomes, Cl—ax],[1—a*1],[1—a], &e. ; that is to say, l—a%taathi—guthitn+ &e. — it gt —&e, —7N1 +a41tn —&c. + &c. when multiplied into the development of [1—a#]~%, manifestly leads to the following formula : D“[m]=D"[1]—S [D*-aht]]+ Ss fomse soa a Bite cli "[1)]] + &e. where, since the powers of x, in (VI.) or (VII.) developed, are to be all positive, no expression of the form (u—m), (u—a—P,), (u—m—Ai—N), &e. is to be negative. Then by giving to D*[1], D*-"1[1], D¥- if), Se, ee their respective values, we obtain the series of expressions : 1 ME ull Dv(1j= Te 1 - [ui =[5 ] where in all the kinds the elements are plural without limit; a for- mula given by Hirsch : Dv[2]= = 4 [we _ [ma] | a fe where the elements A are limited in number to @, but those of the other ie kinds are plural without limit : D*[3]= Shoe 1) — [ay — a8 1 + [a — a, — B81 ee =n —[u,—B,]s- aS where, moreover, the elements B are limited in number to #, but those of the other (s—2) kinds are plural without limit : and so for the rest. The law of the terms being evident, they need not be continued further. Example of ([X.). Given one element of 1 kind, two elements of a 2nd kind, three of a 3rd, and four of a 4th; and let u=5. Then ‘ —4.5.6 er aetie —3.4.5+1.2.3 {u, o} = 1.28 6.7. 8 osehe ¢. 4 era —1.2.3 (VIII*.) 4 7 4. If a2=B=y=K&c., formula (IX.) becomes | iris: 3 % 6 sfl—1 .. {we}= rt, sC) [ (-1) “ar mba] mn ]. (X.) Example of formula (X.) Given seven kinds of elements, and three of each kind; and let u=4. Then a) {u, ¢}= | 5.6.7.8.9.10—7.1.2.8.4.5.6 |=203. 5. If it is required to determine many, or all, of the terms of the » {7,0}, formulas (VIII.) sug- gest the following process for the determination of those terms. An example will best explain the process. Given 1 element of one kind, 2 elements of a second kind, and 3 of a third kind. How many combinations can be formed from these elements, when taken 0, 1, 2, 3, 4, 5, 6 at a time, respectively ? series {0,ct, {l, ov}, {2,o}, pas EE av sixkcnnsesssessescnvegsonasce «9s 0 1 2 3 4 5 6 Coefficients of 2“ inf l—a] —3 ...............00 1 3 6 10 | 15 } 21 28 Multiply by [1—a*]; that is, subtract ...... E os 3 6 | 10 | 15 Coefficients of 2“ in [1—a?][1—z]~* ...... 1 3 5 7 9) 4-18 Multiply by [1—.23] ; that is, subtract ...... Sea) Bete Ble ay bees 3 5 7 Coefficients of 2* inf1—2°][1—23][1-z]—3 1 | 3 | 5 | 6 | 6 | 6 | 6 Multiply by [1 —2*]; that is, subtract ...... wee | nen | ee | eee 1 3 5 Coefficients of 2” in[1—2?] i ad 1 3 5 6 5 3 1 MMU EBGAET. ahs. Jivcsises.ccleccacucccsecaibeces {0,0} {1,0} {2,0} {3,0} {4,0} {5,o}/{6,o} 6. Let a set of elements, S, such as we have been previously con- sidering, consist of two similar sets, T and T’, which do not contain in common any elements of the same kind. If S consists of ¢ ele- ments combining w at a time, and T consists of 7 elements combining v at a time, T’ will consist of (e—7) elements combining (u—v) at atime. Consider wu as constant, for the moment, and v as variable. The author then shows that if by the process described in art. 5, the whole series of terms fv, T} and the whole series of terms {u— v, ¢—7}, have been determined, we can thence determine the whole series of terms {u,o} by means of the formula {u,o}= SiL{e-t}-{u—e, o—T} | PA get Gade) and of this he gives examples. 7. In formula (XI.) substitute (r—z) for u; and develope {u, o} and {*—u,c} in the manner indicated by that formula. By com- paring the Ist, 2nd, 3rd, &c. terms respectively of {u, ¢} with the last, last but one, last but two, &c. terms of {o—u, ¢}, and vice 48 versd, the author shows that {u, 7} will be identical with {«—w, c}, provided {v, 7} is identical with {r—v, 7}, and provided also {u—v, o—r} is identical with {s—r—(u—v),7—T}. But this identity actually exists when T consists of elements of one kind only, and when T” also consists of elements of one kind only. For, in that case, every term of the series {v,7} and every term of the series {u—v,¢—7} is equal to 1. Let the elements of the single kind which T contains, be different from those of the single kind which T’ contains. Then the identity in question will exist, when S con- sists of elements, finite in number, of two different kinds: conse- quently, it exists also when T consists of elements, finite in number, of two different kinds, and T’ consists of elements, finite in number, of one or two other kind or kinds; that is, when S consists of ele- ments, finite in number, of three or four different kinds. And there- fore universally, in the case as well of finitely plural, as of singular elements, the following law obtains : * {uo}=fo—u, oe}. 2 ee. (XII.) Hence it follows that in applying formulas (IX.) and (X.) to parti- cular cases, the labour of computation will be shortened by substi- tuting for the variable the lesser of the two numbers u and o—u. 8. ‘The author next considers how many different combinations can be formed from a given set of elements, when every combination is to be constructed in conformity with a given type; in which ty there are m different kinds containing v elements each, m! other dif- ferent kinds containing v! elements each, m” other different kinds containing v” elements each, and so on; and where, consequently, in each combination, z, the number of kinds, is m-+-m'+m"+ &c.; and uw, the number of elements, is mvu-+m!'v!-+m"v" + &e. The type remaining constant, any combination conformable thereto may be altered, either by changing the particular z kinds which are selected out of the s given kinds; or, the kinds remaining the same, by alter- ing the distribution of the parts v, v, v,... (m)v’, v', v', ... (m')v",v",v”, nae (m") &c., among those kinds. When all the elements are plural without limit, the changes of the former description will be repre- sented by sti-l jail ; and those of the latter description by #1 1™)1, | m'[1, [m1 ee: os and their joint effect by the product s?l-1 1212 rr * jwall [mt wh, * (XIII) But when the elements of all the given kinds are finite in number, class these kinds, so that each kind in class 1 contains not fewer 49 than v elements; each kind in class 2 contains fewer than », but not fewer than v’.elements ; each kind in class 3 contains fewer than v’, but not fewer than v” elements; and so on; and so that the given kinds may in this way be reduced, say, to ¢ kinds containing v ele- ments each + T’ kinds containing v' elements each + T” kinds con- taining v” elements each, &c. Then let¢—m+T'=?'; t'—m! +T'=?; and so on. The given kinds being thus ordered, since we are required to select, lst, m out of ¢ kinds ; then, 2nd, m!’ out of ¢' kinds; then, 3rd, m” out of ¢” kinds; and so on; the number of the different combinations which can be constructed from those kinds in confor- mity with the type, will be . gm|-1 '¢m'i—1—"¢m"|-1 jm * [mj * [m1 If tv+-T'v'+T’v’.. &c. is reduced to a single term, ¢.v; then formula (XIV.) becomes 8 Page iy, Es f#|-1 [oh lH, [mt Example of (XIV.). Given eight elements of 1 kind, seven of a 2nd kind, six of a-3rd, five of a 4th, four of a 5th, three of a 6th, two of a 7th, and one element of an 8th kind, out of which it is re- quired to construct combinations, each consisting of three kinds with five elements each + two kinds with three elements each + one kind with two elements. Of such combinations there can be formed 43|-1 32|-1 -2 132 © y22 * yan = 24. Sie. @, deka 2d OF eagy’y 9. If it be required to determine how many different combinations can be constructed, each containing u elements of z kinds, and the given elements are all finite in number; we must form all the differ- ent z-partitions of w; and each of these partitions being regarded as a type, we must determine, by formula (XIV.) or (XV.), how many combinations correspond to each of these types; and the total num- ber required will be the sum of all these particular determinations. But if the given elements may all be repeated without limit, it follows from formula (XIII.), that the sum of all the particular de- terminations may be represented by 7 yz) jen * S (Gar jm, ym", w.)" Ss 1411 (Gar [mL pm ) denotes how many different permutations can be formed, when, in each different z-partition of u, the parts are permuted z together at a time ; and the number of such permutations is Sif) J=3—[w-1] = od 27111 Now 50 Consequently the required sum is g#l-1 (w—1]?-"-} yar ps eae e . : (XVI.) If in (XVI.) z varies from 0 to u—1, u—1[ sz|—1 [u—1]?7-1 __ sul this summation being a particular case of formula (XI.). The result agrees with Dv[1] formula (IX.), art. 3. 10. When the given elements are all finite in number, we may determine {u, 7}, by taking the sum of all the particular determina- tions that may be obtained pursuant to art. 9, by giving to z the successive values 0, 1, 2,3, &c. If us, the upper limit ofziss; and the number of types to be formed is [u+s,s,]. (See articles 4 and 5, Section I.) But, if the repetition is finite, some of these partitions may fail to yield combinations. 11. If the elements A, B, C, &c. represent different prime num: bers, all the methods and theorems contained in this section will apply, mutatis mutandis, to the composite numbers of which those primes, or the powers of those primes, are divisors. III. On Permutations. 1. Let the given elements be of s different kinds. We can de- termine in two known cases, by an explicit function of w, when the elements are taken u at a time, in how many different ways they can be permuted. The number of the permutations is denoted, when there is but one element of a kind, by s“|-1; and when in all the kinds the elements are plural without limit, by s¥. When the plu- rality is finite, it is only in the particular case of all the elements being permuted at a time, that there is a known formula to express the number of their permutations. 2. Every combination constructed on a given type, u=mu+m'v! +m"v" + &c., will generate the same number of permutations, yu P [ie] Celt jm’ Lem” : &e. =. > Therefore, if the number of the different combinations which can be constructed out of the given elements in conformity with that type, is represented by Q, QP will be the number of the permutations corresponding to the type and to those elements. If the plurality be without limit, s*|-1 jal [ai wees * F will be the number of the permutations. If the given elements be finite in number, as in formulas (XIV.) and (XV.), the number of 51 the permutations corresponding to those elements and to the type, will be gmi—1 fym'l—-1 “gm -1 pm * ym * ym &e. x P. Every different partition of w that may be formed within the limits pointed out in art. 10, Section IJ., will give rise to a similar product, ‘QxP; and the sum of all these particular products, S[Q x P], will show how many different permutations can be formed from the given elements, taken u at a time. The author illustrates this method of computing the number of permutations, by examples. 3. Let P oie denote how many different permutations can be formed when u elements are taken at a time out of s kinds ; and P {u, 7} denote how many different permutations can be formed when u elements are taken at a time out of ¢, a finite number of elements. If all the elements may be repeated without limit, {0} pets ee}ae =D«[mfitet B+. art: en}: Hence the author infers that, if the elements A are limited in num- ber to a, while those of the other (s—1) kinds are plural without limit, r{i}am[men [reer er) that if, moreover, the elements B are limited in number to #, while the other (s—2) kinds are plural without limit, u : : P p \ =p«/ 1 Ctra ee +e+ = +.. ar | [itet rt vier aa |} and so on, until finally, if all the elements are finite in number, and the elements A, B, C, &c. are respectively limited, in point of num- ber, to a, 6, y, &c., P{u, ey=D"[ eit[1+e+.. 3 JID +2 [ite+.. = = HE 4. Hence, if in all the s kinds the elements are dual, (XVII.) becomes im . (XVIL) 52 Pfu, ot=D«| yuif eT a} eo [ Boe: a | . (XVIIL) gi) [eee 7. / 7° 94|2 This is the only addition which the author has been able to make to the cases wherein P ix , or P{u, «} is expressed by an explicit function of u, symmetrical in form. Example. Let there be five kinds of elements, and two of each kind. Let u=3. 15.4.3 , 3.2x5.4_ Pfu, o} = 4 XP = 120. 5. The author gives the following theorem, which is precisely analogous to that of art. 6, Sect. II., formula (XI.), in Combina- tions; viz. ‘ i T yi-) P{u,o}=S [a P{v,r}.Pfu—v, ot} |. (XIX.) 6. By a mode of proof precisely analogous to that employed in art. 7, Sect. IL, he shows that P{e—1,¢}=P{o,o}; that is to say, that — loll 1411, 1411, 17ll, &e. denotes the number of permutations that can be formed with a ele- ments A, (3 elements B, &c. (where [a+B+y+ &c.]=c), as well when «—]1 elements, as when ¢ elements, are taken at a time. Since correcting his paper for publication, the author has had his attention called to the work of Bézout on Elimination (4to. Paris, 1779, p. 469), as containing a formula similar in structure to that numbered VIII*. in the present abstract. Bézout investigates the composition of a polynome function of s quantities, A, B, C, &c., consisting of terms which are of the form A? B¢C’, and of every dimension from 0 to u inclusive. Let [s]” denote such a polynome, complete in all its terms, and N[s]” the number of its terms. Then, Ist, u— (ut1 js, NLS eT ea and 2nd, the number of the terms in [s]” which are not divisible by either A%, or B®, or C7, &c., he expresses by N[s]“—N[s]“—* +N[s]4—4-8 — &e. | —N[s]“—*+ &. — &e, 53 He also observes (p. 39) that when A%, B4, Cy, &c. are the high- est powers of A, B, C, &c. which a polynome, agreeing in other respects with [s]”, contains, the terms of such incomplete polynome agree in point of number with those terms in [s]“ which are not divisible by either A*+1, or B°+!, or Cy+!, &c. The polynomes from which Bézout proposes to eliminate certain terms, contain terms of all dimensions from 0 to u inclusive. The terms which are to remain after the others have been eliminated, and which are enu- merated by means of the condition, that they are not divisible by certain powers of A, or B, or C, &c., may be of all dimensions indis- criminately from 0 to wu inclusive. Bézout’s object is exclusively Elimination, and he makes no allusion to any other application of his formule. The polynomes considered by the author, taken in their entirety, agree in their general structure with those considered by Bézout; but the nature of the author’s inquiries led him to confine his atten- tion to the composition of those particular terms in a polynome which were of the same dimension; and to seek to express the number of the terms, not of all dimensions indiscriminately, but of each particular dimension separately. To show how it has hap- pened that researches, very different at their point of departure, have, as regards one point of investigation, ended in nearly similar formule, the author proceeds to deduce his formula (VIII*.) from the investigations of Bézout. Such a deduction, he conceives, might readily have been made by any one to whom it had occurred to make it ; and the application of such a deduction, when once made, to pro- blems in Combinations, would have been much too obvious to have remained long unnoticed. Expressions of the form above considered are regarded by Bézout as of the nature of Differences; and the truth of this view of the subject may be shown in the following brief manner. If 9(x) generates )(u),[ 1—a=]9(x) will generate }(u)—P(u—«), which we may denote by Ag(w). Consequently [1—x*][1—2«] 9(«), that is to say, i eccr a4 9(2) will generate A? av(u); and so on; the independent variable, u, undergoing, not uniform, but variable decrements, a, 6, y, &c. May 3, 1847. On the Internal Pressure to which Rock Masses may be subjected, and its possible influence in the Production of the Laminated Struc- ture. By W. Hopkins, M.A., F.R.S. If a plane of indefinitely small extent pass through any proposed point in the interior of a continuous solid mass in a state of con- 54 straint, the resultant pressure or tension on this plane will vary with the angular position of the plane, and its direction will not, as in fluid masses, be generally perpendicular to the plane. There are, however, three angular positions in which the direction of the pres- sure does coincide with a perpendicular to the plane. These are called principal directions, and are at right angles to each other; the corresponding pressures are called principal pressures. In these par- ticular positions of the plane there will be no tangential action upon it ; but generally the whole pressure or tension may be resolved into two parts, of which one is normal and the other tangential. In cer- tain positions of the plane these forces assume their maximum or minimum values. The normal action is a maximum, when a per- pendicular to the plane coincides with one of the three principal directions; and a minimum, when it coincides with another, the third of those directions, not corresponding either to a maximum or minimum value. These conclusions have been established by Poisson, Cauchy and others. In this paper the author has investigated the positions of the small plane, when the tangential force upon it isa maximum. ‘There are two of these positions perpendicular to each other, in each of which the plane passes through that principal di- rection which does not correspond to either the maximum or mini- mum value of the normal force, and bisects the corresponding right angle between the other two principal directions—those of the maxi- mum and minimum normal forces. Having established the relative positions of the planes of greatest normal and of greatest tangential action, the author proceeds to examine how far the evidence afforded by the distorted forms of organic remains may justify the conclusion that these forces have had an influence in determining the position of the planes of cleavage in the rocks containing those remains. Conceive one stratified bed placed on another, and acted on by forces tending to give the upper a small sliding motion along the surface of the lower one. A considerable tangential force will be called into action between the beds; and if any object be placed between them, its lower part will be pushed in one direction by the action of the lower bed, while its upper part will be equally pushed in the opposite direction by the action of the upper bed, and thus the object will be twisted from its original form. For example, sup- pose the object be an equilateral shell lying between the two beds, with the plane of junction of the two valves parallel to the surfaces of the beds, and suppose the median line of either valve to be per- pendicular to the direction in which the one bed tends to move along the other. The shell in its distorted form will no longer be equila- teral ; one half of each shell will be crumpled into a smaller space, while the other half will be extended into greater breadth ; so that if there be longitudinal folds on the valve, those on the former half will be pressed together, and those on the latter will be dilated into greater breadth. An exactly similar effect will be produced on both shells ; but the compressed half of one will be opposite to the dilated half of the other. Again, suppose the beds to be acted on by forces tending to com- 39 press them equally in a direction parallel to their surfaces. The shell will then be compressed in the same direction, so that, generally, the ratio of the length to the breadth of the shell will be altered, but without that twisting which will characterize the distorted form in the former case. In the case of this paragraph, the direction of compression will coincide with what has been above termed a prin- cipal direction, and it will also be that of maximum normal pressure. In the previous case, the common surface of the two beds will be the plane of maximum tangential action. If, then, in any stratified mass, we observe the organic remains to be regularly distorted, and twisted from their original forms, as above described, we may conclude that the planes of stratification have nearly coincided with those of maximum tangential action; but if, on the contrary, the distortion consists only in compression of the shells in a given direction along the surface of the bed where they are found, we may conclude that the direction of maximum normal pres- sure has nearly coincided with this direction of compression, and was consequently parallel to the planes of stratification. ‘The masses in which distorted remains have been found, are generally those which have been much disturbed. ‘The disturbing forces are those to which the distortions are to be referred ; and it may be remarked, that in such cases the directions of maximum and minimum pressure at any point would probably lie in a plane perpendicular to the strike of the elevated beds, and that consequently the planes of maximum tan- gential action, which bisect the angles between those directions, will have approximately the same strike as the beds themselves. The bearing of these conclusions on the question of laminated structure is easily seen. Suppose the planes of lamination are ob- served to be nearly coincident with those of stratification, and that the distortion of the organic remains consists in their being twisted from their primitive forms. Then, if the position of the planes of lamination has been due to the internal pressures to which the mass has been subjected, it is to tangential action, and not to direct pres- sure, that the effect is attributable. Again, if the planes of lamina- tion have nearly the same strike as the beds, and are inclined to them at an angle of about 45°, while the organic remains have been dis- torted only by direct compression, the planes of lamination must in this case also have coincided with those of maximum tangential action, and we shall have the same conclusion as in the former case. The direction of compression of the organic forms ought, according to this view, to be perpendicular to the intersections of the planes of lamination and those of stratification. Mr. Sharpe, in a paper recently published in the Journal of the Geological Society, has stated nearly all the evidence hitherto col- lected on this subject ; and it appears that the organic bodies are most twisted from their original forms in those cases in which the planes of lamination coincide most nearly with those of stratification, and that they have generally suffered most direct compression without twisting in those cases in which the planes of lamination are inclined to those of stratification at an angle of 40° or 50°. We must 56 therefore conclude, according to the last paragraph, that the planes of lamination approximately coincide with those which were formerly the planes of greatest tangential action. ; as The author does not regard this mechanical action as the probable primary cause of the laminated structure, but rather as a secondary cause, which may have had its influence in determining the positions of the planes of lamination. He trusts that further evidence will be collected on the subject. May 17, 1847. On the Symbolical Equation of Vibratory Motion of an Elastic Medium, whether Crystallized or Uncrystallized. By the Rev. M. O’Brien, late Fellow of Caius College, Professor of Natural Philo- sophy and Astronomy in King’s College, London. , The object of the author in this paper is twofold : first, to show that the equations of vibratory motion of a crystallized or uncrystal- lized medium may be obtained in their most general form, and very simply, without making any assumption as to the nature of the mo- lecular forces; and secondly, to exemplify the use of the symbolical method and notation explained in two papers read before the Society during the present academical year. First, with regard to the method of obtaining the equations of vibratory motion. This method consists in representing the disarrangement (or state of relative displacement) of the medium in the vicinity of the point xyz by the equation dv dv dv 1 dv dv Spee —— Or +- — by be. 2S hae ee , a ee, e+ ~ James a4. ose ate + dnlg et &e (where v=fa+yP+%y, yf denoting, as usual, the displacements at the point zyz, and ay being the direction units of the three coordinate axes), and in finding the whole force brought into play at the point xyz (in consequence of this disarrangement) by the symbo- lical addition of the different forces brought into play by the several terms of dv, each considered separately. It is easy to see that these different forces may be found with great facility, without assuming anything respecting the constitution of the medium more than this, that it possesses direct and lateral elasticity. By direct elasticity we mean that elasticity in virtue of which direct or normal vibrations take place ; and by Jateral, that in virtue of which Jateral or transverse vibrations take place. The forces due to the several terms of dv are obtained by means of the following simple considerations. Let AB be any line in a perfectly uniform medium, and conceive the medium to be divided into elementary slices by planes perpen- dicular to AB; let OM(=z) be the distance of any slice PP’ from any particular point O of AB, and suppose this slice to suffer a dis- 57 placement equal to =o (c being a constant) in the direction OAB, and the other slices to be similarly displaced. Then it is evident that the medium suffers by these displacements a uniformly increasing expansion in the direction OB, and a uniformly increasing condensa- tion in the direction OA ; the rate of increase both of the expansion and condensation being c. Now in all known substances, whether solid, fluid, or gaseous, a disarrangement of this kind would bring into play on the slice O a force along the line AB proportional to the rate of increase c, i. e. a force Ac, A being a constant depending upon what we may call the direct elasticity of the substance. Again, suppose that the slice PP’ receives a displacement = ca? in the direction OC perpendicular to AB, and the other slices similar displacements. Then the line AB will become curved into a para- bola A’OB’, and all the lines of the medium parallel to AB will be similarly curved, the radius of curvature being equal to J and per- c pendicular to AB. Now in all known substances* a disarrangement of this kind would bring into play upon the slice O a force in the direction OC proportional to the curvature c, 7. e. a force Be depend- ing upon what we may call the /ateral elasticity of the substance. Lastly, suppose that MP=y, and that the point P of the medium receives a displacement cay parallel to AB, and the other points similar displacements. Then the slice PP’ will, in consequence of this kind of displacement,.turn through an angle tan—!(czx) into the dotted position, and the other slices will suffer similar rotations, those on the other side of O, such as QQ’, turning the opposite way. Now it is easy to see that a disarrangement of this kind produces a uniformly increasing expansion in the direction OC, and a uniformly i ine condensation in the direction OC’, the rate of increase both of the expansion and condensation being c. But the expansion and condensation here described are quite different from that pre- viously noticed; since it is produced, not by displacements parallel to C'C, but by Jateral displacements, i. e. perpendicular toC’'C. On this account all that we can assert without further investigation is, that the force brought into play upon an element at O by this dis- arrangement acts along the line C’C, and is proportional to ¢, i. e. equal to Cc, where C is some constant evidently depending in some way both upon the direct and Jateral elasticity of the medium. There is however a very simple way of finding the precise value of the force brought into play by a disarrangement of this kind ; for if we turn the axes of x and y in the plane of the paper through an angle of 45°, it appears that this disarrangement is nothing but a combination of the two kinds of disarrangement previously noticed ; and from this it immediately follows, in the case of an uncrystallized medium, that the force brought into play at O is(A—B)c; in other * Fluids and gases possess lateral elasticity as well as solids, only in a comparatively feeble degree. 58 words, the coefficient C, which must be multiplied into c, in order to give the force brought into play by the disarrangement cay, is equal to the coefficient of direct elasticity (A) minus the coefficient of lateral elasticity (B). . In the case of a crystallized medium, it may be shown that six _ relations, corresponding to the relation C=A—B, are most probably true, and are essential to Fresnel’s theory of transverse vibrations ; that is to say, the medium is capable of propagating waves of trans- verse vibrations if these six conditions hold, but otherwise it is not. In employing the above considerations to determine the equations of vibratory motion, the directions AB and C'C are always taken so as to coincide with some two of the three coordinate axes; and it is this circumstance that makes the method peculiarly applicable to crystallized media. Indeed, if it were necessary to take the lines AB and C'C in any directions but those of the axes of symmetry, the above considerations would not apply without considerable mo- dification. The equations of vibratory motion obtained by: this method for an uncrystallized medium, are the well-known equations involving the two constants A and B. The equations obtained for a crystallized medium are perfectly free from any restriction of any kind, are appli- cable to all kinds of substance, whether we suppose its structure to be analogous to that of a solid fluid or gas, and hold for all kinds of disarrangement, whether consisting of normal or transverse displace- ments, or both. When we introduce the six relations between the constants above alluded to, and moreover assume that the vibrations constituting a polarized ray are in the plane of polarization, we arrive at Professor MacCullagh’s equations*. If, on the contrary, we suppose the vi- brations to be perpendicular to the plane of polarization, we arrive at equations which agree exactly with Fresnel’s theory in every par- ticular+. _ If we introduce these six relations into the equations for crystal- lized media deduced from M. Cauchy’s hypothesis, that the mole- cular forces act along the lines joining the different particles of the medium, it will be found that these equations are immediately re- duced to the equations for an uncrystallized medium. From this it follows that M. Cauchy’s hypothesis cannot be applied to any but uncrystallized media. In fact, it may be easily proved that if the equations derived from this hypothesis be true, a crystallized medium is incapable of propagating transverse vibrations. Secondly, respecting the use of the symbolical method and notation above alluded to. The application of the symbolical method and notation to the subject of vibratory motion is very remarkable, and leads to equations of great simplicity. In the case of an uncrystallized medium, the three : * ig in a paper read to the Royal Irish Academy, Dec. 9, 1839, age 14. : t On this subject see a paper by the late Mr. Greene in the seventh volume of the Cambridge Transactions, p. 121. dy lie 59 ordinary equations of motion are included in the single symbolica- tion — =B. tat +gafet@- Daf rads alta ers If we employ the notation Av'.u, and assume the symbol D to re- present the operation a f +B 5 +7 - the equation of motion becomes = =B(AD.D)v+(A—B)+AD.0; or, by using the sg Du'.u also, it may be put in the form = = {ADAD.—B(DD.)*}v. The symbol written before any quantity U which is a function of xyz, has a very remarkable signification ; the direction unit of the symbol WU is that direction perpendicular to which there is no va- riation of -U at the point ayz, and the numerical magnitude of DU is the rate of variation of U, when we pass from point to point in that direction. The symbols AD.v and DD.v have also remarkable significations. AD.v is a numerical quantity representing the degree of expansion, or what is called the rarefaction of the medium at the point ayz. DiD.v represents, in magnitude, the degree of lateral disarrangement of the medium at the point ryz, and, in direction, the axis about which that displacement takes place. These two symbols may be found separately by the integration of an equation of the form @U_( (@U , @U =): dt* da? * dy? © dz? When the six conditions above alluded to are introduced, the equation of motion for a crystallized medium becomes dv d d d (Aezt Pa, t 7) AB d d de a +p. { (B.7—BF a+(B,¢ —B,E) 6+ (BS a —B', a) yh, where A, A, A, are the three coefficients of direct elasticity with reference to the three axes of symmetry, and B, B,' B, B,' B; B,' the six coefficients of lateral elasticity with reference to the same axes. If the vibrations be transverse, this equation is reducible to the ~ form dy _ Ee = —(DD.) "(uta +d%48-+e%y) = — (DD.)*(a*aAa+ b°BAB + cxyAy)u, 60 assuming the vibrations of a polarized ray to be perpendicular to — the plane of polarization. The well-known condition that a plane polarized ray may be transmissible without subdivision, and: the velocity of propagation may be immediately deduced from this equation. If we assume the vibrations of a polarized ray to be in the plane of polarization, the equation becomes wile —DD.(a°aha+b°*Bap+ cyAy)DD v. This includes Professor MacCullagh’s three equations, PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. beer SLB a December 6, 1847. On the Critical Values of the sums of Periodic Series. By G. G. Stokes, M.A., Fellow of Pembroke College, Cambridge. There are a great many problems in heat, fluid motion, &c., the solution of which requires the development of an arbitrary function of z, f(x), between certain limits as o and a of a, by means of func- tions of known form. The form of the expansion is determined, at least in part, by the conditions to be satisfied at the limits; and it is usually considered that these conditions are satisfied by adopting the form of expansion to which they lead. Thus, if the problem requires that f(0) and f(a) vanish, it is considered that this condition is satisfied by developing f(z) in a series of sines of @” and its mul- a tiples. But since an arbitrary function admits of expansion in such a series, the expanded function is not restricted to vanish at the limits 0 and a. It becomes then a question, how shall we know when the expanded function does really vanish at the limits, and if it does not, how are such expansions to be treated, and are they of any practical importance ? In considering the logic of such developments, the author was led to perceive in what manner the evanescence of f(x) at the limits can be ascertained, or else the values of /(0) and f(a) obtained, from the development itself, even when the series cannot be summed, by ex- =a ‘ < ane ey amining the coefficient of sin —— in the ath term. Ina similar man- a ner the discontinuity of f(x) or any of its derivatives may be ascer- tained, and the amount of the sudden change of the function deter- mined. In such cases the expansions of the derivatives of f(x) can- not be obtained by differentiating under the sign of summation, but are given by formule which the author has considered. The most important case in considering a series of sines, is that in which f(z) is continuous ; but f(o) and f(a), instead of being equal to zero, are given quantities, and the coefficients in the expansion are indeterminate. In this case the coefficients in the expansions of J fz) and f"(«) contain, in addition to the indeterminate coefficients which enter into the expansion of f(r), the given quantities f(o) and No. V.—Procexrpines or THE CAMBRIDGE Pui. Soc. 62 f(a). Thus the expansion in a series of sines is useful, not only when ‘f(o) and f(a) vanish, but also when they are given quantities. In the same way the expansion of f(x) in a series of cosines is useful when/"(0) and f’(a) are given, as well as when they vanish. Thus, to take an example, the permanent temperature in a rectangular parallelepiped, when the temperatures of the faces are any arbitrary functions of the co-ordinates, can be expressed in a double series of sines in- volving any two of the three co-ordinates. 4 ~The author has only considered a series of sines and a series of cosines, with the corresponding integrals; but the methods which he has employed are of very general application. The comparison of different expressions of the same function of two or more inde- pendent variables often leads to very remarkable formule. The de- velopment of arbitrary functions in the way considered by the author is, however, not only curious but useful; for the expressions thus obtained are often much better adapted to numerical computation than those which would be obtained by the developments usually employed. In connexion with these investigations, the author was led to con- sider the discontinuity of the sums of infinite series, or of the values of integrals between infinite limits, which sometimes takes place even when the series or integral remains convergent, and the general term of the series, or the quantity under the integral sine, is a continuous function of some quantity which. is regarded as variable. ‘The author has shown that in all such casés the convergency of the series or integral becomes infinitely slow. The problem of determining the potential due to a given electrical point within a hollow conducting rectangular parallelepiped, and to the electricity included on the surface, is solved by a method which leads very readily to the result. The author thinks that a similar method may sometimes be advantageously employed in other questions. The electricity is first supposed to be diffused over a finite space: this allows of the expansion of the potential V in a triple series of sines. Instead of the equation VV=0, where Y means the same as B. P @ dat . dy® ° dz® Srp it becomes of course necessary to employ the equation VV= —-4rp. The solution having been obtained, the electricity may now be sup- posed to be condensed into a point, and one of the summations may . be effected. The potential is thus expressed in a double series, which appears to be the simplest form that it admits of. May 8, 1848. Supplement to a paper “On the Intensity of Light in the neigh. bourhood of a Caustic.” By G. B. Airy, Esq., Astronomer Royal. 63 _ The author, after referring to the paper printed in a former volume of the Society’s Memoirs, in which he has shown that the expression for the intensity depends on the integral / cos (w’—mw) between the limits w=0, w = infinity, where m is proportional to the distance of the point at which the intensity is required, from the geometrical caustic, and in which he has calculated by quadratures the value of the definite integral for different values of m as far as m= +40, states that he was induced to have recourse to the method of qua- dratures only because every expansion which he attempted made it necessary to rely (for some of the terms) upon definite integrals equivalent to the integral dy cos § from 6=0 to 6 = infinity, and that he was not satisfied with the reasoning upon which some mathema- ticians had given a determinate value to that integral. Professor De Morgan, however, who felt no doubts upon it, had furnished him with a series proceeding by ascending powers of m, and had also - explained in detail (in a letter embodied in this paper) his views on the evidence for the value of the series, and on the method of de- termining’ it. From this series, the values of the definite integral are computed for all the values of m for which the computation had been made by quadratures, and the result is that the two sets of computed numbers are entirely accordant. The computations are also extended to the limit m= +5°6, which is the greatest value to which it is possible to extend the calculations by the use of 10-figure logarithms. May 22, 1848. Some Remarks on the Theory of Matter. By R. L. Ellis, Esq., M.A., Fellow of Trinity College, Cambridge. The question to which these remarks principally relate is this: Can all phenomena, e. g. those of chemistry, be explained mechani- cally? The writer, assuming that this question is to be answered negatively, endeavours to determine what principles of causation, beside mechanical force, may be introduced into physical theories, consistently with the doctrine that the secondary qualities of bodies are to be explained by means of the primary. His conclusion is, that we are at liberty, in constructing an hypothesis as to the mode of action of matter on matter, to introduce a new principle of causa- tion (which he calls (force)?), bearing the same relation to force that force does to velocity; and further, that following the analogy here suggested, we may introduce an indefinite number of such principles, viz. (force)%. . . (force)", &c., all essentially distinct from one another, and from those previously recognised. But, on the other hand, he conceives that it is necessary to reject any modification of qualitative action ; and that consequently physical science, though it may cease to be wholly mechanical, will yet always 64 continue to be cinematical, in the largest sense of which this word (so far as relates to local motion) can admit. June 5, 1848. Methods of Integrating Partial Differential Equations. By Prof. De Morgan. This paper contains a sketch of two distinct methods. In the first (x,y, 2, p,q, 7,8, t, having their usual significations) the given equa- tion is supposed to be of the form ¢(2, y, p, g)=0, and this is made the result of elimination between two equations involving a new variable v. From these two, and their four differentials of the first order, p,q, 7,s,¢ are eliminated, and an equation of the first order results between z,y,v. This last equation is often more manageable than the original one. The process is rendered very simple when the given equation can be reduced to depend on two of the form P=9(2,y,v) g=V(@,y, v)- The second method was completed, Mr. De Morgan states, and out of his hands for transmission to the Society, when he discovered that Mouge had communicated it to the Institute, by which body it was never published. But M. Chasles found it among the manuscripts of the Institute, and stated it a few years ago in one of the notes to his Apercu Historique .... des Méthodes en Géométrie. Its occurrence in the voluminous additions made to a work which itself treats only of geometry, seems to have prevented it from becoming known to any writer on the differential calculus. Certain particular cases appear in the writings of Legendre and Lacroix. Let the equation be (2, y, z,p,q,7,8,t)=0. Change @ into p, y into q, z into pr+qy—z, p into 2, g into y, r into , § into —s ,. r ——., t into fi >-9 rti—s grated, let its solution be Z=(X, Y). Then the solution of the original equation can be obtained by eliminating X, Y, Z from dZ, aZ Z= W(X, = eee WY) > ax of ae In both methods the most effective mode of proceeding is to find what Lagrange calls a primary solution, containing two arbitrary con- stants, and then to use that primary solution. rt —s? If the equation thus resulting can be inte- z=aX+yY—Z. On some new Fossil Fish of the Carboniferous Period. By Fre- deric M’Coy, M.G.S., N.H.S.D. _The author having premised that the species of fish of the carbo- _ niferous limestone enumerated in the third volume of the Poissons Fossiles of M. Agassiz are for the most part still unpublished, being without definitions or figures, states that through the kindness of 65 Capt. Jones, R.N., M.P., &c. he was enabled to study the original specimens of twenty-eight out of the thirty unpublished species from Armagh in M. Agassiz’s list, and is therefore certain of the species described by him being so far distinct from those alluded to. ‘The greater number of the examples here described are in the cabinets of the University of Cambridge (principally collected by the Rev. W. Stokes, of Caius College), and of Captain Jones; a few from the lower carboniferous shales of Ireland are only known in that of Mr. Griffith of Dublin. The descriptions are accompanied by drawings of all the species of the natural size, and illustrations of the microscopic structures; and acknowledgements are made of the kind co-operation of the Rey. Prof. Clark and Mr. Anthony of Caius College, Cambridge, in this part of the investiga- tion, by allowing the use of their large microscopes, and assisting to prepare the transparent fragments for examination. Twelve new genera are proposed :—lst. Jsodus, for a fish of the yellow sandstone, having very numerous teeth with a simple conical pulp-cavity in their upper part, which divides into branches below as in Rhizodus (Owen); but the section is circular, and the teeth are allequalinsize. 2nd. Centrodus, for curved conical teeth with a wide simple pulp-cavity, reducing the base to a sharp edge, and having not only the form but the microscopic structure of a reptile tooth, that is, from the simple pulp-cavity minute calcigerous tubes radiate to the circumference, terminating near the surface in a layer of small calcigerous cells, covered by a layer of true glass-like enamel, pre- senting no trace of structure with a power of 300 diameters, and quite distinct from that dense modification of dentine, which, forming the polished surface of most fish-teeth, has been confounded with true enamel, but which it is here proposed to call ganoine in future descriptions. 3rd. Colonodus, for very long simple teeth with simple pulp-cavity, and their sides indented by transverse wrinkles. 4th. Os- teoplax, for large, flat, polygonal dermal plates, minutely wrinkled on the surface, and allied to Psammosteus (Ag.) of the Old Red Sand- stone ; but while the latter plates are composed of horizontal layers of large cells, the present genus has a very singular microscopic structure, being traversed by vertical branched (Haversian ?) canals terminating in the pores of the surface ; and in the intervening blas- tema are numerous oval Purkinian cells, the radiating tubuli of which do not anastomose. Sth. Hrismacanthus, for a singular Ichthyodo- rulite not uncommon in the Armagh limestone, which, arising from a large compressed base, branches into two portions, one long ante- rior closely tuberculated prop-like portion, and another extruding backwards, short, and resembling a small Ctenacanthus, but with smooth ridges. 6th. Platycanthus, for small spines, extremely wide and compressed, resembling small Oracanths, but arched and with posterior rows of teeth. 7th. Dipriacanthus, for small, curved dorsal spines, which have two rows of denticles pointing downwards on the posterior face, and two rows pointing upwards on the anterior face, reminding us of the recent Pimelodus and Synodontus of the Nile.- 8th. Polyrhizodus, an extraordinary genus of Psammodontoid teeth 66 not uncommon in the Armagh limestone, having the root divided into numerous fang-like lobes, as in a mammalian tooth. 9th. Glos- sodus, for certain tongue-shaped teeth allied to Helodus (Ag.). 10th. Climaxodus, for some palates allied to Pecilodus (Ag.), but instead of — being transversely trigonal and obliquely ridged, they are equilateral, and have the ridges transverse and parallel (like a flight of steps). llth. Chirodus, for little hand-shaped teeth allied to the Ceratodi, but distinguished by the thumb-like lobe projecting from the middle of the long side, and which would prevent the union of the teeth in pairs in the mouth, in the manner of Ceratodus. 12th. Petrodus, small conical ridged teeth resembling limpets, common in the Der- byshire limestone, but presenting, of all known fossil fish, the near- est approach to the microscopic structure of the recent Cesiracion. It is also proposed to divide the genus Holoptychius of M. Agassiz; and instead of considering it and Rhizodus of Owen as synonymous, to limit the latter to those great teeth with an elliptical section so common in some parts of the Carboniferous series, accompanied by large, thin, quadrate scales, marked with concentric lines of growth, and having a fine cancellated structure internally, the Holoptychius Hibberti (Ag.) Rhizodus ferox, (Owen) and H. Portlocki (Ag.) bemg the types ; thus retaining the name Holoptychius for those fish so abundant in the Old Red Sandstone with thick, bony, ovate, longi- tudinally wrinkled scales, and minute teeth with a circular section, having the H. nobilissimus, H. giganteus, &c. as the type. The number of new species described and figured in this paper is forty-one, of which several belong to genera not previously known in rocks of the carboniferous period, many showing a strong affinity to the Devonian type of form. Thus we have two species of Psam- mosteus, one of Chelyophorus, one (doubtful) of Coccosteus, one of Asterolepis, two of Homacanthus, and one of Cosmacanthus, genera hitherto only found in the Old Red Sandstone. On an Absolute Thermometric Scale founded on Carnot’s Theory of the Motive Power of Heat*, and calculated from Regnault’s ob- servationst. By Prof. W. Thomson, Fellow of St. Peter’s College. The determination of temperature has long been recognized as a problem of the greatest importance in physical science. It has ac- cordingly been made a subject of most careful attention, and, espe- cially in late years, of very elaborate and refined experimental re- * Published in 1824 in a work entitled Réfleaions sur la Puissance Mo- trice du Feu,by M.S. Carnot. Having never met with the original work, it is only through a paper by M. Clapeyron, on the same subject, published in the Journal de [ Ecole Polytechnique, vol. xiv. 1834, and translated in the first volume of Taylor’s Scientific Memoirs, that the author has become acquainted with Carnot’s theory.—W. T. + An account of the first part of a series of researches undertaken by M. Regnault by order of the French Government, for ascertaining the various physical data of importance in the Theory of the Steam-Engine, is just published in the Mémoires de ’ Institut, of which it constitutes the twenty- first volume (1847). The second part of the researches has not yet been published. 67 searches* ; and weare thus at present in possession of as complete a practical solution of the problem as can be desired, even for the most accurate investigations. The theory of thermometry is however as yet far from being in so satisfactory a state. The principle to be followed in constructing a thermometric scale might at first sight seem to be obvious, as it might appear that a perfect thermometer would indicate equal additions of heat, as corresponding to equal elevations of temperature, estimated by the numbered divisions of its scale. It is however now recognized (from the variations in the specific heats of bodies) as an experimentally demonstrated fact that thermometry under this condition is impossible, and we are left with- out any principle on which to found an absolute thermometric scale. Next in importance to the primary establishment of an absolute scale, independently of the properties of any particular kind of mat- ter, is the fixing upon an arbitrary system of thermometry, according to which results of observations made by different experimenters, in various positions and circumstances, may be exactly compared. This object is very fully attained by means of thermometers constructed and graduated according to the clearly defined methods adopted by the best instrument-makers of the present day, when the rigorous expe- rimental processes which have been indicated, especially by Regnault, for interpreting their indications in a comparable way, are followed. The particular kind of thermometer which is least liable to uncertain variations of any kind is that founded on the expansion of air, and this is therefore generally adopted as the standard for the comparison of thermometers of all constructions. Hence the scale which is at present employed for estimating temperature is that of the air-ther- mometer ; and in accurate researches care is always taken to reduce to this scale the indications of the instrument actually used, whatever may be its specific construction and graduation. The principle according to which the scale of the air-thermometer is graduated, is simply that equal absolute expansions of the mass of air or gas in the instrument, under a constant pressure, shall indicate equal differences of the numbers on the scale; the length of a “‘ de- gree” being determined by allowing a given number for the interval between the freezing- and the boiling-points. Now it is found by Regnault that various thermometers, constructed with air under dif- ferent pressures, or with different gases, give indications which coin- cide so closely, that, unless when certain gases, such as sulphurous acid, which approach the physical condition of vapours at saturation, are made use of, the variations are inappreciablet. This remarkable circumstance enhances very much the practical value of the air- thermometer ; but still a rigorous standard can only be defined by * A very important section of Regnault’s work is devoted to this object. + Regnault, Relation des Expériences, &c., Fourth Memoir, First Part. The differences, it is remarked by Regnault, would be much more sensible if the graduation were effected on the ty fo rasa that the coefficients of expansion of the different gases are equal, instead of being founded on the principle laid down in the text, according to which the freezing- and boiling- points are experimentally determined for each thermometer. 68 fixing upon a certain gas at a determinate pressure, as the thermo- metric substance. Although we have thus a strict principle for con- structing a definite system for the estimation of temperature, yet as reference is essentially made to a specific body as the standard ther- mometric substance, we cannot consider that we have arrived at an absolute scale, and we can only regard, in strictness, the scale actu- ally adopted as an arbitrary series of numbered points of reference sufficiently close for the requirements of practical thermometry. _ In the present state of physical science, therefore, a question of extreme interest arises: Js there any principle on which an absolute thermometric scale can be founded? It appears to me that Carnot’s theory of the motive power of heat enables us to give an affirmative answer. The relation between motive power and heat, as established by Carnot, is such that quantities of heat, and intervals of temperature, are involved as the sole elements in the expression for the amount of me- chanical effect to be obtained through the agency of heat; and since we have, independently, a definite system for the measurement of quantities of heat, we are thus furnished with a measure for intervals according to which absolute differences of temperature may be esti- mated. To make this intelligible, a few words in explanation of Carnot’s theory must be given; but for a full account of this most valuable contribution to physical science, the reader is referred to either of the works mentioned above (the original treatise by Car- not, and Clapeyron’s paper on the same subject). In the present state of science no operation is known by which heat can be absorbed, without either elevating the temperature of matter, or becoming latent and producing some alteration in the physical condition of the body into which it is absorbed; and the conversion of heat (or caloric) into mechanical effect is probably impossible*, certainly undiscovered. In actual engines for obtaining mechanical effect through the agency of heat, we must consequently look for the source of power, not in any absorption and conversion, but merely in a transmission of heat. Now Carnot, starting from universally acknowledged physical principles, demonstrates that it is by the letting down of heat from a hot body to a cold body, through the medium of an engine (a steam-engine, or an air-engine for instance), that mechanical effect is to be obtained ; and conversely, he proves that the same amount of heat may, by the expenditure of an equal amount of labouring force, be raised from the cold to the hot body (the engine being in this case worked backwards) ; just as mechanical * This opinion seems to be nearly universally held by those who have written on the subject. A contrary opinion however has been advocated by Mr. Joule of Manchester ; some very remarkable discoveries which he has made with reference to the generation of heat by the friction of fluids in motion, and some known experiments with magneto-electric machines, seeming to indicate an actual conversion of mechanical effect into caloric. No experiment however is adduced in which the converse operation is exhi- bited; but it must be confessed that as yet much is involved in mystery with reference to these fundamental questions of natural philosophy. 69 effect may be obtained by the descent of water let down by a water- wheel, and by spending labouring force in turning the wheel back- wards, or in working a pump, water may be elevated to a higher level. The amount of mechanical effect to be obtained by the trans- mission of a given quantity of heat, through the medium of any kind of engine in which the ceconomy is perfect, will depend, as Carnot demonstrates, not on the specific nature of the substance employed as the medium of transmission of heat in the engine, but solely on the interval between the temperatures of the two bodies between which the heat is transferred. Carnot examines in detail the ideal construction of an air-engine and of a steam-engine, in which, besides the condition of perfect ceconomy being satisfied, the machine is so arranged, that at the close of a complete operation the substance (air in one case and water in the other) employed is restored to precisely the same physical con- dition as at the commencement. He thus shows on what elements, capable of experimental determination, either with reference to air, or with reference to a liquid and its vapour, the absolute amount of mechanical effect due to the transmission of a unit of heat from a hot body to a cold body, through any given interval of the thermo- metric scale, may be ascertained. In M. Clapeyron’s paper various experimental data, confessedly very imperfect, are brought forward, and the amounts of mechanical effect due to a unit of heat descend- ing a degree of the air-thermometer, in various parts of the scale, are calculated from them, according to Carnot’s expressions. The results so obtained indicate very decidedly, that what.we may with much propriety call the value of a degree (estimated by the mecha- nical effect to be obtained from the descent of a unit of heat through it) of the air-thermometer depends on the part of the scale in which it is taken, being less for high than for low temperatures *. The characteristic property of the scale which I now propose is, that all degrees have the same value; that is, that a unit of heat descending from a body A at the temperature T° of this scale, to a body B at the temperature (T—1)°, would give out the same me- chanical effect, whatever be the number T. This may justly be termed an absolute scale, since its characteristic is quite independent of the physical properties of any specific substance. To compare this scale with that of the air-thermometer, the values (according to the principle of estimation stated above) of degrees of the air-thermometer must be known. Now an expression, obtained by Carnot from the consideration of his ideal steam-engine, enables * This is what we might anticipate, when we reflect that infinite cold must correspond to a finite number of degrees of the air-thermometer below zero ; since, if we push the strict principle of graduation, stated above, suf- ficiently far, we should arrive at a point corresponding to the volume of . air being reduced to nothing, which would be marked as —273° (“3p if 366 be the coefficient of expansion) of the scale; and therefore —273° of the air-thermometer is a point which cannot be reached at any finite temperature, however low. 70 us to calculate these values, when the latent heat of a given volume and the pressure of saturated vapour at any temperature are experi- mentally determined. ‘The determination of these elements is the principal object of Regnault’s great work, already referred to, but at present his researches are not complete. In the first part, which alone has been as yet published, the latent heats of a given weight, and the pressures of saturated vapour at all temperatures between 0° and 230° (Cent. of the air-thermometer), have been ascertained ; but it would be necessary in addition to know the densities of saturated vapour at different temperatures, to enable us to determine the latent heat of a given volume at any temperature. M. Regnault announces his intention of instituting researches for this object ; but till the results are made known, we have no way of completing the data necessary for the present problem, except by estimating the density of saturated vapour at any temperature (the corresponding pressure being known by Regnault’s researches already published) according to the approxi- mate laws of compressibility and expansion (the laws of Mariotte and Gay-Lussac, or Boyle and Dalton). Within the limits of natural temperature in ordinary climates, the density of saturated vapour is actually found by Regnault (Hiudes Hygrométriques in the Annales de Chimie) to verify very closely these laws; and we have reason to believe from experiments which have been made by Gay-Lussac and others, that as high as the temperature 100° there can be no consi-. derable deviation; but our estimate of the density of saturated va- pour, founded on these laws, may be very erroneous at such high temperatures as 230°. Hence a completely satisfactory calculation of the proposed scale cannot be made till after the additional experi- mental data shall have been obtained ; but with the data which we actually possess, we may make an approximate comparison of the new scale with that of the air-thermometer, which at least between 0° and J00° will be tolerably satisfactory. The labour of performing the necessary calculations for effecting a comparison of the proposed scale with that of the air-thermometer, between the limits 0° and 230° of the latter, has been kindly under- taken by Mr. William Steele, lately of Glasgow College, now of St. Peter’s College, Cambridge. His results in tabulated forms were laid before the Society, with a diagram, in which the comparison between the two scales is represented graphically. In the first table, the amounts of mechanical effect due to the descent of a unit of heat through the successive degrees of the air- thermometer are exhibited. The unit of heat adopted is the quan- tity necessary to elevate the temperature of a kilogramme of water from 0° to 1° of the air-thermometer; and the unit of mechanical effect 1s a metre-kilogramme; that is, a kilogramme raised a metre high. In the second table, the temperatures according to the proposed scale, which correspond to the different degrees of the air-thermo- meter from 0° to 230°, are exhibited. [The arbitrary points which coincide on the two scales are 0° and 100°. ] Note.—If we add together the first hundred numbers given in the 71 first table, we find 135-7 for the amount of work due toa unit of heat descending from a body A at 100° to B at 0°. Now 79 such units of heat would, according to Dr. Black (his result being very slightly corrected by Regnault), melt a kilogramme of ice. Hence if the heat necessary to melt a pound of ice be now taken as unity, and if a metre-pound be taken as the unit of mechanical effect, the amount of work to be obtained by the descent of a unit of heat from 100° to 0° is 79 x 135°7, or 10,700 nearly. This is the same as 35,100 foot-pounds, which is a little more than the work of a one-horse-power engine (33,000 foot-pounds) in a minute; and consequently, if we had a steam-engine working with perfect ceco- nomy at one-horse-power, the boiler being at the temperature 100°, and the condenser kept at 0° by a constant supply of ice, rather less than a pound of ice would be melted in a minute. # yi Arse %t % Bente oH fies Ray PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. —_-_-<$__—_—_ May 17, 1847. A Theory of the Transmission of Light through Transparent Media, and of Double Refraction, on the Hypothesis of Undula- tions. By Professor Challis. The object of the author in this, as in two preceding communica- tions on Luminous Rays and on the Polarization of Light*, is, to esta- blish the undulatory theory of light on hydrodynamical principles, by means of a system of ray-vibrations, the motions in which are mathematically deduced from hydrodynamical equations. In ap- plying these views to the transmission of light through transparent media, it is assumed that the ether is of the same uniform density and elasticity within any transparent medium as without; and that the diminished rate of propagation in the medium is owing to the obstacle which its atoms oppose to the free motion of the ethereal particles. Considering the proximity of the atoms to each other, and that the retarding effect of each atom at a given instant extends through many multiples of its linear dimensions, it is presumed that the mean retardation, though resulting from the presence of discrete atoms, may be regarded as continuous. It is also supposed that the mean effect of the presence of the atoms is to produce an appa- rent diminution of the elasticity of the ether, the motion in all other respects being the same as in free space. By the application of these principles, it is first shown that the surface of elasticity, that is, the surface whose radius vector drawn in any given direction repre- sents the elasticity in that direction, is in general an ellipsoid. This being ascertained, the velocity of a ray in any given direction is inves- tigated; and the result is, that the surface whose radius vectors drawn in any given direction represent the velocities of propagation -of two oppositely polarized rays in that direction, is precisely the wave-surface in Fresnel’s theory of double refraction. March 6, 1848. A Mathematical Theory of Luminous Vibrations. By Professor Challis. This paper is intended to be supplementary to three former com- * See Lond, Ed. and Dubl. Phil. Mag. vol. xxx. p. 365. No. VI.—Procerepines or THE CAMBRIDGE Paix, Soc. 74 munications in which the undulatory theory is treated on hydrody- namical principles, and to elucidate or confirm results previously arrived at. In particular the author enters more at length into the mathematical theory of ray-vibrations, which, according to his views, correspond to rays of light. The principal theoretical deductions are,—(1.) that the longitudinal vibrations of a ray are defined by a 2 function of the form sin = (2-at ea , A being the V vy breadth of the undulation, and a, ¢ certain constants ; (2.) that light ; 4 eX from any source is in general composed of rays for which a and a are the same and A different; (3.) that light coming immediately from its origin is common light, whatever be the nature of the cause producing it, and that to become polarized light, it must be acted upon by reflexion, refraction, &c.; (4.) that light coming imme- diately from its origin is seen in all directions. ee ee November 18, 1848. Second Memoir on the Fundamental Antithesis of Philosophy. By W. Whewell, D.D. This memoir is a continuation of a former one in which the anti- thesis of thoughts and things, of ideas and facts, of subjective and objective, were shown to be at bottom the same antithesis, and to be a fundamental antithesis, the union of the two elements entering into all knowledge, and their separation being the test of all philo- sophy. The present memoir is employed in illustrating the proposi- tion that the progress of science consists in the transfer of some truth from the factorial to the ideal side of the antithesis, or as it may be termed, in the idealization of facts. This is exemplified in mecha- nics, astronomy, botany and chemistry. In a note, the author remarks on certain German systems of phi- losophy with reference to this antithesis. The Sensatorial school having reduced all knowledge to facts, Kant re-established the neces- sity of Ideas, which Fichte made almost the exclusive element. Schel- ling founded his philosophy upon the absolute, from which he derives both facts and ideas, but which a wiser philosophy shows us that we can never reach; and Hegel took the same foundation, but in a cer- tain degree rightly pointed out that the progress towards the identity of fact and idea is to be traced in the history of science; which view, however, he has carried into detail by rash and blind conjecture. On the Elements of Plane Geometrical Trigonometry, applicable to Trigonometrical Formule. By the Rev. F. Calvert. The object of this paper is to define as distinctly as possible the elementary terms of trigonometry, and to explain the conventional use of the negative sign in expressing such simple functions of angles as the sine, cosine, tangent, &c. 75 November 27, 1848. - Ona Difficulty suggested by Professor Challis in the Theory of Sound. By Robert Moon. In a paper by Professor Challis contained in the Supplementary Number of the 32nd Volume of the Philosophical Magazine, I find the following :— “The difficulty respecting the augmentation of the velocity of sound by the development of heat, cannot be so summarily disposed of as Mr. Airy appears to imagine. I shall perhaps succeed better in conveying my meaning by using symbols. If § be the tempera- ture where the pressure is p and density p, and §, the temperature in the quiescent state of the fluid, we have, by a known equation, p=ep(1+a.d—6,). la A — _ias dd aes ahs ara(9 a ea. . (1.) “The usual theory explains how the third term of the right-hand side of this equation may be in a given ratio to the first; but my difficulty is to conceive how the same can be the case also with the second term, since it changes sign with the change of sign of §—8,.” I conceive that the explanation, according to the usual theory to which Professor Challis here alludes, depends upon the principle, “« that for very small condensations of air, the rise of temperature will be proportional to the increase of density.” (Vide Herschel On Sound, Hncyc. Met., art. 72.) ‘Thus we may put 6—8,=A(1—p), where & is a constant, and 1 is put for the density of equilibrium : on which hypothesis it is obvious that the third term of equation (1.) will be a multiple of the first, as described by Prof. Challis. It also follows that the second term vanishes, since it has (1—p) for a fac- tor, and in reducing (1.) to the ordinary form of the differential equation of sound the difference between p and 1 is neglected. It thus, I think, appears that the difficulty suggested by Prof. Challis has no real existence. Observations of the Aurora Borealis of Nov. 17, 1848, made at the Cambridge Observatory. By Professor Challis. These observations relate principally to the corona, or point of apparent convergence of the streamers, the remarkable display of Nov. 17 being peculiarly favourable for noting the position of this critical point. They were taken partly by estimation of distances from stars, and partly by a small altitude and azimuth instrument (called by the author a meteoroscope), which is furnished with a bar, eighteen inches long, carrying at one end a rectangular piece whose edges are horizontal and vertical, by looking at which through an eyelet-hole, about the size of the pupil of the eye, at the other end, the collimation is performed. Each observed position is compared Hence 76 with the point of the heavens to which the south end of the dipping- needle was directed at the time of observation. This point was ascertained by means of observations of declination, horizontal force, and vertical force, taken at the Greenwich Observatory during the prevalence of the aurora by Mr. Brooke’s photographie pro- cess, the results of which were communicated to the author by the Astronomer Royal. It is assumed that the magnetic declination and dip at Cambridge differ from those at Greenwich at any given time by ‘certain constant quantities, whether the magnet be disturbed or not. ‘These constant differences were derived from the following formulee :— | faiths V—V, = 0°142518A + 071595481 D—D, = 0°027713A + 0:5135231, in which V and D are the declination and dip at a place not very distant from Greenwich, V, and D, the contemporaneous declina- tion and dip at Greenwich, \ the longitude of the place west, in seconds of time, and I the eacess of its latitude in minules above that of Greenwich. These are merely formule of interpolation by sim- ple differences derived from the following data :— Lat. Long. West. pas oT ° 1 m. & . ° ; Greenwich ....51 286 00°0 2817759 69 1:9 Makerstoun....55 34°7 10 3°5 25 22°85 71 25°0 Dublin........53 210 25 40 27 9°87 70 41°3 The above are very accurate contemporaneous values of the decli- nation and dip at the three places, and the formule derived from them will probably apply with considerable accuracy to any place in the United Kingdom at any date not very remote from 1843. For the Cambridge observatory V—V,= + 87 and D—-D,= +22'0. The mean result from 24 observations of the position of the corona is, that it was situated 5!’ further from the astronomical zenith, and 1° 14! nearer to the meridian than the point of the heavens to which the south end of the dipping-needle was directed. The places of the corona given by the different observations ex- hibit considerable discrepancies, which are accounted for by saying, that as the formation of the corona is merely an effect of perspective, its position varies, since the streamers are not exactly parallel, with the locality from which they rise; also with any variation of their direction at a given locality; and, supposing the course of the streamers to be somewhat curved in their ascent, it will vary with the height to which they rise. Accordingly, as appeared to be the fact, the corona would be continually shifting its position within certain limits. Prof. Challis has made a similar comparison with observations of the position of the corona of the same aurora made at Haverhill, at Darlington, and at Bath ; also with observations at Whitehaven of the aurora of Oct. 18, 1848, and of that of Oct, 24, 1847, at Cam- 77 bridge. From a consideration of all the results derived from the discussion of observations made on different occasions and at differ- ent places, the following conclusions seem to be established :— First, that the corona of an aurora borealis is formed near the point of the heavens to which the south end of the dipping-needle at the place of observation is directed. Secondly, that the observations, while they indicate no decided difference of altitude between the two points, show with great pro- bability that the corona is the more westerly by about 14° measured on an are perpendicular to the meridian. The paper concludes with a particular description of the aurora borealis of Nov. 17 as observed at the Cambridge Observatory, and with three tables of the observations of declination, horizontal force, and vertical force, made at Greenwich, and used in the calculations. These observations present so striking an instance of great magnetic disturbances occurring simultaneously with an extraordinary display of the aurora borealis, that the connexion in some way of the two kinds of phenomena must be regarded as a physical fact. On Clock Escapements. By E. B. Denison, Esq., of Trinity College. The object of this paper is, first to point out the real cause of the general excellence of the dead beat escapement; and secondly, to show that in a gravity, or remontoir escapement, in which the pen- dulum raises an arm carrying a small weight, from an angle y up to its extreme semiarc a, which follows the pendulum down again to an angle G (either + and less than y, or = — y), there is a certain pro- portion between a, 8, and y, which will cause the errors of the clock for small variations of a to be much smaller than in the dead escape- ment, and in fact inappreciable. The author adopts the equations obtained by Mr. Airy in his paper on this subject in vol. iii. of the Transactions of the Society, and shows that the increase of the time of an oscillation d@ 3da =—4 6 SS ? ¢ where A is the difference between the time of oscillation of a free pendulum and one affected by this escapement (which in clocks of the best construction he shows will amount to about 1 second a day) ; @ is the angular accelerating force of the escapement on the pen- dulum; dg the variation in this force due to the variation of the friction of the train and of the state of the oil on the acting part of the pallets; da the variation of the arc from the same causes, and also from the state of the oil on the dead or circular part of the pallets. It appears therefore that the two causes of error have a tendency to correct each other; and in practice it is found that = is generally not far short of 2 , Which is the reason of these clocks going so well. In a gravity escapement there is no variation of the force; and 78 : dA the author shows from Mr. Airy’s equations that 77 =0 ifa=yVv2 in that escapement where the remontoir weight is taken up at y and follows the pendulum again to—y; and in the other kind of gravity escapement 77a, = 0 when am 2Var—y Va? — pe. This last construction however is barely practicable, if this con- dition is to be satisfied, on account of the small difference between B and y which is allowed by the deduction of the value necessary for «—+, the angle in which the unlocking of the escapement is effected ; although this is the construction which has been used in nearly all the gravity escapements that have been tried ; and of course the proper condition has been very far from satisfied, and the clocks have failed. In a supplement to this paper the author proposes, chiefly fo turret clocks, a new construction of a spring remontoir on the axis of the escape-wheel. The object of such remontoirs is to remove from the escapement (of any ordinary kind) the great inequalities of force caused by the varying friction of the heavy train and dial-work, and by the action of the wind on the hands; and also to cause the minute-hand to move only at visible intervals, such as 4a minute, and the striking to take place exactly at the right second. The Royal Ex- change clock, made under the superintendence of the Astronomer Royal, has a gravity remontoir in the train introduced for these purposes ; but it is too complicated and expensive for ordinary use, and has a good deal of friction, from which the proposed remontoir is free. Spring remontoirs winding up at similar intervals have been tried in France, but without success, from defects in their con- struction. December 11, 1848. On the Formation of the Central Spot of Newton’s Rings beyond the Critical Angle. By G. G. Stokes, M.A., Fellow of Pembroke College, Cambridge. It has long been known that when Newton’s rings are formed be- tween the under surface of a prism and the upper surface of a lens, or of a second prism, so as to allow of increasing the angle of inci- dence at pleasure, the rings disappear when the critical angle is passed, but the central spot remains. The existence of the spot under these circumstances has even been attributed to the disturbance in the second medium, which, when the angle of incidence exceeds the critical angle, takes the place of that disturbance which at a smaller incidence constitutes the refracted light; but the expression for the intensity has not hitherto been given, so far as the author is aware. The object of the author in the present er is t 1 this deficiency. P Pepe a ee 79 The author has not adopted any particular dynamical theory, but has deduced his results from Fresnel’s expressions for the intensities of reflected and refracted polarized light. When the angle of inci- dence becomes greater than the critical angle these expressions be- come imaginary. When the imaginary expressions are interpreted in the way in which physical considerations show that they must be interpreted, it becomes easy to obtain the expression for the intensity of the light, whether reflected or transmitted, in the neighbourhood of the spot. When the first and third media are of the same nature, the following expression is obtained for the intensity (I) of the re- flected light, the incident light being polarized in the plane of inci- dence, and its intensity being taken for unity, 27D =a (1—9*)? a > Ve in? i=] T= Ga pypagmy Whereg=e kT” In this expression yu is the refractive index of the first medium, 7 the angle of incidence on the surface of the second medium, or inter- posed plate of air, D the thickness of that plate at the point con- sidered, A the length of a wave in air, 24 the acceleration of phase due to total internal reflexion. When the light is polarized perpen- dicularly to the plane of incidence, it is only necessary to replace 2 6 by 2¢, the angles §, ¢ being those so denoted in Airy’s Tract. The intensity of the transmitted light is obtained by subtracting that of the reflected light from unity. From the expression for the intensity, the author has deduced the following results, all of which he has verified by observation. The spot is comparatively large near the critical angle, and becomes smaller and smaller as the angle of incidence increases. Near the critical angle the fainter portion, or ragged edge, of the bright spot seen by transmission is broad; at considerable angles of incidence the light decreases with comparative abruptness. ‘Towards the edge of the spot there is a predominance of the colours at the red end of the spectrum, causing the ragged edge to appear brown. Near the critical angle the spot is larger for light polarized perpendicularly to the plane of incidence than for light polarized in that plane: at con- siderable angles of incidence the order of magnitude is reversed. The difference is far more conspicuous in the former case than in the latter, and in that case consists principally in the greater extent of the ragged edge. When the incident light is polarized at an azimuth of 45°, or thereabouts, and the transmitted light is analysed so as to extinguish the light transmitted near the point of contact, there is seen a central dark patch surrounded by a luminous ring. February 26, 1849. On a New Method of finding the Rational Roots of Numerical Equations. By Robert Moon, Esq. The author proposes to found a new experimental method of find- 80 ing the integral roots of numerical equations upon the following theorem. If the equation a" + pa" +-poa" + &e. +pa"'+ &e.+p,=0 has a positive and integral root m, we shall have —p, equal to m terms of the following series :— A,-atl. Ayo$1.2. Ay-3 +1.2.8A, 4+ &e. +1.2...(n—i+]) A; + &c. +1.2...(n—1) Ay +A,_:+2. A,-o+2.3 A,3+2.3.4A,,+ &c. +2.3...(n—i+2) A; + &c. +2.3...2 Ay +A,-1+3.A,-2+3.4 A,_3+3.4.5 A,_y+ &c. +3.4...(n—i+38) A, + &c. +3.4...(m+4+1) Ay +A, +4. A,-2+4.5 A,3+4.5.6 A, 4+ &c. +4.5...(n~—i+4) A; + &c. +4.5...(n +2) Ay &c. &e. where A;=(n—i)(p;—hy pj + he pia + &e. + (— 1), pj-, + &e. thj_api Fh) and h,=the sum of the natural numbers 1,2,3...... (n—i). hz=the sum of the homogeneous products of the same quantities of two dimensions. hz==the sum of the homogeneous products of the same quantities of three dimensions, and so forth. . From the above formula for A; may be determined all the coeffi- cients A except the first, which is determined from the equation ~ Ay-1=Pa-1— Pn-2 +pn-3— &e. +pi+ E: Having determined the quantities A in any particular case, let them be substituted in the first line of the series. If the sum of that line be equal to —p,, unity is a root of the equation. Let the second line be then written down and added to the first. If the sum of the two equals—p,,, 2 is a root of the equation, and so by adding successive lines we shall ascertain whether the successive integers 3.4., &c. are or are not roots of the equation. The quantities 4 in the expression for A; depend upon the number of the coefficient and the number of the dimensions of the equation. The author proposes that these should be calculated and tabulated for equations of all dimensions up to a certain limit, by which means we should be in possession of so many skeletons of equations, ready for application in any particular case, and the calculation in particular instances would be thus greatly facilitated. It will be observed that each successive line is derived from that preceding by a simple division and multiplication of the separate terms of the latter, and thus each succeeding trial facilitates those which follow ; contrary to what obtains in the ordinary method by successive substitutions, in which each attempt proceeds de novo. If the addition of a term makes the series from being greater than | Pp less than it, or vice versd, a fractional or surd root will lie between 81 the number expressing the number of the term so added and the number next below it. If all the roots are impossible, the series will be either always greater or always less than p,, whatever be the number of terms taken. For an example take the cubic a+pa°+qe+r=0. Here —r=3 x 1.24+2(p—3)l+q—p+1 +3 x 2.3+2(p—3)2+q—p+l +3 x3.4+2(p—3)3+q—ptl + &c. to x terms, if x is a positive integer. _ The method in common with other experimental methods applies to the discovery of all roots, possible or impossible, which do not involve surds. March 12, 1849. On the Intrinsic Equation to a Curve, and its application. By the Master of Trinity. ‘The author remarked that the expressions for the lengths of curves, their involutes and evolutes, in the ordinary methods, are complex and untractable, which arises in a great measure from the properties of evtrinsic lines being introduced, namely, coordinates. But a curve may be represented without any such additions, by an equation between the length and the angle of flexure, which is therefore called the ixtrinsic equation. ‘This equation gives, with remarkable facility, the radii of curvature ; involutes and evolutes of most curves. It also expresses very simply what may be called running curves; namely, curves which run like a pattern along a strip of ornamented work. A very simple equation expresses, for instance, the inclined scroll pattern so common in the antique, and by altering the constants, gives to this pattern an endless variety of forms. If s be the length of the curve and ¢ the angle, the intrinsic equation to the circle is s=ag; to the cycloids=asing. The equation to an epicycloid or hypocycloid is s=asin m¢, according as m is less or greater than unity. The equation to an undulating pattern is ¢=™m sin s, which assumes very various shapes by varying m. ‘The method was also used in proving that if we take the suc- cessive involutes of a curve an indefinite number of times, the re- sulting curve (with certain limitations) tends to become the equi- angular spiral if the unwrapping be always in the same direction, and tends to become the cycloid if the unwrapping be alternately in opposite directions. The latter proposition had already been dis- covered by Bernouilli and proved by Euler. 82 April 23, 1849. On the Variation of Gravity at the Surface of the Earth. By G. G. Stokes, M.A., Fellow of Pembroke College, Cambridge. In the theory of the figure of the earth on the hypothesis of ori- ginal fluidity, a simple expression is obtained for the variation of gravity along the surface, which contains the numerical relation between the ellipticity and the ratio of polar to equatoreal gravity, known as Clairaut’s theorem. The demonstration, however, of this expression does not require the hypothesis of original fluidity, if the spheroidal form of the surface and its perpendicularity to the direction of gravity be assumed as results of observation. On the hypothesis merely that the earth consists of nearly spherical strata of equal density, Laplace has established a connexion between the form of the surface, regarded as a surface of equilibrium, and the variation of gravity along it; and in the particular case in which the surface is an oblate spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation, the expression which results for the variation of gravity is identical with that which is obtained on the hypothesis of original fluidity. The object of the author in the first part of this paper is to obtain the general connexion between the form of the surface and the variation of gravity along it, by an application of the doctrine of potentials, without making any hypo- thesis whatsoever respecting the distribution of matter in the interior of the earth. The latter part of the paper was devoted to the consideration of the irregularities produced in the variation of gravity by the irregular distribution of land and sea at the surface of theearth. ‘The author has shown why gravity should appear less on continents than on small islands situated at a distance from any continent, which is a circumstance that has long since been observed. The result is ac- counted for by the elevation of the sea-level produced by the attrac. tion of a continent, in consequence of which a station on a continent is further removed from the centre of the earth than it appears to be. It is shown also that the numerical value of the earth’s ellipticity, which has been deduced from pendulum experiments, is somewhat too great, in consequence of the undue proportion of oceanic stations in low latitudes, among the group of stations at which the observa- tions were made which have been employed in the discussion. The author has given formule whereby observed gravity may be corrected for the irregularities of the earth’s surface. These formule require a knowledge, or at least an approximate knowledge, of the height of the land and the depth of the sea throughout the earth’s surface. The sign and magnitude of the difference between observed gravity, and gravity calculated on the hypothesis of the earth’s ori- ginal fluidity, appears on the whole to depend on the insular or con- tinental character of the station at which the observation has been taken. This circumstance renders it probable, that if observed gra- vity were corrected for the irregular attraction due to the irregular 83 distribution of sea and land throughout the whole surface of the earth, the result would agree far better with gravity calculated on the hypothesis of original fluidity. May 7, 1849. Additional Note toa Memoir on the Intrinsic Equation of Curves. By Dr. Whewell. This note contained an extension of a theorem discovered by John Bernouilli, and demonstrated by Euler, to this effect: that if from any rectangular curve a string be unwrapped, and from the curve so described again a string unwrapped, and so on perpetually and alter- nately in opposite directions, the curves constantly tend to the form of the common cycloid. The extension is to this effect: thatif the original curve be not rectangular, the curves perpetually tend to the form of an epicycloid or hypocycloid, according as the angle is greater or less than a right angle. ee May 21, 1849. Discussion of a Differential Equation relating to the breaking of Railway Bridges. By G. G. Stokes, M.A., Fellow of Pembroke College. In August 1847 a Royal Commission was appointed “for the purpose of inquiring into the conditions to be observed by engineers in the application of iron in structures exposed to violent concussions _ and vibration.” Among other branches of inquiry, the members of the Commission have lately been making experiments on the motion of a carriage, variously loaded in different experiments, which passed with different velocities over a slight iron bridge; the object of the experiments being to examine the effect of the velocity of a train in increasing or decreasing the tendency of a bridge over which the train is passing to break under its weight. The remarkable result was obtained, that the deflection is in some cases much greater than the central statical deflection, and that the greatest deflection takes place after the body has passed the centre of the bridge. In in- vestigating the theory of the motion, reducing the problem to the utmost degree of simplicity by regarding the moving carriage as a heavy particle, and neglecting the inertia of the bridge, Professor Willis, who is a member of the Commission, was led to a differential equation of the form @y 2 ee dz® (2exr—22)?’ where 2, y are the horizontal and vertical co-ordinates of the moving body, 2c is the length of the bridge, and a, 6 are certain constants. Professor Willis requested the author’s consideration of this equation, with a view to obtain numerical results, and to determine, if possible, the velocity which produces a maximum deflection. The author has expressed y in a series according to ascending 84 powers of 2, which is convergent when #< 2c. The convergency, however, becomes very slow when 2 approaches the limit 2e; and the series does not point out the law according to which /(x) or y approaches its extreme value 0 as a approaches 2c. When the con- stant term in the second member of the preceding equation is omit- ted, the equation may be integrated in finite terms ; and consequently the variables can be separated in the actual equation, so that /(a) can be expressed explicitly by means of definite integrals. In this way the author has obtained /(2c—#)—/(®) in finite terms, so that the numerical value of /(v) may readily be obtained from w=e to x==2c, after it has been calculated from the series from 7=0 to r=c: and between these limits the series is very convergent, being ulti- mately a geometric series with a ratio = The author has also in- vestigated a series proceeding according to ascending powers of e—z, which converges more rapidly than the former when x approaches c. By the use of these two series, f(x) may be calculated by means of series which are ultimately geometric series, with ratios ranging from 0 to }. The unsymmetrical form of the trajectory, and the largeness of the deflection produced by the moving body, come out from the in- vestigation. By means of the numerical values of f(x) the author has drawn a figure representing the trajectory for four different ve- locities. The expression for the central deflection, however, becomes infinite when « becomes equal to 2c, which shows that it is neces- sary to take into account the inertia of the bridge; although, if the bridge be really light, the solution obtained when the inertia of the bridge is neglected may be sufficiently exact for the greater part of the body’s course. On Hegel’s Criticism of Newton’s Principia. By Dr. Whewell. Parts of Hegel’s Hncyclopedia are here examined with the purpose of testing the value of his philosophy, not of defending Newton. Hegel says that the glory due to Kepler has been unjustly transferred to Newton; confounding thus the discovery of the laws with the discovery of the force from which the laws proceed, in which latter discovery Kepler had no share. Hegel pretends to derive the New- tonian ‘‘ formula” from the Keplerian law, thus ;—by Kepler’s law, 3 is constant: but A being the distance, and T the periodic time, T?2 Ao i, : Newton (Hegel says) calls pa Universal gravitation, whence universal gravitation is inversely as A?:—a most absurd misrepresentation of the course of Newton’s reasoning. In the same manner Hegel criti- cises, and utterly misrepresents Newton’s explanation, for the ellip- tical orbit, of the body’s approaching to and receding from the centre; and of the reason why the body moves in an ellipse. Hegel also offers his own explanation of Kepler’s laws from his own @ priori assumptions. He says that the motion of the heavenly bodies is not a being pulled this way or that, as is imagined by the Newtonians; they go along, as the ancients said, like blessed gods. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. —————>___—_ November 26, 1849. On the Dynamical Theory of Diffraction. By Professor Stokes. The problem of diffraction is treated mathematically by conceiving each wave of a series incident on a small aperture, or passing the edge of a diffracting body, broken up on arriving at the aperture or diffracting edge, regarding each element of the wave as the centre of an elementary disturbance, which diverges spherically from that element, and finding by integration the aggregate disturbance at any point in front of the primary wave. With the exception of one case of diffraction, which will be mentioned further on, the illumination in front of an aperture is insensible except in the immediate neigh- bourhood of a normal to the primary wave drawn through a point in the aperture. Consequently we are only concerned with the law of disturbance in that part of a secondary wave which lies very near the normal to the primary wave; the nature of the disturbance in other directions does not affect the result, since the secondary waves neutralize each other by interference. Now it has been shown by others, by indirect methods, that if ¢ be the coefficient of vibration in the incident light, dS an element of the area of the aperture, 7 the radius of a secondary wave diverging from dS, A the wave length, the coefficient of vibration in the secondary wave will be <= and r the phase of vibration must be accelerated by a quarter of an undula- tion; or in other words, ; must be subtracted from the retardation due to the radius r. These results, however, according to what has been already remarked, only apply to that portion of a secondary wave which lies immediately about the normal to the primary. The object of the author in this paper was to determine, on purely dyna- mical principles, the law of disturbance in any direction in a se- condary wave. The author has treated the ether as an elastic solid; and as such it must be treated in considering light, if the theory of transverse vibrations be not rejected. The object which he had in view re- quired the solution, in the first instance, of the following problem ;— to determine the disturbance at any point of an elastic medium, and No. VII.—Procrepines or tae CamBripGE- Put. Soc. 86 at any time, due to a given small arbitrary disturbance confined to a finite portion of the medium. This problem was solved long ago by Poisson ; but the author has given a totally different solution of it, which appears to be in some respects simpler than Poisson’s. In the course of the solution, the author was led incidentally to the fol- lowing very general dynamical theorem. Let any material system whatsoever, in which the forces actin depend only on the positions of the particles, be slightly disturbed from a position of equilibrium, and then left to itself: then the part of the disturbance at any time which depends on the initial displace- ments will be got from that which depends on the initial velocities by differentiating with respect to the time, and replacing the arbi- trary functions, or arbitrary constants, which express the initial ve- locities by those which express the corresponding initial displace- ments. Particular cases of this theorem are of frequent occurrence, but the author is not aware of any writing in which the theorem is enunciated in all its generality. The problem above-mentioned has been applied by the author to the case of diffraction in the following manner. Conceive a series of plane waves of plane-polarized light propagated in a direction perpendicular to a fixed mathematical plane P. According to the principle of the superposition of small motions, we have a perfect right to consider the disturbance of the ether in front of the plane P as the resultant of the elementary disturbances corresponding to the several elements of P. Let it be required to determine the dis- turbance which corresponds to an elementary portion only of the plane P. In this consists the whole of the dynamical part of the theory of diffraction, if we except the case of diffraction at the com- mon surface of two different media; the rest is a mere question of integration. Let the time ¢ be divided into equal intervals, each equal to 7. The disturbance which is propagated across the plane P during the first interval 7 occupies a layer of the medium having a thickness vr, if v be the velocity of propagation, and consists of a certain velocity and a certain displacement. By the problem above mentioned, we can find by itself the effect of the disturbance which occupies so much only of this layer as corresponds to a given element dS of P. By doing the same for the 2nd, 3rd, &c. intervals 7, and then making the number of such intervals increase and their magni- tude decrease indefinitely, we shall get the effect of the disturbance which is continually transmitted across dS. The result is a little complicated, but is much simplified when certain terms are neg- lected which are only sensible when the radius of the secondary wave is comparable with A, and which are wholly insensible in the physical applications of the problem. The result thus simplified may be enunciated as below:—In the enunciation, the term diffracted ray is used to denote the disturbance in an elementary portion of a secondary wave, diverging in a given direction from the centre ; the plane containing the incident and diffracted rays will be called the plane of diffraction, the supplement of the angle be- tween these two rays the angle of diffraction, and the plane passing 87 through a ray of plane-polarized light and containing the direction of vibration the plane of vibration. The incident ray being plane-polarized, each diffracted ray will be plane-polarized, and the plane of polarization will be determined by the following law :—The plane of vibration of the diffracted ray is parallel to the direction of vibration of the incident ray. The direction of vibration being thus determined, it remains only to specify its magnitude. Let t=csin oe (vt—2x) be the displacement in the case of the incident light, ¢' the displace- ment in the case of the diffracted ray, ¢’ being reckoned positive in the direction which makes an acute angle with that in which ¢ is reckoned positive. Let 7 be the radius of the secondary wave diver- ging from dS, and let r make angles 4 with the direction of propa- gation of the incident ray, and ¢ with the direction of vibration; then cdS 2Ar When an arbitrary function of vt—zx, f(vt—z) occurs in %, it is not f(vi—r) but f/'(vt—r) that appears in ¢’, where /’ denotes the derivative of f, and accordingly in the particular case in which J(u) = sin u the sine in { is replacedin ¢’ byacosine. It may readily be verified, that if the formula (a.) be applied to determine by inte- gration the disturbance which corresponds to the whole of the plane P, the disturbance in front is the same as if the wave had not been supposed broken up, and no disturbance is propagated backwards. The law obtained for determining the position of the plane of po- larization of the diffracted ray seems to lead to a crucial experiment for deciding between the two rival theories between the directions of vibration in plane-polarized light. Suppose the incident light po- larized by transmission through a Nicol’s prism mounted in a gra- duated instrument, and let the diffracted light be analysed in a similar manner. By means of the graduation of the polarizer, we can turn the plane of polarization of the incident ray, and consequently the plane of vibration, which is either parallel or perpendicular to the plane of polarization, round through equal angles of say 5° or 10° atatime. According to theory, the planes of vibration of the dif- fracted ray will not be distributed uniformly, but will be crowded towards the perpendicular to the plane of diffraction. But experi- ment will enable us to decide whether the planes of polarization are crowded towards the plane of diffraction or towards the perpendicular to the plane of diffraction, and we shall accordingly be led to con- clude, either that the vibrations are perpendicular, or that they are parallel to the plane of polarization. In ordinary cases of diffraction, the illumination, in consequence of interference, is insensible beyond a small angle of diffraction. It is only by means of a fine grating that we can procure light of con- siderable intensity that has been diffracted at a large angle. The v= (1+ cos 9) sin ¢ cos = (vt—r) phan tic ae 88 author has been enabled to perform the experiment, or rather a mo- dification of it, by the kindness of his friends Professors Miller and O’Brien; of whom the former lent him a fine glass-grating, con- sisting of a glass plate on which parallel and equidistant lines had been ruled with a diamond at the rate of 1300 to the inch, and the latter lent him the graduated instruments required. The theory does not quite meet the case of a glass-grating, in which the diffrac- tion takes place at the common surface of two media, but it leads to a definite result on each of the two extreme suppositions :—I1st, that the diffraction takes place before the light reaches the grooves ; 2nd, that it takes place after the light has passed them; and the results are very different according as one or other of the two rival theories is adopted. In the principal experiments, the plane of the plate was placed perpendicular to the incident light, and the light observed was that which had been diffracted by transmission through the plate. The angle of diffraction, by which is meant the angle mea- sured in air, ranged in the different experiments from about 20° to 60°. The result obtained was, that when the grooved face was turned towards the eye, there was a very sensible crowding of the planes of polarization of the diffracted light towards the plane of diffraction. When the grooved face was turned towards the incident light, there was a considerable crowding in the same direction, much more than-in the other case. Since the effect of refraction, con- sidered apart from diffraction, would be to crowd the planes in the contrary direction, the result seemed decisive in favour of Fresnel’s hypothesis, that the vibrations are perpendicular to the plane of po- larization. On the other hypothesis, diffraction would have con- spired with refraction to produce a large crowding in a direction contrary to that in which the observed crowding took place. The amount of crowding, in both positions of the plate, was nearly what would be given by theory, on adopting Fresnel’s hypothesis, and supposing that the diffraction took place before the light reached the grooves, but appeared in both cases a little less. The difference, how- ever, was comprised within the limits of uncertainty depending upon the errors of observation and the error in the assumed value of the refractive index of the glass plate. December 10, 1849. Impact on Elastic Beams. By Homersham Cox, Esq., B.A., Jesus College. Among the experiments instituted by the Royal Commission ap- pointed to inquire respecting the use of iron in railway structure, was a series relating to impact on beams. These experiments were undertaken by Professor Hodgkinson, and were conducted in the following manner. The two ends of the beam were fixed in a hori- zontal position, and the blow was given against one of its vertical sides in a horizontal direction. The instrument for giving the blow 89 was a heavy iron ball, hanging down, when.at rest, from a point of suspension vertically above the centre of the beam. The ball was raised through different arcs, and after descending by its own gra- vity, struck the beam. The deflection corresponding to different ares of descent were carefully noted by a graduated scale. The object of the present paper is to show that the results might have been predicted by known theoretical principles with consider- able precision and confidence. The problem is divided into two parts :— lst, to estimate the amount of velocity lost by the ball at the first instant of collision ; 2nd, to ascertain the effect of the elastic forces of the beam in destroying the vis viva which the whole system has immediately after collision. In the first part of the investigation, a general formula, derived from the combination of D’Alembert’s principle and that of virtual velocities, is given for the motion of any material system subject to impact. The requisite geometrical condition required for the application of this general formula to the present case is obtained by the assumption, that immediately after impact the form of the beam is a gradual and tolerably uniform curve, such as, for example, the elastic curve of equilibrium. In this way it is determined that about one-half the inertia of the beam is ef- fectively applied at the instant of collision to retard the ball. The vis viva of the whole system thus computed is destroyed by the elastic forces of the beam developed by deflection. These, in the second part of the problem, are assumed to vary as the amount of central deflection. By the principle of vis viva a formula is easily obtained, connecting the amount of total deflection with the vis viva of the system immediately after collision. Tables are given in which the theoretical and experimental results are compared. The correspondence is of the closest and most satis- factory nature. Indeed the theoretical result generally differs less from the mean of several experiments than those experiments differ among themselves. Both in the theoretical and experimental in- quiries, every possible variation of the elements of the investigation —the relative masses of the beam and ball—the velocity of the latter —the rigidity and dimensions of the former—have been included. February 11, 1850. A paper was read by the Master of Trinity, “ Criticism of Aris- totle’s account of Induction.” The passage criticised was Azalyt. Prior. 11. 25, and is by Aris- totle illustrated by this example. Elephant, horse, mule, &c.,are long- lived; but elephant, horse, mule, &c. have no gall-bladder. If we suppose that the latter proposition may be converted and put in this form, ‘‘ all animals which have no gall-bladder are as elephant, horse, mule, &c.,” we may draw the conclusion that all animals which have no gall-bladder are long-lived. ‘This convertibility and generalization of the second proposition are the necessary conditions for translating 90 induction into syllogism. And Aristotle really contemplated such a generalizing induction. He did not contemplate what has been called inductio per enumerationem simplicem, which is really no induc- tion at all. ‘This was shown to be so by reference to the case, often used as an example of induction, of the inference of Kepler's laws from the observation of the separate planets. It may be objected that the reasoning in such cases is inconclusive; and to this it is replied, that induction, as reasoning, is inconclusive. It is a source of truth different from reasoning; of first truths, the bases of rea- sonings, as Aristotle has remarked. February 25, 1850. On the Symbols of Logic, the theory of the Syllogism, and in particular of the Copula, and the application of the Theory of Pro- babilities to some questions of evidence. By Professor De Morgan. This paper, which is in continuation of the one published in vol. viii. part 3 (read Nov. 9, 1846), and of subsequent additions contained in the author’s work on Formal Logic, is divided into six sections. Section I. On the approximation of logical and algebraical modes of thought. —The subjects of this section are,— Ist, some development of the idea that the oppositions of logic have affinities which may one day lead to a connected theory, making use of a common instru- ment, just as the oppositions of quantity which are considered in algebra are connected by the general theory of the signs 4+ and —; and 2nd, some remarks on the resemblance of the instrumental part of inference to algebraic elimination. ics Ten such instances as affirmative and negative, conclusive and inconclusive, &c., are compared with the logical distinction of uni- versal and particular; and it is pointed out, in all the cases in which it is not already acknowledged, that it would be possible to use any one of the ten in place of the last. Section II. On the formation of symbolic notation for propositions and syllogisms.—Exclusive of remarks on the Aristotelian notation and on notation in general, and a statement for comparison of Sir William Hamilton’s notation, this section contains the following matters. 1. A pictorial or diagrammatic representation of syllogistic infer- ences, being after the method pursued by Lambert, with such addi- tions as will enable the system to represent all the cases in which contraries are used. 2. An abbreviated and arbitrary method of representing proposi- tions and syllogisms. Following Sir William Hamilton in making the quantity of both subject and predicate matter of symbolic expression, Mr. De Morgan gives his system of notation two new features. First, he dispenses with the representatives of the terms (except when it may be con- 91 venient to introduce them for the time), and represents the proposi- tion by the symbols of quantity only, and the presence or absence of asign of negation. Secondly, instead of making the symbols of uni- versal and particular absolute, he gives one symbol, ), to a universal subject and a particular predicate, and another, (, to a particular subject and a universal predicate: a dot [.] signifying negation. Thus X)-(Y, or simply )-(, represents ‘No X is Y’: X(-(Y, or (-(, represents ‘Some Xs are not any Ys:’ X()Y represents ‘ Some Xs are Ys.’ Of the second circumstance above mentioned, Mr. De Morgan believes that it makes the rules easier, and knows that it makes the notation more suggestive. Retaining in mind the order XY, YZ, XZ, which is the only figure used in the classification (being the first with inverted order of pre- mises), the syllogism is to be denoted by the junction of the propo- sitional symbols. Thus ))))=)) denotes ‘Every X is Y, every Y is Z, therefore every X is Z.”. When this is to be read in any figure, the subject-quantities are to have their symbols thickened, the second premise being read first: thus in the four figures, in order, will be seen such symbols as ||}, |\\[. {Ill IIL Section III. On the symbolic forms of the extension of the Aristo- telian system in which contraries are introduced.—This system is the one which was completed and published to the Society before any correspondence with Sir William Hamilton. Mr. De Morgan re- marks that it contains (incidentally, not designedly) every distribu- tion of quantifications ; and gives his reasons for not dwelling on this fact while the controversy was unfinished, with his statement that it had not struck him when the controversy began. Mr. De Morgan frequently distinguishes this system from Sir W. Hamilton’s by calling the former that of introduction of contraries, the latter that of inven- tion of predicates. For distinctness, it may be stated that Mr. De Morgan’s other, or numerically definite system (the one concerned in the discussion), does not appear in the present paper, except as matter of allusion. The forms of predication in this system are as follows, with refer- ence to the order XY, z and y being not X and not Y. Universals. ¢ A, ~X)jy¥ Every X is Y Affirmative { Al x))yy or X((Y Every Yis X Nevati E, X))y or X)-(Y NoXis Y egatiye 9 mi x))Y or X(-)Y Everything is X or Y or both. Particulars. eniweye ft ROY Some Xs are Ys mative { I' a(y or X)(Y Some thingsare neither Xs nor Ys arep O, XQy or X(-(Y Some Xs are not Ys egative 10} a()Y or X)*)Y Some Ys are not Xs. Various rules of connexion are given, being all translations of those in the work on Formal Logic, except a classification of the par- ticulars by probability, answering to that of universals. Thus of X))Y and X(-)Y, each makes the other impossible: of their con- 92 traries X(-(Y and X)(Y, each, so far as it affects the other, reduces its probability. It appears that a quantified term has a quantified contrary : that of ‘Every X’ is ‘some ws, &c. The symbolic canon of validity is:—if both middle parentheses turn the same way, there need be one universal proposition ; if dif- ferent ways, two. Thus )))( and (-))-( both have inferences ; and so has )*()); but )-(( has none. The symbolic canon of inference is ;—erase all signs of the middle term, and what is left (two nega- tions, if there, counting as an affirmation) shows the inference. Thus from X(-)Y)-(Z we infer X(*-(Z or X((Z: more simply, from (+))+( we infer ((. Section IV. On the symbolic forms of the system in which all the combinations of quantity are introduced by arbitrary invention of forms of predication (Sir W. Hamilton’s). The modes of predication peculiar to this system have the same symbols, )( and (+), as the peculiar propositions of the system of contraries ; but with very different significations, as follows :— Contraries. (-) Universal negative with particular terms, and affirmative form in common language. All things are either Xs or Ys. )( Particular affirmative with universal terms, andnegative form in common language. Some thingsareneither Xsnor Ys. Invention of predicates. (-) Particular negative with particular terms, not used in common language. Some Xs are not some Ys. )( Universal affirmative with universal terms, being declaration of identity in common language. All Xs are all Ys. Mr. De Morgan argues that Sir William Hamilton’s system cannot be called an extension of that of Aristotle, in the sense in which that word is used, The forms of predication are as follows :— A,+A!)( All Xs are all Ys E, )*( No Xs are Ys I, () Some XsaresomeYs — (*) Some Xsarenotsome Ys A, )) All Xs are some Ys O! )>) No Xs are some Ys A! (( Some Xsare all Ys O, (-( Some Xs are no Ys. Previously to entering upon the forms of syllogism, Mr. De Morgan repeats and reinforces the objections brought forward in his Formal Logic; namely, that )( is a compound of )) and ((, and has no sim- ple contradiction in the system; and that (-) not only has no simple contradiction, but cannot be contradicted except when the terms are singular and identical. He then proceeds to propose one mode of remedying these defects. Calling the ordinary proposition cumular, he proposes to make it exemplar, as asserting or denying of one in- stance only. In the universal proposition, the example is wholly in- definite, any one ; in the particular proposition it is not wholly indefinite, some one. ‘The defects of contradiction are thus entirely removed, as in the following list, in which each universal proposition is followed by its contradiction. 93 )( Any one X is any* one Y (( Some one X is any one Y (-) Some one X is not some one Y | )-) Any one X is not some one Y )) Any one X is some one Y ):( Any one X is not any one Y (-( Some one X is notany one Y | () Some one X is some one Y In both systems there are thirty-six valid syllogisms, and in both the canon of validity is,—one universal (or wholly indefinite) middle term, and one affirmative proposition. But the symbolic canons of inference differ as follows (with reference to the order XY, YZ, XZ). Exemplar system.—Erase the middle parentheses, and the symbol of the conclusion is left: thus ())-) gives (-). Cumular system.—Erase the middle parentheses, and then. if both the erased parentheses turn the same way, turn any universal paren- thesis which turns the other way, unless it be protected by a mark of negation. Thus )-(() gives )-), ())( gives (), and ())-( gives (-(. Section V. On the theory of the copula, and its connexion with the doctrine of figure.—In his work on Formal Logic, Mr. De Morgan had analysed the copula, and abstracted what he calls the copular conditions of the relation connecting subject and predicate. These are, transitiveness, seen in such copule as support, govern, is greater than, &c., ex. gr. if A govern B, and B govern C, A governsC: and convertibility, seenin such copule as is acquainted with, agrees with, &c.; ex. gr. if A agree with B, B agrees with A. Mr. De Morgan’s position is, that any mode of relation which satisfies both these conditions has as much claim to be the copula as the usual one, is, which derives its fitness entirely from satisfying the above conditions. So far the work cited. In the present paper the correlative copula is introduced, as is supported in opposition to supports, &c., and every system of syllogism is thus extended. Ifa copula be taken which is only trans- itive, but not convertible, every syllogism remains valid, provided that the correlative of that copula be used instead of it, when needful. And in this consists, according to Mr. De Morgan, the root of the doctrine of figure. If + represent affirmative, and — negative, the four figures are connected with ++, +—,—+, and —— (in the system of contraries, where negative premises may have a valid con- clusion, the fourth figure has equal claims with the rest, though the conditions of all the figures are singularly altered). These forms do not require the correlative copula: thus + — in the second figure (as Camestres and Baroko among the Aristotelian forms) are as valid when the copula is ‘ supports’ or ‘is greater than,’ as when ‘is’ is employed. But in every other case the rule for the proper intro- duction of the correlative copula is as follows :—The preceding being called the primitive forms of the four figures, when one premise of a primitive form is altered, the necessity of a correlative copula is thrown upon the other; when both, upon the conclusion. Thus, the primitive form of the second figure being + —, and Cesare showing —-+, it is only valid with the copula ‘ governs,’ by making ‘is not governed by’ the copula of the conclusion, as follows :— No Z governs any Y Every X governs a Y Therefore no X is governed by any Z. * So that there can be but one X and one Y, and that X is Y. 94 By an additional letter (g) introduced into the usual words of syllogism, the places of the correlative copula may be remembered, as in Barbara, Celagrent, &c.: a g being made to accompany any member of the syllogism in which the correlative copula must be employed. This theory is applied equally to the Aristotelian system, to Sir William Hamilton’s (though not of universal application in the eu- mular form), and to Mr. De Morgan’s system of contraries. The extensions required by the use of a merely transitive copula, in the last-mentioned system, are discussed; and mention is made of the tricopular system, in which the leading copula and its correlative have an intermediate or middle relation, equally connected with both; as in > = and < of the mathematicians. The next step is the assertion that it is not necessary that any two of the three copule of a syllogism should be the same; all that is requisite is that, in affirmative syllogisms, the copular relation in the conclusion should be compounded of those in the premises. The instrumental part of inference is described by Mr. De Morgan as the elimination of a term by composition (including resolution) of relations, which leads to the conclusion that whenever a negative premise occurs, there is a resolution of a compound relation. ‘This resolution is shown in a case (among others) of the ordinary copula, in which, however, _ it would hardly strike the mind more forcibly than would the pro- perties of powers in algebra if every letter represented unity. Mr. De Morgan shows (in an addition) that in some isolated cases of in- ference which are not reducible to ordinary syllogism, logicians have had recourse to what amounts to composition of relations. Mr. De Morgan next points out that the copular relation, in affirmative propositions, need not be restricted as applying to one instance only of the predicate; and shows that the removal of this usual restriction entirely removes all his objections to Sir William Hamilton’s form of his own system. Section VI. On the application of the theory of probabilities to some questions of evidence.—This inquiry was suggested by the apparent (but only apparent) error of the logicians, who seem to lean towards the maxim that, when the subject and predicate are unknown, the universal and particular propositions ‘ Every X is Y,’ ‘Some Xs are not Ys,’ are @ priori of equal probability. The difficulty is one which occurs in the following case:—If a good witness, drawing a card from a pack, were to announce the seven of spades, his credit would not be lowered, though he would have asserted an event against which it was 51 to 1 a priori. A common person gives the true answer, * Why not the seven of spades as well as any other?’ Many readers of works on probability would be inclined to say ‘That is not the question ; why the seven of spades rather than some one or another of the fifty-one others?’ The retort is fallacious: it rubs out the distinctive marks from the other fifty-one cards, and writes on each of them ‘not the seven of spades’ as its only exponent. Laplace has chosen two problems, in one of which the distinctive marks exist, and not in the other; and, neglecting the consideration of the first one, has founded his remarks upon the deterioration of evidence ' 95 by the assertion of an improbable event, entirely upon the second. The object of this section is, by a closer examination of the mathe- matical problem of evidence, to ascertain the accordance or non- accordance of the results of usual data with usual notions. The result of the examination is, that common notions, as in other cases, are found closely accordant with theory. For instance, if there be n possible things which can happen, so that the mean probability of an event is 4 a witness of whom we know no particular bias towards n one mode of error rather than another, asserting an event of which the @ priori probability is a, has his previous credit raised, unaltered, or lowered, according as a— ti. positive, nothing, or negative. So n that though the 2 priori probabilities were distributed among a mil- lion of possible and distinguishable cases, yet a witness asserting one of them against which it is only 999,999 to 1, would have as good a right to be believed as though there had been but two equally pro- bable cases, of which he had asserted one. March 11, 1850. On the Numerical Calculation of a class of Definite Integrals and Infinite Series. By Professor Stokes. In a paper ‘“‘ On the Intensity of Light in the neighbourhood of a Caustic,” printed in the sixth volume of the Cambridge Philosophical Transactions, Mr. Airy, the Astronomer Royal, has been led to con- sider the integral ad TT w=" cos — (w’—mw)dw, 0 2 and has tabulated it from m=—4 to m= +4 by the method of qua- dratures. In a supplement to the same paper, printed in the fifth part of the eighth volume, Mr. Airy has extended the table as far as m= +-5°6, by means of a series proceeding according to ascending powers of m. This series, though convergent for all values of m, however great, is extremely inconvenient for numerical calculation when m is large, and moreover gives no information as to the law of the progress of the function for large values of m. ‘The author has obtained the following expression for the:calculation of W for large, or even moderately large, positive values of m: W=2 (3m)-# { R cos (e- 7)+s sin (°—3) iz where R=) — 2-9:7-11 yp E670) ABT P 19. 2S 1.2(72¢) 1.2.3.4(729)* 21.5 1.5.7.11.18.17 | 1.72¢ 1.2.3(72¢)s eee r=n(3) When m is negative, and + mw is written for —mw in the integra 96 W, so that in the altered form of the integral m is positive, there — 14 1.5.7.11 2 ei pie Ppived 3 5.7.11 aaa ens } 1.729 | 1.27998 } By means of these expressions, W may be calculated with great facility when m is at all large. The author has given a table of the roots of the equation W =0, from the second to the fiftieth inclusively, calculated by a formula derived from the former of the above expres- sions. This formula was not sufficiently convergent to give the first root to more than three places of decimals; but this root may be obtained more accurately from Mr. Airy’s table. . The method by which the author has treated the integral W ap pears to be of very general application, and he has further exem- plified it by applying it to the infinite series T a xt x8 2° oe '~ at ae ae TO = Md Sy which occurs in a great many physical investigations, as well as to the integral which occurs in investigating the diffraction produced by a screen with a small circular aperture, placed in front of the object-glass of a telescope through which a luminous point is viewed. Curvature of Imperfectly Elastic Beams. By Homersham Cox, B.A. Jesus College. The equation to the curve of an elastic deflected beam is usually deduced from the assumption,—1, that the longitudinal compression or extension of an elastic filament is proportional to the compressing or extending force; 2, that for equal extension and compression the compressing and extending forces are equal to each other. These hypotheses are not quite correct in practice. All substances appear to be subject to a defect of elasticity, i. e. their elastic forces of restitution increase in a somewhat less degree than in proportion to the extension or compression. If the forces be taken as functions of the latter quantities expressed by a converging series of their ascending integral powers, the terms after the third may in general be neglected as of inconsiderable magnitude. If then e be the exten- sion of a uniform rod of a unit of length and a unit of sectional area, the longitudinal force producing that extension is ae+ Ber+ B'es, where a, B, PB! are empirical constants. Similarly, if ¢ be the compression of a similar rod, the force pro- ducing that compression is yet éc2+ d'cs, where y, 6, 5 are three other empirical constants. These formule are to be applied to a uniform beam of rectangular section, resting on horizontal supports and slightly deflected at its centre. For this purpose, the compression and extension of every filament of the beam are expressed in terms of the radius of curva- ture and the distance from the neutral axis. Analytical expressions are thus obtained for the elastic forces developed in any transverse section of the beam ; and the position of the neutral axis is obtained 97 from the integrals of these expressions by the principle, that the sum _of all the horizontal forces above is equal to the sum of all the hori- zontal forces below the neutral axis. Next, the sums of the moments of the elastic forces about the neu- tral axis are obtained; and the sums are equated to the moment about that axis of the pressure (P) of the fulcrum, the latter moment being the product of half the deflecting weight by the distances (x) of the fulcrum from the point of the neutral axis here considered. This equation involves the fadius of curvature, and is solved with respect to the reciprocal of that quantity. It is to be observed, that this equation, and also the preceding one determining the neutral axis, are not of such a form as to admit of direct solution, and are therefore solved by an ordinary method of approximation. The reciprocal of the radius of curvature of a point (2, y) ofa curve is equal to (the second differential of y with respect to r)+ (a quantity which becomes equal to unity when, as here, the incli- - nation to the axis of v of the tangent at any point of the curve is comparatively very small). _ Making the substitution indicated, and integrating twice the equa- tion last obtained, we obtained finally for the equation to the neutral line of a rectangular beam of vertical depth d, and horizontal breadth pw, and length 2a, ines bxtxt , (2b°—c)xox> CS _ bx2as 4 Pps): fam. 8.4 4.5 ale, 4 where b= : d( B+ 8a%y-2)(1 + aby—4)-2 = 82 4p aty-4)-0(8'4 Haby~8) Ros et eS Hoes | dn,—3 )2 % ear 'y: If, according to the ordinary hypotheses of perfect elasticity, we put a=y and neglect terms depending on 8, f', 0, 3’, this equation to the elastic curve coincides with that given by Poisson and others. If we put e=a, the value of the deflection at the centre of the beam is xaS _-bx2at 4 (262—c)x8a5 3 4 5 Whence it may be seen that the deflection is greater than it would be if the elasticity were perfect. On the Knowledge of Body and Space. By H. Wedgwood, M.A. No part of the great metaphysical problem chalked out by Locke has been more assiduously laboured, and none has attained a less satisfactory solution, than that which relates to the origin of the idea of space and its subordinate conceptions, figure, position, mag- nitude. It was seen that the exercise of the muscular frame must somehow be instrumental in making us acquainted with the material and ex- 98 tended world, but all hopes of a logical explanation of the process by which that effect is produced seemed cut off at the outset by a preliminary objection. The knowledge of motion, it was said, ob- viously involves the knowledge of the body moved. The conscious- ness of the motion of the hand therefore implies the conception of the hand itself, an object of certain shape and size. The attempt to account for the notions of shape and size from the motion of the hand was thus apparently stranded in a hopeless paralogism ; and so insurmountable was the difficulty taken to be, that philosophers were driven to imagine a second source of elementary ideas, in addition to the simple apprehension of the thing conceived in actual existence, maintaining that space is known to us as the condition under which we perceive external things, or, as others express it, that the notion of space arises in the mind on the first apprehension of body by a principle of necessary judgement, which impresses upon us the con- viction that all body is contained in space. In the paper laid before the Society, an attempt is made to show the utter barrenness of this hypothesis of a necessary origin (as it is called) of the idea of space; and the main object of the paper is to rest the idea on a more solid foundation, by showing the adequacy of muscular exertion, in conjunction with the sense of touch, to fur- nish us with complete knowledge of the material and extended world by the ordinary way of actual experience. There are two kinds of action; one instinctive, immediately in- duced by the physical constitution of the agent independent of the understanding ; the other rational, induced by the discernment of some object of desire in the end to be accomplished, and of course implying a previous conception of the action in question. Familiar instances of instinctive action are then pointed out, from whence it would appear that the sensations of touch felt on contact of any part of the living frame with a foreign body operate as motives to instinctive exertion through the instrumentality of that part of the muscular frame on which the sensible impression is made, in- stinctively impelling the sentient being to muscular reaction against the material cause of the sensation, or leading him to shrink from it if the sensation is of a painful nature. Attention is directed in particular to the action of an infant in- stinctively closing his hand upon a finger placed within his palm; and it is argued that the effect of such an action on his understanding will be the direct apprehension of body, a complex object consisting of surface (undeveloped as yet in form and magnitude) apprehensible by tactual sensation; and substance, revealed by resistance to mus- cular exertion, constituting a new kind of being essentially different from any of those discerned by means of the five senses. The relation between body and space is illustrated by comparison with the case of light and darkness, the second of the two correla- tives belonging in each case to Locke’s class of positive ideas from negative causes. As he who has once apprehended light is subse- quently enabled to look for that phenomenon in a direction from whence no rays actually penetrate the eye, so, it is argued, will he who has once made use of his hand in the apprehension of body be (lil pet elem pars Dal te: Pago is to experience of the itself im concrete existence. The subsequent enlargement of the idea, so as to comprehend the space occupied by the solid substance of bodies and that which stretches away to infinity in all directions around us, is duly accounted for on the same principle; and that i Gheomediring the destruction of any portion of space, on which 20 much stress has been laid as establishing the necessity of a deeper- seated origin than simple experience, is shown to be the natural consequence of the negative foundation of the idea as explained by . the analogy of light and darkness. April 15, 1850. On the Mathematical Exposition of some Doctrines of Political Economy. By the Master of Trinity. The of this paper was to solve algebraically certain pro- Be Gee have book acieed by Mr. 5.5 Mill and others by means of numbers, taken as examples; the principles of these writers being taken for granted in the algebraical solution. Mr. Mill has rightly observed, that instead of saying that prices are determined by the ratio of demand and supply, we ought to say that they are determined by the equation of demand and supply. This equation may be thus stated. vr 7 fant sede rt ee teem Ge 27 and sold at that price becomes p’, let g become q'; and p’ being equal to pl), Nef =re lametrniys this is the equation of demand and supply. For different commodities, we have different values. ‘There are such classes of commodities as these: (A.) Con- ventional necessaries, for which m=1: of these the same quantity is bought whatever be the price. (B.) Articles of fixed expenditure, for which m=0: on these the same sum is always expended, a smaller quantity being bought in proportion as they aredearer. (C.) Common necessaries, in which m is between 1 and 0: in these, when the price falls, the consumption is increased, but the money ex- pended diminished. (D.) Popular lururies, im which m is negative : in these, when the price falls, the consumption is so much increased that the money expended on them is increased alto. For corn, the mean value of m seems to be about 5 : on this supposition a failure of one-fourth in the supply would double the price. The quantity m measures the susceptibility of the price to change when the supply changes, and also the intensity of the demand. 100 Another division of commodities is, according to the cost of pro- duction. These are (a) commodities of fixed and limited supply ; (8) commodities of fixed cost; (y) commodities of increasing cost for increasing supply, as for instance corn in a given limited district. The equation of price for the last case was given. The like methods were applied to solve certain problems concern- ing international trade, treated by Mr. Mill. If the relative value of two commodities, C and D, in England and Germany be different, there will be a saving in exporting each from where it is cheaper to where it is dearer; and the question is, at what point prices will settle. We must introduce here the principle of the uniformity of international prices; namely, that when the trade is established, the relative prices of C and D will be the same in the two countries: the principle of the equality of imports and exports in each country ; and the equation of demand and supply already stated. By combining these principles, the problem of the resulting price is solved. But it is found that there is no solution possible (that is, no solution in which both countries gain by the trade), except the mutual demand for the interchange of commodities be nearly equal. This limitation of the solution is given by the algebraical method, and seems to have been overlooked by previous writers. The same methods were extended to a greater number of ex- ported and imported commodities; and finally, it was remarked that these calculations are all founded on principles of equilibrium, whereas a state of equilibrium is never attained; and thus the theory may be very imperfectly applicable, like the equilibrium theory of the tides. Second Memoir on the Intrinsic Equation of Curves. By the Master of Trinity. The intrinsic equation of curves, according to which any curve is expressed by means of an equation between its length (s) and its angle of deflection (¢), may be conveniently used for many purposes. When a curve is so represented, the portion of the length which comes after a cusp must necessarily be taken asnegative. This had appeared anomalous to some mathematicians, on the ground that a cusp is in all cases the limit of a loop. To clear up this point, the author adduces two cases. (1.) Thecurve of which the equation is s=a?+06 sin ¢, which is a looped curve when 6 is less than a, and a cusped curve otherwise. But in this curve it appears that a loop arises from the vanishing of two cusps, and of the intervening nega- tive portion of the arc. (2.) The case of the ordinary trochoid, which is a looped curve when the describing point is exterior to the rolling ~ circle, and becomes a cusped curve (a cycloid) when the point is in the circle. But in this case the length of the trochoid is equal to the length of an elliptical arc, which, in the case of the cycloid, coincides with the major axis, and becomes negative beyond the vertex of the ellipse. Other equations were examined, which give running pattern curves with cusps, cusped curves with infinite diver- ging spirals at the extremities, and sinuous curves with infinite con- verging spirals at the extremities; and certain integrals which oc- curred in the former memoir on this subject were discussed. 101 May 13, 1850. Results* connected with the theory of the singular solution of a Differential Equation of the first order between two variables. By Professor De Morgan. By a singular solution of a differential equation is here meant any solution which can be obtained by differentiation only, whether it be a case of the primitive by integration or not. By acurve is meant all that is included under one equation, whether resoluble into what are commonly called complete curves or not. Thus, the equation (z—y) (2? +y°—1)=0 belongs to a curve, having a rectilinear branch and a circular one. By such a symbol as v, is meant the partial differential coefficient ao when obtained from an equation in which v is explicitly expressed in terms of x and (it may be) other variables. - Let haGe y, c)=0 be the nin primitive of the differential equa- tion y'=x(2, y). _ O(x, y; ¢) belongs to two distinct classes of curves :— . 1. Continuous curves derived from such values of c, real or ima- ginary, as will enable ¢=0 to exist for points infinitely near to one another. 2. Systems of points, derived from S A(a, y,%,B)=0, Bia, y, a, 8)=0, where (x, y,a+8 /—1)=A(za, y, a, B)+B(a, y, a, B)./—1. When a curve is such that the points on one side of it are on curves of the first kind, and those on the other side are part of systems of the second kind, let that curve be called a separator; and the same when it separates points of both kinds from points which belong to one kind only. No solution of the differential equation can be formed by combining all those systems of the second kind in which a@ and # are connected by a real relation. The curve which has at every point of it, either $2 — @, Py = &, Pe Pe or Pz — oo, Py finite, «= const., Pe Pe * This communication is the abstract of a part of a paper not yet com- pleted, and was forwarded to the Society for the purpose of ascertaining whether any examples could be produced destructive of the perfect gene- rality of the results, 102 or fr = &, $e finite, y = const., Pe Pe is a singular solution, And in the above are contained all the sin- gular solutions. jean an Every branch of a singular solution is either— A separator, only. A curve, every point of which has a contact of the first order at least with some one real primitive, only. Or both. Or neither. If the first or last, it is a case of the complete primitive, And such cases may be introduced at pleasure into the singular solution, by writing the primitive in the form 9 (x, y,fc)=0. A branch of a singular solution has at the utmost » contacts with each primitive which it touches (m being determined by the nature of the equation), and all of the first order, generally. Or, p, of the first order, p, of the second, &c., p,+2p.+38p3;+... being 2 or 2— (an even number); or some of these cases for some primitives, others for others, including the possibility of some cases giving none at all, when z is even. D fos The branches of the singular solution which have contact with ordinary primitives (whether themselves ordinary primitives or not) to the exclusion of the branches which are only separators, may be determined from the differential equation by the following test. Let y'=x(2, y) be the differential equation; whence SXF Ky -% Find the curves which satisfy either of the following sets of con- ditions :— X= @ Xy=.& y'' finite, or X_= % #= const. y"' finite, or X= % y= const. y" finite. _ Every such curve does satisfy the differential equation, and is singular solution having contact with some one primitive at every point. And other such singular solutions there are none except those designated by r= o, or y= o, or both. But if _ es TP sient . X,= © and Xg= © y =o, X,= z= const. y= a, or X,= @ y= const.. y"= a, y then the differential equation may or may not be satisfied; but the curve passes through the singular points of the primitives, with or without contact, according as the differential equation is or is not satisfied. An evolute is such a pseudo-singular solution to all the involutes, passing through their cusps. . PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. —_—— November 11, 1850. On the Mathematical exposition of certain Doctrines of Political Economy. By the Master of Trinity. Third Memoir. The object of this memoir was to point out some of the laws of in- ternational trade, taking into account the effect of the import or ex- port of money, i. ce. of metallic currency. It was stated that when the balance of imports and exports is deranged by the import of a new commodity, previously produced at home, the effect is, to diminish the annual import of gold and silver; hence, to lower the scale of prices in general; hence, to increase the exports, and thus, to pro- duce a new condition of equilibrium: and the necessary supposi- tions being made, the amount of depression in prices arising from such a cause was calculated. The Master of Trinity also made a communication relative to a new kind of coloured fringes. He stated that he had, many years ago, remarked that if we hold a candle before a dusty looking-glass at a distance of six or eight feet, so that the image of the candle is near to that of the eye, the image of the candle is seen in the middle of a patch of coloured bars, which are perpendicular to the plane passing through the candle and the eye, normal to the looking-glass. This remark was communicated to M. Quetelet, and published by him. Attention has recently been drawn to this observation, at the Con- gress of Swiss men of science, held at Aarau, in August of the pre- sent year. M. Mousson of Zurich pointed out, at that meeting, the differences between the stripes noticed by Dr. Whewell, and the rings on specula observed by Fraunhofer. Among these differences are,—1st, Fraunhofer’s rings depend upon the first surface of the spe- culum, the stripes upon both ; 2nd, the rings are not produced except the dust be particles of uniform size; the stripes are produced by dust of irregular and various particles; 3rd, the rings depend for their size on the size of the particles of dust ; the stripes do not. Some discussion took place as to the manner in which these stripes arise from the theory of interferences, and upon their relations to Newton’s “ colours of thick plates.” No. VIII.—Procrepines or THE CAMBRIDGE Putt. Soc. 104 December 9, 1850. On the effect of the internal friction of Fluids on the Motion of Pendulums. By Professor Stokes. It has been acknowledged for some time that the results which follow from the common theory of fluid motion relative to the effect of a fluid on the time of vibration of a pendulum do not agree well with observation. The volume of the Philosophical Transactions for 1832 contains .the results obtained experimentally by the late Mr. Baily relating to the effect of air in altering the time of vibration of a great variety of pendulums. The experimental results are exhibited by the value cf 2, the factor by which the correction for buoyancy must be multiplied in order to give the whole effect observed. With pendulums composed of spheres suspended by-fine wires, Baily found n=: 1'864 for spheres a little less than 14 inch in diameter, and n=1°'748 for spheres about 2 inches in diameter. The result which follows from the common theory is »=1°5, as was first shown by Poisson. The value 1°864 was the mean of 16 pair of experiments, giving a mean error 0°023, and 1°748 was the mean of 12 pair, which gave a mean error 0°014, so that the difference between the two re- sults, and between either of them and the common theory, is far too’ large to be attributed to errors of observation. — The chief object of this paper was, to apply to the calculation of the motion of a pendulum the general equations of motion which are arrived at when the internal friction of the fluid is taken into account, and to compare the resulting formule with the experiments of Baily and others. The general equations, simplified, first, by neglecting the square of the velocity, secondly, by neglecting the compressibility. of the fluid, the effect of which in the present instance is in fact quite insignificant, thirdly, by omitting the external forces, the effect of which may be taken into account separately, are L(t ao p dx dat dy2t das wilt The second and third of the general equations are not written down, because they may be supplied by symmetry. In these equations pis the density, p the mean of the normal pressures in the direction of any three rectangular planes passing through the point of which 2, y, z are the coordinates ; wu is the velocity in the direction of 2, ¢the time, and y’ a certain constant, depending upon the nature of the fluid, which the author proposes to call the index of friction. The author has succeeded in obtaining the solution of equations (1.) in the two cases of a sphere and of an infinite cylinder. The’ latter may be applied to the case of a pendulum consisting of a long cylindrical rod, by treating each element of the rod as belonging to an infinite cylinder oscillating with the same linear velocity. ‘The following is the solution in the case of a sphere, so far as relates to the resultant action of the fluid on the sphere. Let £ be the abscissa of the centre of the sphere, measured in the Bees 6 ER eG G.) 105 direction of the tina a the radius of the sphere, 7 the time of vi- bration, M’ the mass of the fluid displaced, F the resultant force of ‘the fluid on the sphere, so that —F is the resistance; then | —F= am! SE v= sie aks. eon: Se where 1 9/fplr\3 3 ; oe § t se Git, " 27° 2\2ra? rae 5 ie yond se (3) The effect of a fluid on the time of vibration depends on the term which involves k; the effect on the arc of vibration depends on the term which involves #’. ‘ The expression for F has precisely the same form (2.) in the case of a cylinder, but-* and &’ are certain transcendental functions of (w! r)? a—! (a here denoting the radius of the ta i which the author has tabulated. _ The value of yw’ having been determined for air, or any given fluid, by one experiment giving the effect of the fluid either on the time of vibration, or on the arc of vibration, of any one pendulum consisting either of a sphere suspended by a fine wire, or of a long cylindrical rod, or of a combination of a sphere and a rod, the formule which follow from (2.) ought to make known the effect of the fluid both cn the time and on the arc of vibration of all pendulums of the above forms. The agreement of theory with the experiments of Baily re- lating to the effect of the air on the time of vibration of pendulums is remarkably close. Even the rate of diminution of the arc of vibration, the observation of which held quite a subordinate place in Baily’s experiments, agreed with the rate calculated from theory as closely as could reasonably have been expected. The value of the index of friction of water was deduced by the au- thor from some experiments of Coulomb’s on the decrement of the are of oscillation of discs which performed extremely slow oscilla- tions in their own plane by the force of torsion. When this value was substituted in the expression for the time of vibration of a sphere, the result was found to agree almost exactly with Bessel’s experiments on the time of vibration of a sphere swung in water. As a limiting case of the problem of a ball pendulum, the author has deduced the resistance of a fluid to a sphere moving uniformly under such circumstances that the square of the velocity may be neg- lected. ‘The resistance thus determined proves to be proportional, not to the surface, but to the radius of the sphere; and therefore the quotient of the resistance divided by the mass increases very ra- pidly as the radius decreases. Accordingly, the terminal velocity of a minute globule of water descending through the air depends almost wholly on the internal friction of air. Since the index of friction is known from Baily’s pendulum experiments, the terminal velocity can be calculated numerically for a globule of given diameter. The ve- locity thus calculated proves to be so small, in the case of globules 106 such as those of which we may conceive the clouds to be formed, that the suspension of the clouds does not seem to offer any diffi- culty. Had the pressure been strictly equal in all directions in airin the state of motion, the terminal velocity of such globules would have been far larger, and consequently the quantity of water which could have existed in the air in the state of cloud would have been im- mensely diminished. It appears therefore that these small and hi- therto almost unrecognized forces, which depend on internal friction, are essential to the fertility of at least the tropical regions of the earth. The author has also applied the theory of internal friction to the calculation of the subsidence of a series of oscillatory waves. On substituting for the index of friction in the resulting formula the nu- merical value deduced from the experiments of Coulomb, it appears — that in the long swell of the ocean the effect of friction is insignifi- cant, whereas in the case of the short ripples excited on a small pool by a puff of wind the subsidence due to friction is very rapid. Accord- ingly, short ripples of this kind quickly die away when the breeze that excited them ceases to blow. February 24, 1851. On some points of the Integral Calculus. By Professor De Morgan. é - Some time ago, Mr. De Morgan communicated to the Society an abstract of some unfinished views on the connexion between the or- dinary and singular solution of a differential equation. ‘The present paper completes those views, and also contains sections on the solu- tion of differential equations by elimination, on the proof of the number of constants which a solution may contain, and on the cri- terion of integrability of a function of «, y, and differential coefficients of y. 1. On singular solutions.—As to equations of the first order, the tests obtained in this paper may be described as follows :— Mr. De Morgan means by a singular solution any one which is obtained by other process than integration, whether it be contained in the integrated primitive, or not. When the singular solution is not contained in the primitive, he calls it an extraneous solution. Let 9(2, y, c)=0 be the primitive equation, giving c=(a, y). The differential equation then is So. d® '=—_* say= (z, {e == y ®, y x(@ y) x dx &e and o.=—= “aPa &,= — 22 Pe . Every relation between 2 and y which satisfies either of the fol- 107 lowing collective conditions, is a solution of the equation; and, by~ definition, a singular solution. . 1. ©, and ©, both infinite. 2. , only infinite, and e=const. 3. ®, only infinite, and y=const. And all possible finite solutions of the differential equation are given either by the original primitive, or by these relations. Let tet eX. Then all relations between x and y which Xy satisfy either of the following collective conditions are solutions of the differential equation, and are singular solutions. UR Xe and Xe both infinite, and X=0. 2. X, only infinite, z=const., and X=0. 3. x, only infinite, y=const., and X=0. _ But when one of these sets fails only in that X does not vanish, the curve so obtained, instead of having contact with a primitive curve at every one of its points, passes through the points of infinite curvature of the primitives; and the differential equation which is satisfied is y’=y—X. Every evolute is related in this manner to its involutes, passing through all their cusps. The above tests do not give the possible case in which c=, or y= o, is a singular solution. Mr. De Morgan proposes the following geometrical illustration of the connexion between the primaries and the singular. Let ¢ be the third ordinate of a surface (usually denoted by z) having the equation @(z, y,c)=0. The projections upon the plane of zy of sections parallel to that plane are the primaries: the singular solution is the base, upon the plane of zy, of a cylinder perpendicular to that plane, and which always touches the surface. By means of this illustration, it may be made manifest that certain cases of singular solution which have always been discarded as unmeaning, are limiting cases of the kind which are admitted in analysis so soon as the way up to the limit is clearly seen. Taking a general equation with two arbitrary constants, so that a relation between those constants selects and designates a family of curves, it is shown generally (without examination of exceptional cases) how to find the families which have with their singular curves contact of the second order. The equation of these singular curves is a differential equation of the first order: but ifs singular solution is the singular curve of a family of curves which haye with it a con- tact of the third order. ~ 2. Solution of differential equations by elimination.—This is an idea derived from the method which Mr. De Morgan communicated (vol. viii. part 5) relative to partial differential equations, and which he found, after his paper was finished, had been given by M. Chasles, as he supposed, from knowledge of the results of Monge. But it afterwards appeared that the authority for Monge having obtained 108 such results is only a candid supposition of M. Chasles himself, and that no memoir on the subject, written by Monge, has been traced. _All that M. Chasles had to proceed on was the f¢itle of a memoir - mentioning a certain mode of generating conjugate surfaces, from which he thought it very likely that the solution of partial differential equations which he himself thence found, had really been found by Monge. Under these circumstances, Mr. De Morgan is of opinion that the method must be attfibuted to M. Chasles as its first dis- coverer, at least until something further appears. Mr. De Morgan proceeds to make use of the equation Si P y= py” —Ply"-» a ee, to form various cases of equations which can be reduced to lower orders, and which can finally be solved by elimination. Of these, the most simple specimen, being the one suggested by thinking on the method above alluded to, is as follows :— A If e=Y' and y=XY'—Y, Y being a function of X, whence y is a function of x, we have the following sets of correlative equations :— r= Y! X=y' y=XY'—Y Y=ay'—y y’=X Visser 7-1 1 Y— y’=—— Yy’ " sais ae Ul ois yl"! ieialgothe 4 gti and soon. If, then, ¢(z,y, y', y", y!",...)=0 be a given differential equation, and if it be found that 1 yu iff o(¥ XY'—Y; xX, yi — srg) = can be solved; it is seen that the original equation can be solved by eliminating X between e=Y' and y= XY'—Y. The general method of which this is a particular case, is as follows. Let f(x, y, X, Y)=0 have its differential equations of the first order formed on two suppositions: first, that X and Y are constant; se- ood that z and y are constant. Let these differential equations e X=0(2, Y; y') x=9(X, +i wy Y=a, y, y') y=U(X, Y, Y’). These equations may be used instead of the first two pairs of cor- relatives in the preceding example: and each differential coefficient of Y is expressible by means of the same and lower differential coefficients of y; and vice versd. To get convertible forms, as in the instance above, f(x,y, X, Y) must be chosen so that # and y are simultaneously interchangeable with X and Y. Mr. De Morgan gives a similar extension of the method as applied to partial differential equations. . 8. On the constants of a primitive equation.—It is usually left to 109 be collected from induction that the equation of the nth order has n constants, and no more, in its complete primitive. Mr. De Morgan proposes an @ priori proof of this point, on which, as in all such cases, it would be presumptuous to decide until it has been tho- roughly examined. He further proposes an extension of the meaning of the term so- lution, in the case of all the primitives intermediate between the differential equation and the original*primitive. ‘Thus, supposing an equation of the third order, of which the admitted primitives of the second order are _ U,=const., U,=const., U;=const., he maintains that the general primitive of the second order is J(U,, U;, U;)=0, where f is any function whatsoever: and, starting from this last equation, he determines a general primitive of the first order in a similar way. This view is supported by the reduction of a common differential equation of the zth order to a partial differential equation of the first order with n independent variables. 4. On the criterion of integrability of (2, y, y', y",...).—lf we denote the differential coefficients of y by p, g, r, s, &c., it is well known that the condition which is both necessary and sufficient, in order that V=¢(z, y, p, q,...) may be integrable without reference to relation between y and 2, is V,—V,'+V,!'—V"+ ...=0, the accent denoting complete differentiation with respect toz. This has usually been established, either by the calculus of variations, or by a process of elaborate expression of the actual result in terms of definite integration with respect to a subsidiary variable. Mr. De Morgan, after some remarks upon the manner in which certain » proofs of the necessity of the criterion fail, gives a very simple ele-' mentary proof founded upon the following theorem. If U be any function of z, y, p, &c.,—as far say as s, for an instance,—then (U),=U/,, (U),=U,!, (U'),=U,'+U,, (U!),=U,'+U,, (U'),=U,'+U,, (U),=U,'+U,, (U'),=U,. Mr. De Morgan takes it to have been hitherto unnoticed that the formule V,—V,'+V,"—..., V,—V,/+..., 80 much used in this sub- ject, are, when V is integrable, nothing but the differential coefficients of f Vdx, with respect to y, p, &c. [But since the paper was communicated, Mr. De Morgan has found the above theorem, and its consequences, in a memoir by M. Sarrus, apparently belonging to the Journal de [Ecole Polytechnique, and printed in 1824. No notice is taken of this method by MM. Ber- trand, Binet, or Moigno, who have written on the subject since M. Sarrus. ] tn 110 May 5, 1851. Of the Transformation of Hypotheses in the History of Science. By W. Whewell, D.D. . The author remarks that new theories supersede old ones, not only by the succession of generations of men, but also by transfor- mations which the previous theories undergo. Thus the Cartesian hypothesis of vortices was modified so that it explained, or was sup- posed to explain, a central force: and then, the Cartesian philoso- phers tried to accommodate this explanation of a central force to the phenomena which the Newtonian principles explained ; so that in the end, their theory professed to do all that the Newtonian one did. The machinery of vortices was, however, a bad contrivance to pro- duce a central force; and when it was applied to a globe, its defect became glaring. Still however, the doctrine of vortices has in it nothing which is absurd anterior to observation. The “ nebular hypothesis” is a hypothesis of vortices with regard to the origin of the system of the universe, and is now held by eminent philosophers. — Nor is the doctrine of the universal gravitation of matter at all in- consistent with some mechanical explanation of such a property ; for instance, Le Sage’s. We cannot say therefore that if the planets are moved by gravitation, they are not moved by vortices. ‘The Carte- sians held that they were moved by both; by the one, because by the other. Like remarks may be made with respect to the theories of mag- netism and of light. May 19, 1851. On the Colours of Thick Plates. By G.G. Stokes, M.A., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge. By the expression “colours of thick plates” is usually understood the system of coloured rings, discovered by Newton, which are formed on a screen when the sun’s light is transmitted through a small hole in the screen, and received perpendicularly upon a concave mirror of quicksilvered glass, placed at such a distance from the screen that the image of the hole is at the same distance from the mirror as the hole itself. The brilliancy of the rings, as was afterwards discovered, is greatly increased by tarnishing the surface of the mirror; and it is also advantageous to use a lens to collect the sun’s rays, and to place the screen so that the small hole may be situated at the focus of the lens. These rings were first explained on the undulatory theory by Dr. Young, who attributed them to the interference of two streams of light, of which the first is scattered at the tarnished surface of the mirror, and then regularly reflected and refracted, while - the second is regularly refracted and reflected, and then scattered in coming out of the glass. ‘The theory has been worked out in detail 111 by Sir John Herschel, who has investigated the case in which the two surfaces of the glass belong to a pair of concentric spheres, and © the hole in the screen is situated in the common centre of curvature. A set of coloured bands has since been observed by Dr. Whewell ina common plane mirror. These bands are seen when a candle is held near the eye, at the distance of several feet from the mirror, and is viewed by reflexion. It is necessary that the first surface of the glass should be a little tarnished. The theory of these bands had not been worked out, and it had even been doubted by some philosophers whether they were of the nature of the coloured rings of thick plates. In this paper the author gave a general investigation, which in- eludes as particular cases the theory of the rings formed on a screen in Newton’s experiment, and that of the bands which Dr. Whewell had observed in a plane mirror, and which are not thrown ona screen, but viewed directly by the eye. He also exhibited to the meeting a variation of Newton’s experiment, in which an extremely beautiful system of rings is very easily produced without sunlight. The face of a concave mirror of quicksilvered glass was prepared by pouring on it a mixture consisting of one part of milk to three or four of water, and then holding the mirror vertically in front of a fire to dry. When the flame of a taper, or of an oil-lamp with a small wick, is placed in front of a mirror thus prepared, in such a position as to coincide with its inverted image, a beautiful system of rings is seen encompassing the flame. These rings appear to have a definite position in space, like a bodily object. The rings thus formed, which are evidently of the nature of Newton’s coloured rings of thick plates, may be made to pass in a perfectly continuous manner into the coloured bands observed by Dr. Whewell. The author has compared theory and experiment in various par- ticulars, and has found the agreement perfect. It will be sufficient to mention here one result of theory, which is of great generality and of considerable elegance. It applies to the system of rings seen by reflexion in a mirror, either plane or curved, when a luminous point is placed anywhere near the axis, and the eye occupies any other position likewise near the axis. The result is as follows :— Join the eye with the luminous point, and likewise with its image, whether it be real or virtual, and find the points in which the join- ing lines, produced if necessary, cut the mirror. Describe a circle having for diameter the line joining these two points. This circle will be the middle line of the bright colourless fringe of the order zero, and on each side of it the colours will be arranged in descend- ing order. June 2, 1851. On a new Elliptic Analyser. By Professor Stokes. After mentioning some of the inconveniences and inaccuracies attending the use of a Fresnel’s rhomb in the analysis of elliptically- polarized light, and alluding to some other methods which had — 112 employed for the purpose, the author proceeded to describe a new instrument which he had invented, and which he exhibited to the meeting. In the construction of this instrument he had aimed at being independent of the instrument-maker in all important points except the graduation. The construction is as follows :-— A brass rim or annulus is mounted so as to stand with its plane vertical when placed on atable. Within this rim turns a brass gra- duated disc ; and the angle through which it turns is read off by means of verniers engraved on the face of the rim, and reading to tenths of a degree. ‘This disc is pierced at the centre, and carries on the side turned towards the incident light a retarding plate of selenite, of such a thickness as to give a difference of retardation in the oppositely polarized pencils amounting to about a quarter of an undulation. In front it carries a hollow cylinder, turned on the lathe along with the disc itself. Round this cylinder there turns a collar containing a Nicol’s prism, and carrying a pair of bevel-edged verniers, by which the angle may be read off through which the prism has been turned. Thus the retarding plate moves in azimuth carrying the prism along with it, and the prism has likewise an in- dependent motion in azimuth. In observing, the light is extinguished by a combination of the two movements, in which case the elliptically-polarized light is con- verted by the retarding plate into plane polarized, which is then ex- tinguished by the Nicol’s prism. On account of chromatic yaria- tions, the light is not, strictly speaking, extinguished, unless homo- | geneous light be employed, but only reduced to a minimum. There are two principal positions of the retarding plate and Nicol’s prism in which the light is extinguished, or at least would be extinguished if the incident light were homogeneous; and for each principal position there are four subordinate positions, since either the retard- ing plate or the Nicol’s prism may be reversed by turning it through 180°. The mean of the four subordinate positions may be taken for greater accuracy. Let R, R’ be the readings of the fixed, r, r' those of the moveable > verniers in the two principal positions; I the index error of the fixed verniers, that is, the azimuth of the major axis of the ellipse de- scribed, measured from a plane fixed in the disc; 7 the index error of the moveable verniers, that is, the azimuth of the principal plane of the prism, measured from a fixed plane in the disc; @ the angle whose tangent. is equal to the ratio of the axes of the ellipse de- scribed ; p the difference of retardation of the oppositely polarized pencils transmitted through the plate, measured as an angle, at the rate of 360° to one undulation. Then the unknown quantities I, i, @, and p are given in terms of the known quantities R, R/, r, and r/ » _ by the following formule, which happen to be extremely convenient for numerical calculation :— 1 oe I=—(R/+R); =< (r'+r); 5 (R’+R); ak Blk he : pe: Kee gee tan (r'—r) sin (r'—r) pe tan (R’—R). CON Dig se Ne gin (BIR) a 113 The author stated that he had already observed with this instru- ment, and after a little practice had found that it worked in a very satisfactory manner. When the light of the clouds was reflected horizontally by a mirror, and modified so as to produce ellipti- cally-polarizéd light in which the ratio of the axes was about 3 to 1, it was found that the mean error of single observations amounted to about a quarter of a degree in the determination of the azimuth of the major axis, about three or four thousandths in the determination of the ratio of the minor to the major axis, and little more than the thousandth part of an undulation in the determination ofp. Since the magnitude of p depends upon the length of wave, or, what comes to the same, the refrangibility of the light, it follows that a knowledge of the former leads to a knowledge of the latter. It may thus be said that the instrument determines the azimuth and excentricity of the ellipse described, and the refrangibility of the light. An error of the thousandth part of an undulation in the de- termination of p would correspond to an error in the place in the spectrum assigned to the light operated on amounting to less than the twentieth part of the interval between the fixed lines D and E. Now by the use of absorbing media it is possible, without too much reducing the intensity of the light employed, to alter greatly its mean refrangibility ; and yet for each medium the refrangibility may be determined very accurately by means of the value of p. Accord- ingly, the instrument is specially adapted for investigations relating to the dispersion of metals, and for other similar researches. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. December 8, 1851. On the Oscillations of Suspension Bridges. By J. H. Réhrs, ' Esq., M.A. In this paper the oscillations of a chain suspended at two points were discussed, with a view to explain the causes of fracture in sus- pension-bridges, by vibration arising from the tramping of troops, gusts of wind, &c., as well as to suggest means for obviating the mischief under those circumstances. The following were some of the most remarkable results arrived at :— Ist. That if the tension at the ends of the chain where it is sus- pended be kept constant by allowing play at those points, the varia- tion of tension due to vibration at any other point of the chain will be but small. 2ndly. That if the chain be tied at the points of suspension so that it can have no motion there, a slight extent of vibration will produce comparatively a great increase of tension. 3rdly. That periodic forces, such as may be taken, for instance, to represent the effect of tramping in time of troops moving across the bridge, are dangerous in the extreme, as if they happen to coincide in period with any of the possible types of vibration, the extent of vibration will increase continuously, till it ceases to be represented approximately by a linear or even an equation of the second order ; in this case, the chain will be divided by nodal points where there is no vertical motion. 4thly. That the mere transit, without tramping, of ordinary loads at an ordinary pace would not cause sensible vibration in a bridge of wide span ; but that terms not periodic might be introduced by the variable pressure of wind sweeping in rapid gusts along the platform. February 16, 1852. On the Composition and Resolution of Streams of Polarized Light from different Sources. By Professor Stokes. In this paper the author investigates the nature of the light result- ing from the union of several independent streams of polarized light. No. [X.—Procrepines or THE CamBRiIDGE Pai. Soc. 116 The refrangibility of the several streams is supposed to be the same, and the polarization to be of the most general nature, that is, to be elliptic. The following proposition is established. When any number of independent polarized streams, of given refrangibility, are mixed together, the nature of the mixture is com- pletely determined by the values of four constants, A, B, C, D, de- fined in the following manner :—Let J be the intensity of one of the elliptically-polarized streams, a the azimuth of its plane of maximum polarization, tan 8 the ratio of the axes of the ellipse described by the ethereal particles; then A=X(J); B=X(J sin 28); C=X(J cos 28 cos 2a) ; D=X(J cos 26 sin 2a). . Two groups of polarized streams, of the same refrangibility, which are such as to give the same values to each of the four constants A, B, C, D, are defined to be equivalent; and the author has shown, that if two equivalent groups be transmitted through any optical train, and be afterwards analysed, they will present exactly the same — appearance ; so that equivalent groups may be regarded as optically identical. It readily follows from the above theorem, that any group of polarized streams is equivalent to a stream of common light com- bined with a stream of elliptically-polarized light from a different source. If J, J' be the intensities of these streams, a! the azimuth of the plane of maximum polarization of the latter, tan 8! the ratio of the axes of the characteristic ellipse, J=A— V(A2+B2+C%); J’ /(A2+B240%); “ ; tan 2a! =P WV (A?+ Be+ C2)’ Cc’ The author has applied these formule to a few examples, and has likewise shown, from the general principles established in the paper, that the changes which are continually taking place in the epoch and intensity of the vibrations of polarized light may be of any nature. In the case of common light, the author contends that there is no occasion to suppose the transition from a series of vibrations of one kind to a series of another kind to be abrupt, but that it may be of any nature. sin 23'= : Professor Miller made a communication on the Artificial Forma- tion of Crystallized Minerals. March 1, 1852. Mr. Hopkins, F.R.S. &c., gave a Lecture on the Influence of In- ternal Heat, Stellar Radiation, and Configuration of Land and Sea in producing Changes of the Earth’s Superficial Temperature. 117 March 15, 1852. Professor Miller made a communication on different improvements in the Reflective Goniometer; and a description of a New Reflective Goniometer. . Professor Stokes concluded a paper on the Composition and Reso- lution of Streams of Polarized Light from different Sources (see the abstract under the date Feb. 16, 1852, Phil. Mag. vol. iii. p. 316). He also made a communication on Haidinger’s Brushes. Also on the Optical Properties of a New Salt of Quinine. The salt alluded to is that which had recently been discovered by Dr. Herapath (Phil. Mag. vol. iii. p.161). The substance of this communication formed the subject of a notice of the properties of the salt which the author read at the Meeting of the British Association at Belfast, which will be found in the Report of the Transactions of the Sections. ; April 26, 1852. The Rev. Mr. Kingsley gave an account of the application of Pho- tography to the Microscope. - The earliest attempts in photography were directed both by Sir H. Davy and Mr. Fox Talbot to the fixing upon prepared paper the images of objects by the solar microscope, and the latter gentleman succeeded completely, as far as his instrument allowed, in obtaining pictures of minute structures. Shortly after the publication of Mr. Talbot’s process, various attempts were made to apply the oxyhy- drogen microscope to the same purpose, as that instrument had superseded the solar. The result however was, that it was aban- doned on account of the great time that was found to be necessary for impressing an image; and after a great variety of trials by Prof. Owen, Dr. Carpenter, Dr. Leeson and others, the use of the instru- ment for this purpose was given up. The discovery of the collodion process, so much more sensitive than that of Mr. Talbot, led the author to think, as soon as he be- came acquainted with it, that we were in possession of the means of impressing microscopic objects by means of artificial light without any great trouble. A friend of his had an oxyhydrogen microscope of the common form, and on making a trial with it, he found that by using a very sensitive kind of collodion, he could obtain images by about a minute’s exposure. On examining the instrument, how- ever, he saw that its form must be completely changed, in fact, that an entirely new kind of instrument was required to obtain the best effect. The two points to be regarded as the peculiar principles of this microscope are, lst, that none of the radiant light be lost, or as little as possible; 2ndly, that the magnifying power be obtained by such means as would not place the screen for receiving the image beyond such a distance from the object, that the motions of the 118 instrument could be governed at the same time that the image was closely inspected. The first of these objects is secured by giving a very large angular aperture to the system of lenses used for col- lecting the light, and by using another set of lenses for condensing it again on the object, and so arranging their focal length in pro- portion to the focal length of the object-glass, as to cover the plate to be acted upon, and that space only: the second, by using a sort of eye-piece for enlarging the image formed by the object-glass. The lenses divide themselves into four groups, as represented in the figures, in which the light is supposed to proceed’ from the left hand to the right. The first set for collecting the light is composed of three large lenses, a meniscus, plano-convex and double convex, being a combination of three lenses similar in effect to Herschel’s doublet; the second set for condensing the light on the object is a (@) . (3) A (4) qT Vv similar set of lenses, but of much shorter focal length, and turned the other way; between these two sets is a plano-convex or plano- concave lens placed at its focal length from the convergence of the rays from the first group, so as to make the rays pass to the con- densers in a state of parallelism, and so do away with the necessity of changing the distance between the collectors and condensers for each adjustment of the latter: the third group forms the object- glass, which must be so corrected as to have the rays of the spec- trum between the fixed lines G and H as much as possible brought to a point, as these rays are those that produce the maximum ac- tion on the silver salts used in photography ; this will require the red rays to be left untouched, just in the same way as Fraunhofer left those of the blue end of the spectrum dispersed in correcting an object-glass for light. The fourth group is the common eye-piece left under-corrected. A rather better form for this is a Ramsden’s eye-piece with the first lens partially achromatized, by making it a compound lens with the radius of curvature of the common surface nearly double that of the surface that would render it’ achromatic. 1d This form of eye-piece gives a better correction of the oblique pen- cils than the common negative. The time of exposure to obtain an intense negative six inches diameter, on a collodion plate prepared as below, is about a minute ; a positive is obtained in a fraction of a second. The collodion is formed by dissolving gun-cotton in sulphuric ether, and adding to it a small portion of iodide of silver dissolved in iodide of potassium, and also a very small portion of bromide of iron, or of iodide or bromide of arsenic. The image is developed by protonitrate of iron, or by a solution of pyrogallic acid in acetic acid and water, and fixed by a solution of hyposulphite of soda. By taking out the two first lenses of the collectors, the instrument is adapted for using sunlight. - Note.—At the time that this communication was made to the Society, Prof. Stokes had kindly made known to the author the re- sults of his discoveries with regard to the rendering visible the che- mical spectrum, but as he had not then made them public, the author of this communication could not state the use that Prof. Stokes’s discovery enabled him to make of a screen composed of uranium glass, or of infusion of horse-chestnut bark, for finding the focal di- stance of the chemical image, or of arranging the lenses of the con- denser so as to produce the maximum of chemical action. Also, since the communication was made, it has been found that the instrument described gives light enough to impress an image on any of the ordinary papers or Daguerreotype plates in periods ran- ging between one and five minutes, with the oxyhydrogen and lime light; and with direct sunshine the impression is almost instantane- ous; of course sunlight is much better than any artificial light when it can be procured, both as regards speed and the clearness of the picture produced. May 10, 1852. Professor Miller gave an account of a new method of adjusting -the Knife-edges of a Balance. Also ofa method of determining the height of clouds by night. May 24, 1852. Professor Stokes gave a Lecture on the Internal and Epipolic Dispersion of Light. November 8, 1852. Mr. Adams, F.R.S. &c., gave an account of some Trigonometrical Operations to ascertain the difference of geographical position he- 120 tween the Observatory of St. John’s College and the Cambridge ‘ Observatory. The observations, especially those of eclipses and occultations, which were made during many years by the late Mr. Catton at the Observatory of St. John’s College, and which have recently been reduced under the superintendence of the Astronomer Royal, render it a matter of some importance to determine the exact geographical position of that Observatory. The simplest and most accurate means of doing this appeared to be, to connect it trigonometrically with the Cambridge Observatory. For this purpose, a base was measured along the ridge of the roof of King’s College Chapel, by means of two deal rods terminated by brass studs, the exact lengths of which were determined by comparison with a standard belonging to Professor Miller. The extremities of the base were then connected by a tri- angle, with a station on the roof of the Observatory at St. John’s, from which, as well as from the two former points, a signal post on the roof of the Cambridge Observatory could be seen. The angles at the extremities of the base, combined with the corresponding ones at the station at St. John’s, furnished two determinations of the distance of the Cambridge Observatory, which served to check one another. The meridian line of the transit instrument at St. John’s passes through King’s College Chapel, so that by observing the point at which it intersected the base, the azimuths of the sides of the triangles could be immediately found. The result thus obtained is, that the transit instrument of the. Cambridge Observatory is 2313 feet to the north, and 4770 feet to the west of that at St. John’s College. Hence it follows that the difference of latitude is 22!'-8, and the difference of longitude 5!°10 ; and the latitude of the Cambridge Observatory being 52° 12! 51/8, and its longitude 23-54 east of Greenwich, we have finally for the geographical coordinates of the Observatory of St. John’s College, Latitude. . 52° 12! 29-0 Longitude 0° 0! 28!'64 E. of Greenwich. These operations, of course, furnish incidentally, a very exact determination of the orientation of King’s College Chapel. The line. of the ridge of the roof points 6° 20'3 to the north of east. November 22, 1852. Professor Challis made a communication on the recent return of Biela’s Double Comet. December 6, 1852. Professor Stokes gave an account of M. de Sénarmont’s Researches 2 was to the Doubly-refracting Properties of Isomorphous Sub- stances. 121 February 7, 1853. An addition was read to a paper by Professor De Morgan on the Symbols of Logic, the Theory of Syllogism, &c. A paper was read by Mr. Denison on some Recent Improvements in Clock Escapements. The object of this paper was to explain the construction of a new remontoire or gravity escapement invented by the author, which has now been in action for some time on the pendulum of the great clock for the houses of parliament, and is in course of application to others, both turret clocks and astronomical. But by way of introduction to this, which may be called the three- legged gravity escapement (from the form of the scape-wheel), Mr. Denison gave a description of another, which would similarly be called the three-legged dead escapement, and had been previously in- vented by him for the purpose of giving the impulse to the pendu- lum with far less friction than usual. He found that it required only 3th of the force which a common dead escapement had required to make the pendulum swing the same arc. And therefore, as com- pared with a gravity escapement in which there is no sensible fric- tion on the pendulum, there must be still more than #ths of the force in a common dead escapement wasted, in first producing fric- tion on the pendulum, and then overcoming it by an increased im- pulse. The time of the pendulum would be much more disturbed than it is by the inevitable variations of this large amount of friction, as well as that of the clock train, but for a fortunate tendency of the different errors, which are caused by these variations of force and friction, to correct each other. But the amount of this self-correction is uncertain, and some- times one set of errors preponderates and sometimes the other; and so a dead escapement clock sometimes gains and sometimes loses simultaneously with either an increase or a decrease of the are of vibration. And, consequently, none of the contrivances for iso- chronizing a pendulum for different arcs can secure isochronism of the clock; and no further material improvement in clocks can be expected, but from the solution of what has long been known as the great problem of clock-making, viz. the invention of a simple escape- ment which will give a constant impulse to the pendulum without any sensible friction. Mr. Denison showed that his new gravity escapement satisfies all the requisite conditions, mechanical, mathematical, and ceconomical. Its principal features are, that the scape-wheel has only three pins, not far from the centre, which lift the pallets or gravity-arms, and three long teeth which are locked by stops on the arms. The velo- city of the scape-wheel, which usually produces tripping, if the force of the train is increased beyond what is just enough to lift the arms, is moderated by a fan-fly set on the axis of the scape-wheel. The arms are necessarily longer in this than in any other gravity escape- ment, and this also gives a greater depth of locking within a given angle, and therefore a still further security against tripping. And if an arm is by accident lifted a little too high, the tooth does not 122 escape, and the arm falls down again to its proper height until the pendulum carries it off, the pressure of the long teeth on the stops not being enough to hold it up. For these reasons also there is no difficulty in satisfying the mathematical condition investigated by Mr. Denison in a paper read before the Society in 1848, viz. that y (the angle at which the pendulum leaves one arm and takes up the other) should => or at any rate not be less than > (a being the extreme arc of vibration). ‘The escapement requires no oil in the parts affecting the pendulum ; and it contains ne delicate work, and is very easy to make; and as a highly finished train will be no longer necessary, astronomical clocks may be made on this plan much cheaper, as well as better, than heretofore. In turret clocks an escapement of this kind supersedes the necessity for a remontoire in the train to equalize the force on the scape-wheel, and also of long and heavy pendulums, which are expen- sive when compensated, and are sometimes difficult to fix. It will also allow cast-iron wheels to be used throughout the clock (which Mr. Dent has now used for several years in connexion with Mr. Denison’s spring remontoire for the train), as the frietion of the train can no longer affect the pendulum. February 21, 1853, Professor Challis gave a Lecture on Halos, Parhelia, and Para- selene. March 7, 1853. Professor Stokes gave an account of some further researehes relating to the Change of Refrangibility of Light. April 11, 1853. The Rev. Mr. Pritchard, F.R.S., gave an account of the Processes requisite to render Quicksilver tremorless for Astronomical Obser- vation. The great improvements recently introduced, and especially by the present Astronomer Royal, in the construction and methods of using astronomical instruments, require a far more extended use of reflexion from mercury than heretofore. Unfortunately, however, both the convenience and the accuracy of these methods have been greatly limited and impaired by the tremors to which mereury is liable. Many attempts have been made both in France and in Ger- many to remove or obviate these tremors, but hitherto by no means with perfect success. The Rev. C. Pritchard, of Clapham near London, has proposed a method which appears fully adequate to the requirements of astronomy. It consists in the adoption of a silver- plated or amalgamated copper vessel of a peculiar form, admitting the use of a very thin stratum of mercury without the necessity of an inconvenient amount of shallowness in the vessel itself. Mercury, 123 however, placed in an amalgamated vessel after a short time becomes covered with a singular film of amalgam, which impairs the reflect- ing power of the surface, and if at all agitated, soon entirely destroys it. And this is the case even when the vessel is made of amalga- mated platina. The most important, and by far the most difficult part of Mr. Pritchard’s experiments, consisted in the invention of a method by which these films can be easily and practically removed. The details, many of which are curious and interesting, would here | occupy too much space, but they are fully explained in a memoir recently read to the Royal Astronomical Society of London; and it may be added, that the process has been adopted at the Royal Obser- vatory at Greenwich, and is now in progress of trial at the Observa- tories of Paris and Cambridge. April 25, 1853. Professor Challis gave a lecture on the Adjustments of a Transit Instrument. A paper was read by Professor De Morgan on the Principle of Mean Values, and an addition to a paper on the Symbols of Logic, &c. in vol. ix. part 1 of the Society’s Transactions. Though the heading of this paper describes one of its main results, yet it might with equal propriety have been styled a discussion of some points of algebra, with reference to the distinction of form and matter. This distinction, it is contended, is more extensively applied in algebra than in logic, though more recognized in logic than in algebra. Looking at the disputed points which exist in the higher parts of mathematics, and feeling satisfied that they will never be settled until the separation of form and matter is both visible and complete, the author makes a first attempt towards the examination of the question how far this yet remains to be done. A number of comparisons are made between algebraical and logical process, in the course of the inquiry, illustrative of the opinion entertained by Mr. De Morgan, that logic, as treated, requires the interposition of the algebraist, and cannot, except by aid of algebraical habits, be rendered a complete exposition of the forms of thought. In digress- ive notes, he combats the opinion that a generalization of the quan- tity is, as asserted, a new material introduction. He argues against the too mathematical tendency of some of the logicians who have endea- voured to extend the ancient system, especially the attempt of some to make the logical import of the proposition nothing but a comparison of more and less, and an equation or non-equation of quantities. He points out that the proposition has been formalised in nothing but its terms, subject and predicate; and gives an instance of the method in which a failure of general maxims is answered by the sole asser- tion that the mode of expression which brings about the failure is useless. He refers to what were called sophisms, contrasting the neglect of them by the logicians with the use which the algebraists have made of their corresponding difficulties, as in the case of nega- tive and imaginary quantities, the fraction $,&c. He argues against the assertion of more than one eminent writer on logic, that the - 124 identity of two terms, X and Y, expressed as in ‘‘all X is all Y,” is not a complex proposition—is not the union of Every Y is X with Every X is Y. In an appendix to a former paper on the symbols of logic, he refers to a complaint of misrepresentation made by Sir W. Hamilton of Edinburgh, to whom certain technical phrases had been attributed. Mr. De Morgan makes the requisite correction, affirms that he had good reason for attributing such phraseology, and points out what that reason was: he then proceeds to answer two new charges of plagiarism against himself, from the same quarter; giving as his reason for addressing such answer to the Society, that Sir W. Hamilton makes the appearance of the asserted plagiarisms in the Transactions his principal ground of notice. Finally, as to the logical part of the communication, Mr. De Morgan, reverting to his complex syllogism, in which each premise and the conclusion contain two ordinary propositions, generalizes the premises into the numerical form, and, giving terms and quantities algebraical designations, points out the mode of producing all pos- sible inference. The immediate occasion of this introduction is as follows :—Sir W. Hamilton, in a recent publication, one tract of which is directed against Mr. De Morgan’s last paper on syllogism, affirms that a proposition, as to its logical force, is merely an equa- tion or non-equation of quantities, from which the declaration of coalescence or non-coalescence of terms into one notion is a conse- quent. Mr. De Morgan maintains the converse; namely, that the proposition is a declaration of coalescence or non-coalescence, of which the equation or non-equation of quantities is an essential. In treating the complex syllogism, under definitely numerical quan- tities, he has to search for the properties of the equation of coales- cence, as distinguished from the equation of quantity; and, having made the former the means of arriving at inference, he invites those who can to try if the same result can be produced by means of the latter alone. To pass to the algebraical part of the paper. It is first contended that the states infinity and zero, whether represented by distinctive symbols attached to 0 and o, or by negative and positive powers of de, must be formally distinguished, as being each, not a value, but a staius, containing an infinite number of corrational values, just as happens in finite quantity. In order to lay down the formal laws of connexion of these different states, it is necessary to examine the formal use of the symbol =. After pointing out instances in which the laws of algebra are by many declared invalid, as by those alge- braists who admit and interpret 22=2, but cannot give permission to divide both sides by 2, the following laws are suggested. The symbol = is to be read with an index, asin =,, which has reference to the order o, or O_,, or as in =_,, which has reference to On or to On. The equation A=,B is normally satisfied when A and B are of the order x, and A—B of a lower order. It is supernormally satisfied if A and B be both of any (the same or different) higher order than the nth, and subnormally if both be of any lower order. Among the most conspicuous rules which follow, are that AC=4,BD is normally satisfied, if A=,,B and C=,,D are so; and that when an 125 equation is multiplied or divided by a quantity of the order x, the index of equality must be increased or diminished by 2. Various cases are given in which such results as now present anomalies are reduced under formal law, and others which would be absolutely rejected are shown to be capable of consistent interpretation. The formal law of connexion of the different states, of which fini- tude (with the index 0) is only one, is that the order 0,, stands to finite quantity in all respects as finite quantity to om. Hence, so far as 1 and 1+ 0 are simultaneous as well as equal, so fur w and +a are simultaneous as well as equal. And if ¢(1)=o9(1 +9) be a uni- versal law, so must be ¢(©)=,)9(«o +a). Further, o—o must be, formally speaking, wholly indeterminate, even when it is a case of z—z. In relation to such indeterminate forms as o—o, 2, &e., Mr. De Morgan contends that their formal and @ priori character is that of indeterminateness ; and that the choice between determinate and in- determinate character, which so often occurs, is dictated by the matter of the problem, the determinate value being dictated by the laws of algebra. The index of equality, for instance, may be the means of decision: an example is given in which one equation belongs to two different problems, but with different indices of equality ; in one $ is determinate, in the other wholly indeterminate. In assigning © or 0 as values, it is often necessary to assign rela- tions of order. When a quantity passes from positive to negative, or the converse, through 0 or © , it passes through every order of 0 or ©; and this even when the passage is from one phase of 0 or © to another, of different signs. Thus, the orders being powers, # cannot pass from —a.0”™ to +a.0”, without passing through even 0%. Mr. De Morgan insists upon one of twothings: either, the aban- donment of the separate use of 0 and o, except only in the retention of the former symbol to represent A—A;; or, the introduction of dif- ferent orders, and the free use of the comparisons of those orders. For himself, he prefers and adopts the latter alternative. The principle of limits is considered as a formal law of algebra, but not to the exclusion of every other result. If a constant, for instance, have the value A up to #=a exclusive, it has A for one value when x=a. If the constant be transitive, that is, if it be always =B after r=a, then r=a gives both A and B for the con- stant, and, as a fact hitherto observed, its value from calculation is 3(A+B). This observed fact Mr. De Morgan believes he connects with the principle of limits, making it a necessary consequence of the universal truth of that principle ;. and hence he holds that it may be stated as a theorem, under the name of the principle of mean values. Various uses of this principle are given. Further, in assu- ming the free use of the orders of 0 and o, it is shown that it is cor- rect to say that the constant passes from A to B while A, x being a+h, passes through the phases of 0. So that, for instance, at an epoch of transitiveness the value of ¢(a+0) is dependent upon the form of 0. The brevity of an abstract prevents-the statement of those cautions under which such use of language is intreduced. One 126 result, however, may be brought forward. When the functidn fx is transitive (or, as commonly said, discontinuous) at «=a, the equa- tion ¢(fa)=(@f )a no longer necessarily exists. But this, as is pointed out, is what may happen at any value of « which makes a differential coefficient infinite. On the question of sin 2 and cos, Mr. De Morgan deduces their observed values, sin o=0 and cos #=0, both from the prin- ciple of mean values, and from the formal truth of the equation o(% +a)= fe. From the same principles follows the equation (—1)* =0.. In this case, however, and in all which come under the principle of mean values, the absolute necessity of the results is not affirmed. ‘They are the alternatives of indeterminateness. But in thus representing them, Mr. De Morgan does not concede more than he conceives must be conceded with respect to 2 —o, %, and the like. On the question of series, Mr. De Morgan contends that all the uncertainty and danger of divergent series belongs equally to con- vergent series, in every case in which the envelopment is unknown. On this part of the subject he adds to the arguments of a former paper, and insists upon the superior safety of the alternating series, in which the terms are alternately positive and negative. Without going further into details, the purport of this paper may be stated as follows. Algebra, using the term in the widest sense, ought to be, and is approaching towards, a science of investigation, and a symbolic art of expression, of which the laws are strictly and without exception incapable of failure, suspension, or modification. The formal laws under which such a result is to be obtained, though laid down in the first instance by extensive induction, of Which many steps are accompanied by difference of opinion, will at last be received and admitted as parts of the definition of the science, @ priori. The existing defect of the science is an imperfect formalization, arising from the want of views of sufficient extent, and leading to material distinctions, that is, to exceptions dictated, @ posteriori, by the re- sults of particular cases. Such exceptions have in many instances been brought within rule by further consideration ; and it is con- ceived that the same thing will happen at last in all cases. The paper is an attempt to examine the principal outstanding difficulties (those connected with the definition of integration excepted) with reference to the question how far they may arise from imperfect conception of formal laws. That there is to be a formal science, is positively assumed, and made the basis of the attempt: how far any suggestion contained in the paper is a valid step towards it, is treated with doubt and left to opinion. May 9, 1853. Mr. Hopkins, F.R.S. &c., the President of the Society, gave an account of some experiments for the determination of the tempera- ture of fusion of different substances under great pressure; and on the application of the results to ascertain the state of the interior of the earth. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. November 14, 1853. A paper was read by Mr. Dobson on the Theory of Cyclones. See Philosophical Magazine, vol. vi. p. 438. Also, on the Storm-tracks of the South Pacific Ocean. See Phi- losophical Magazine, vol. vii. p. 268. A communication was made by Mr. C. C. Babington on the use that has been made of the mode of growth to distinguish nearly allied Species. November 28, 1853. A paper was read by Mr. Wedgwood on the Geometry of the first three books of Euclid, synthetically demonstrated from premises consisting exclusively of definitions. In a treatise* published by the author a few years ago, definitions founded on relations of direction were indicated as exhibiting the ultimate analysis of the conceptions of straightness and parallelism in lines, and of planeness in surface; and in proof of the adequacy of these definitions as the basis of a complete system of geometry without the aid of axioms or any other assumption whatever, they were employed in demonstrating the principal propositions necessary to place the student on the ground occupied by the definitions and axioms of the ordinary system. If the basis thus built in underneath the old foundations of the science had been complete in every nook and corner, nothing more would have been required in order to rest the entire demonstration on the single principle of definitions. So long, however, as any step in the process, however subordinate, was left to be supplied by others, there always would be room for sus- picion that the assumption in reasoning which was speciously plas- tered over in one place might be secretly undermining the system in another. The reform, moreover, of the premises in geometry is a problem on which such an infinity of thought has been spent, and * The Principles of Geometrical Demonstration deduced from the ori- ginal conception of Space and Form. Taylor and Walton. 1844. No. X.—ProcrgepinGés or THE CAMBRIDGE Putt. Soc. 128 to which so many answers, more or less plausible, have been offered, that nothing short of a complete exposition of a consistent scheme of demonstration can be expected to-carry conviction in the validity of a fresh solution. The object of the present paper is accordingly to complete the task undertaken in the foregoing publication by a formal statement of the other definitions required in connexion with those of straight and parallel lines and plane surface, and by a rigid demonstration from these premises of the steps intervening between those and the premises of the ordinary system; and in additional proof of the fundamental character of the proposed analysis, the de- monstration is carried on through the geometry of the three first books of Euclid by direct reasoning, without resort to the compara- tively unsatisfactory method of er absurdo proof, which, although equally conclusive as to the necessity of the result, yet always leaves a hankering in the mind for an answer why the case must be as the demonstration shows that it cannot avoid being. In the execution of the foregoing plan, the whole of the problems of Euclid are omitted as irrelevant to the demonstration of the other propositions. The grounds on which they were adopted in the system of Euclid appear to be these. It frequently happens that it is necessary in the course of demonstration to make some new con- struction not included in the figure which forms the original subject of the proposition, and it was evidently thought that the geometer would not in strictness be entitled to take such a step until he had demonstrated the means of executing it with exactitude. The stu- dent was accordingly in the postulates put in possession of a ruler and a pair of compasses; and wherever any additional construction was required in the proof of a proposition, a problem was premised, showing the means by which the construction might be made by the aid of those implements. But it should be recollected that the figure by which the demon- stration is commonly accompanied is not the actual subject of the reasoning, but a mere illustration to aid the imagination and the memory, the exactitude of which is matter of comparative indiffer- ence. Moreover, the principle on which the problems are introduced is not consistently carried out to its legitimate conclusion even in Euclid. There is no difference in the reasoning between the figure which forms the original subject of the proposition, and the addi- tional construction which is made in the course of demonstration ; and therefore if it were necessary for the validity of the conclusion to demonstrate the means of executing the latter figure, it would be equally necessary in the case of the former. The student would not be entitled to move a step in the demonstration of the equality of two triangles having two sides and the included angle equal, until he had been taught how to construct two such triangles, and con- sequently how to describe an angle equal to a given angle. The demonstration in Euclid begins with perfect legitimacy. ‘ Let ABC, DEF be two triangles in such and such conditions,” without the necessity of indicating the means by which those conditions may be mechanically executed, or indeed of their possibility of actual exist- 129 ence; and it may with equal legitimacy proceed to exemplify in like manner any further construction which may be found necessary in the course of demonstration. The question of motion has commonly been considered so essen- tially distinct from that of position, that all reference to the former subject has rigorously been excluded from the field of geometrical inquiry. But the position of every point must ultimately be deter- mined by motion from points antecedently known, and to the inci- dents of motion we should accordingly look for the original source of the relations of position. Now motion (in as far as it influences position) admits of variation in two ways; viz. in the direction of the motion at each indivisible instant of time, and in the length of the track accomplished in a finite period; whence it has been said by Sir John Herschel that space (which is primarily known as the receptacle of motion) is reducible in ultimate analysis to distance and direction. The relations of extent ¢ are simply thoi of equal, greater, and less, with respect to which it will be necessary only to define the test by which they are respectively to be demonstrated in concrete figure. The relations of direction are of a much more complicated nature. The different phases of this elementary attribute of motion are di- stinguished, not, like those of colour, by a permanent character inde- _ pendently cognizable in each individual, but more like musical notes, by their relative position on a peculiar scale which may be made to rest on any individual as an arbitrary basis. The scale by which directions are compared is founded on the elementary rélations of opposition and transverseness. In whatever direction we suppose ourselves to be traversing space, we recognize the possibility of returning to the same position from whence we set out by motion in a different direction, the relation of which to the original is that of opposition ; or the two may be classed together as the positive and negative modifications of a common direction. Again, if we fix our thoughts upon any given direction, we find a series of others in each of which it is possible to traverse space with- out advance in the original direction or in the one opposed to it. The directions so marked out by negation of progress in a certain direction are said to be transverse to the normal or direction in which no progress is made by the observer while advancing in the direction of any of the transverse series. If now we start afresh from any of the individuals of the latter series, it will be found that the series includes the opposite direction, as well as one direction and its opposite transverse to the former two. Every other individual of . the series will be recognized as partaking in different proportions of the nature of these coordinates, or transverse directions, adopted as the basis of the scale. In other words, it will be found that distance in any intermediate direction is essentially composed of distance in the direction of each of the coordinates in different proportions, vary- ing from all of the one and none of the other, to all of the latter and none of the former, with every modification arising from taking each of the coordinates in both a positive and a negative sense. B 130 In like manner, as each intermediate direction is transverse to the original normal, a secondary series of directions with a differ- ent normal will arise from the combination of these coordinates in every proportion, and the whole expanse of space around the observer will be recognized as consisting of distance in every pos- sible combination of proportions in the direction of three coordinates, of which the first may be taken at pleasure in space, the second may be identified with any of the series transverse to the first coordinate, and the third will be the single direction transverse to each of the former two. Within the sphere of three directions so related to each other we are entirely shut in. Whatever may be the particular direction in which the coordinates be laid, we can conceive no fourth direction essentially differing in nature from the former three, and therefore can conceive no possible direction which cannot be derived from some combination of three coordinates, or in which a given distance cannot be resolved into equivalent distances in the direction of the three coordinates. : {| We have thus in the relations of transverseness and opposition, and in the conception of intermediate directions arising from the combination of transverse coordinates in different proportions, a uniform scale by which, when applied to known directions in space, the position of any other direction may be accurately defined inde- pendent (it must be observed) of any reference to the notion of angular magnitude, of which as yet no mention has been made. When two directions only are known in a system, they must be considered as members of the series transverse to a common normal ; and one of the two being identified with the first coordinate of the scale, the position of the second will be completely determined by the proportion in which it partakes of the nature of the second co- ordinate or transverse direction of the series. . The directions commonly adopted as the basis of the scale,are the up and down, fore and aft, and right and left lines marked out (in any given position of the observer in a system) by the constitution of his bodily frame; and thus (in any given position of our bodies) a particular direction is defined in our thoughts by the proportion in which it partakes of the nature of those coordinates, that is to say, by the proportion in which distance. in the direction in question is essentially composed of distance up or down, distance to the front or rear, and of distance to the right or left. For the sake of simplifying the question, we will now confine our thoughts to motion in a plane surface, or to directions having refer- ence to two transverse coordinates. Now although, in the actual apprehension of a figured system, the observer must be supposed to traverse the entire outline, and thus continually to change his place, yet he must be capable of doing so without rotation on his own axis, as he would otherwise acquire no notion of the configuration of his track in the external system. He will accordingly carry with him throughout the fundamental conceptions of front and back, right and left, and by reference to these coordinates will be able to compare and to identify directions in any part of the system. 131 It is in virtue of this complex scheme of relation between direc- tions, that we are enabled to conceive the possibility of reaching the same point by different tracks from a common starting-point. We are indeed so much in the habit of thinking of points as marked out by physical phenomena (as by the letters in a geometrical illustra- tion), that it is by no means obvious where the difficulty of the con- ception lies. But it must be remembered that points in geometry are distinguished solely by position, while the position of a given point is determined by the nature of the track by which it is reached from a point antecedently known. It is plain, therefore, that there would be no means of identifying points attained by tracks differing in any respect from each other, if the precise combination of distance and direction’ by which they were respectively attained were the ultimate test of their position. But now the knowledge of the fun- damental scheme of relationship above explained makes us regard the space traversed in each successive instant of time in the track by which the position of a point is determined (and consequently the whole space traversed in the entire track), as equivalent toa . certain distance in the direction of each of the two coordinates of the scale. The aggregate character (in respect of distance and direction) of the space traversed in different tracks (by which the position of the terminal points is governed) will thus be made to depend on the aggregate distance advanced in the direction of the two coordinates, a question to be tried by simple superposition. When the distance advanced in the direction of each coordinate is the same, the positions finally attained will be recognized as iden- tical, and the points will coincide whatever may be the amount of intermediate divergence in the tracks by which they have actually been reached. From the same principle it may be shown, that a straight line may be drawn from a given point to any other point in space. Because the space traversed in the track by which the second point must be supposed to have been determined, will be equivalent in distance and direction to a certain distance in each of the two standard directions of the system. Now inasmuch as the series of directions intermediate between any pair of transverse directions includes individuals partaking in every conceivable proportion of the nature of both the transverse directions between which they lie, it will always be possible to select one of the series a certain distance in which will be equivalent to given distances in each of the two transverse directions, and there- fore the distances in the direction of the coordinates of the system under consideration, into which the space traversed in the original track has been resolved, may again be exchanged for an equivalent distance in a single direction duly related to each of the coordinates ; in other words, the same position may be attained by motion in a single continuous direction as by a track of any other description, or what amounts to the same thing, a straight line may be drawn from a given point to a point determined by a track of any other de- scription. As soon as a straight line is known ‘as lying in a single continuous B2 132 direction, it becomes the most obvious means of marking the direc- tion so exhibited throughout a finite extent of line. The series of directions transverse to a given normal may then be represented by two straight lines crossing each other at right angles, and an inde- finite number of other straight lines diverging from the point of intersection, and dividing the plane surface round that point into as many parts as there are diverging lines. If now we take two of these lines, like the hands of a clock, and suppose one to remain fixed while the other revolves from left to right, it will pass success- ively through all the directions intermediate between left and front, while the quantity of plane surface intercepted between the hands abutting on the point of intersection will continually increase as the difference in their direction becomes greater, or in proportion as distance in the direction of the moveable hand contains a greater proportion of distance in the direction transverse to that of the fixed one. Thus we are taught a new mode of estimating the relation between the direction of straight lines diverging from a common point ; not by a proportion which addresses itself to the understand- ing merely, but by a quantity admitting of measurement by bodily . comparison, viz. by the quantity of plane surface intercepted between the diverging lines and abutting on the point of intersection, or by the magnitude of the included angle. Professor Challis gave an account of a luminous appearance ob- served at the time of the perihelion passage of Klinkerfue’s comet. Professor Stokes read a paper on the Optical properties of Light reflected from Crystals of Permanganate of Potash. The substance of this paper is embodied in a paper on the Metallic Reflexion exhi- bited by certain Non-metallic Bodies, published in the Philosophical Magazine, vol. vi. p. 393. December 12, 1853.. Professor Fisher read the first part of a paper, entitled ‘‘ Researches, ak and Pathological, on the Development of the Vertebral ystem.” After having explained what he meant by the term vertebral system, he stated (and he illustrated what he described by drawings) that the spinal marrow, at a particular stage of growth of the human embryon, exhibits indications of segmental development correspond- ing to that of the spinal column; that is to say, that each of its halves offers on its external surface a series of symmetrical spaces defined by transverse lines, each of which spaces corresponds to the roots of a single spinal nerve; and again, that each half presents in its internal structure, a double series, one anterior, the other poste- rior, of symmetrical areas, two of which appeared to equal in extent one of the external spaces just spoken of. Professor Fisher also 133 Stated that the spinal marrow offers, at the period of development in question, several other peculiarities, some of them bearing likewise a segmental character; but he reserved a detailed description of them for a future communication. February 27, 1854. A paper was read by Professor Challis, entitled ‘‘ A direct Method of obtaining by Analysis the mean motions of the apse and node of the Moon’s Orbit.” See Philosophical Magazine, vol. vii. p. 278. Also a paper by Mr. J. B. Phear on some parts of the Geology of Suffolk, particularly with reference to the Valley of the Gipping. The deposits which constitute what is often termed the glacial formation, but which the present state of our knowledge hardly allows us to designate by a name significant of a common origin, present.so much confusion to the inquirer, and impose upon him so much laborious research by the extent and the unconnected character of their distribution, that they have hitherto met with less attention than their importance deserves. The county of Suffolk seems to be a district where a portion of these deposits is manifested with more than usual distinctness, and is capable of being studied with comparative facility. ‘The county is separated from Norfolk on the north by the well-marked valleys of the Ouse and the Waveney, is bounded on the east and south by the sea and valley of the Stour, and is bordered by chalk uplands on the north-west; the whole central portion is thickly covered with a mass of blue drift-clay, cut into abrupt undulations by a network of val- leys. This clay is totally without any symptom of stratification, and is full of fragments of all rocks of the secondary period, including specimens of granite and other igneous rocks. Wells sunk in different parts of the county show this drift-clay to have a thickness varying from 200 feet to a few inches; it seems to thin off from the northern and western parts of the county towards the coast, and only exists in the shape of outliers beyond 4 line passing through Sudbury, Hadleigh, Bramford, Woodbridge, and Saxmundham ; a line, it may be remarked, nearly coinciding with the edge both of the London clay and of the crag, and approximately passing through the heads of the tidal estuaries of the Orwell, Deben, Ore and Alde. The clay is almost universally underlaid by an un- fossiliferous sand; and there is reason to conjecture that this sand, of a prevailing red colour, passes out beyond the just-mentioned line, and covers in many places the surface of the strip of land between it and the sea. A detailed examination of the Gipping valley reveals a well- marked and connected line of sand cliffs fringing it, and its Codden- ham tributary in particular, at a high level on both sides; the sand is generally pure white, though often red, horizontally stratified and 134 capped with an unrolled gravel, which evidently owed its existence to the quiet washing away of the drift-clay from its insoluble con- tents. Above Needham Market the valley is channelled in drift- clay, but between Needham and Bramford it is cut through chalk ; and it should be remarked, that the line of sand-hills does not extend up the valley with any great distinctness beyond the chalk. The phenomena seen at Creeting are not consistent with this sand lying beneath the drift-clay ; and the inference is, that it constitutes the remains of an estuary deposit formed in the valley subsequent to its excavation in the drift-clay. f All the other streams west of the Gipping have chalk for their floor during the middle part of their course, thus manifesting the existence of a ridge of chalk running beneath the drift accumu- lations nearly due west and east from Sudbury to Bramford. Dis- turbances evidenced in this ridge, and perhaps due to its elevation, are partaken of by the London clay and crag deposits which overlie it on the east and south. In Norfolk the drift-clay attains a greater thickness than in Suf- folk, and towards the north of the county is overlaid by a sand and gravel formation which may be appropriately termed upper drift. The gradual disappearance of this towards the south, together with the thinning away and final extinction of the drift-clay in the same direction, point to a region of greater denuding activity ; it may be an interesting question whether such denudation be in any degree connected with the upheaval of the before-mentioned chalk ridge, or again, whether the sands of the Gipping valley bear any relation to the upper drift of Norfolk. March 13, 1854. A paper was read by Prof. Challis on the Eccentricity of the Moon’s Orbit; supplement to a former communication on the mean motions of the Apse and Node. See the former paper, Phil. Mag. vol. vii. p. 278. Also a paper by Mr. J. Clerk Maxwell on the Transformation of Surfaces by Bending. : The kind of transformation here considered is that in which a surface changes its form without extension or contraction of any of its parts. Such a process may be called bending or development. The most obvious case is that in which the surface is originally a plane, and becomes, by bending, one of the class called ‘‘ developable surfaces.” Surfaces generated by straight lines, which do not ulti- mately intersect, may also be bent about these straight lines as axes. In this way they may be transformed into surfaces whose generating lines are parallel to a given plane, just as the former class are trans- formed into planes. In both these cases, the bending round one straight line of the 135 system is quite independent of that round any other; but in those which follow, the bending at one point influences that at every other point. The case of a surface of revolution bent symmetrically with — to the axis is taken as an example. e remainder of the paper contains an elementary investigation of the conditions of bending of a surface of any form. _ The surface is considered as the limit of the inscribed polyhedron when the number of the sides is increased and their size diminished indefinitely. A method is then given by which a polyhedron with triangular facets may be inscribed in any surface; and it is shown, that when a certain condition is fulfilled, the triangles unite in pairs so as to form a polyhedron with quadrilateral facets. The edges of this polyhedron form two intersecting systems of polygons, which become in the limit curves of double curvature; and when the condition referred to is satisfied, the two systems of curves are said to be “conjugate ’”’ to one another. . ~The solid angle formed by four facets which meet in a point is then considered, and in this way a “‘ measure of curvature” of the surface at that point is obtained. It is then shown that if there be two surfaces, one of which has been developed.from the other, one, and only one, pair of systems of corresponding lines can be- drawn on the two surfaces so as to be conjugate to each other on both surfaces. This pair of systems completely determines the nature of the transformation, and is called a double system of “lines of bending.” By means of these lines the most general cases are reduced to that of the quadrilateral poly- hedron. ‘The condition to be fulfilled at every point of the surface during bending is deduced from the consideration of one solid angle of the polyhedron. It is found that the product of the principal radii of curvature is constant. ; By considering the angles of the four edges which meet in a point, we obtain certain conditions, which must be satisfied by the lines of bending in order that any bending may be possible. If one of these conditions be satisfied, an infinitesimal amount of bending may take place, after which the system of lines must be altered that the bend- ing may continue. Such lines of bending are in continual motion over the surface during bending, and may be called ‘‘ instantaneous lines of bending.”” When a second condition is satisfied, a finite amount of bending may take place about the same system of lines. Such a system may be called a ‘‘ permanent system of lines of bending.” Every conception required by the problem is thus rendered per- fectly definite and intelligible, and the difficulties of further investi- gation are entirely analytical. No attempt has been made to over- come these, as the elementary considerations previously employed would soon become too complicated to be of any use. For the analytical treatment of the subject the reader is referred to the following memoirs :— 1. “Disquisitiones generales circa superficies curvas,” by M. C. ae ae F. Gauss (1827).--Comm. Recentiores Gott. vol. vi.; and in Monge’s « Application de l’Analyse a la Géométrie,” edit. 1850. . 2, Sur un Théoréme de M. Gauss, &c.,” par J. Liouville.— Liouville’s Journal, 1847. . 3. ‘‘ Démonstration d’un Théoréme de M. Gauss,” par M. J. Ber- trand.—Liouville’s Journal, 1848. 4, «Démonstration d’un Théoréme,”’ Note de M. Diguet.—Liou- ville’s Journal, 1848. 5. “ Sur le méme Théoréme,” par M. Puiseux.—Liouville’s Jour- nal, 1848. . And two notes appended by M. Liouville to his edition of Monge. March 28, 1854. Prof. Miller gave an account of the relation between the physical characters and form of crystals of the oblique system as established by the observations of Mitscherlich, Neumann, De Senarmont, Wiedemann and Angstrom. A paper was read by Prof. De Morgan on some Points in the theory of differential equations. 1. The words primordinal, biordinal, &c. are used in abbreviation of the phrases ‘ of the first order,’ ‘ of the second order,’ &c. The symbol for a differential coefficient, U, for = , &c., long used by the author, is thus extended. By Uz p,q is meant dU: dr with reference to z as contained in p and q, as well as explicitly. Thus Usip,q means U,+U,p,+U,¢,; and Usjy means U,+U,y/. Differentiations are sometimes expressed thus: d,U=U, dz, dy, YU=U,de+U,dy. When it is only requisite to express functional relation, without specification of form, (#, y,z)=0 or z=(a,y) may signify an equa- tion between z, y, and z. A letter may be used as its own functional symbol: thus u=wa(2, y, z) may signify that w is a function of 2, y, z. And in ‘ for u write u(z,y, z)’ there is a convenient abbreviation of ‘for u substitute its value in terms of 2, y, z.’ 2. When, as so often happens, a variable enters under relations which destroy the effect of its variation upon the form of differential coefficients, it is called self-compensating. Thus ¢(a, y, a) =0, $,(@; y,@)=0, contain the self-compensating variable a. Similarly, when 9(2,y, a,6)=0 is accompanied by ¢,da+¢,db=0, a and 6 are mutually compensative, and primordinally. The addition of Pacely) da + bowl y) d6=0 makes a and b biordinally compensative. 3. When a finite change in 2 makes an infinite change in y, it makes an infinite change in y!:y, in y”:y', &c. When either or 137 both P and Q become infinite, P: Q-and P,: Q, are both nothing, both finite and equal, or both infinite; provided that the infinite form is produced by substitution for x. If u=(v,w, ...), any rela- tion which makes u, infinite either makes u,, infinite, or is indepen- dent of w. And if u,=« be produced by a relation containing »v, then u,dv+u,dw+ ...=0 and u,~dv+u,~dv+ ...=0 are relations of - identical meaning. 4. From the last it follows that U=const. is solved by making any factor of dU either O orm. In dU=M(Pdzr+ Qdy), singular solutions are obtained, as is known, from M=o: it ought to be asked whether M=0 does not give singular exceptions, that is, cases in which U=const. arises otherwise than from P+Qy'=0. It is found more convenient to treat these cases without actual separation of the factor; that is, from d U=U,dr+U,dy. 5. In a former paper, the author insisted on the arbitrary func- tions which enter the intermediate primitives: maintaining, for ex- ample, that the primordinal of y”=0 is ¢(y', zy'—y)=0, for any form of g. Lagrange, he has since found, notices this extension, and rejects it, because it leads to y’=a, xy'—y=6, as necessary consequences of its ordinary solution. Mr. De Morgan maintains his opinion, and observes that Lagrange’s reason would make it imperative to reject one of the two, y/=a, zy!—y=8, since either is the necessary consequence of the other. 6. In order to avoid the ambiguous use of the word singular, a singular solution is defined as any one which, by the mode of obtain- ing it, cannot have the ordinal number of constants: it is further styled intraneous or extraneous, according as it is or is not a case of the general solution. If y=y(2, a) or a=A(z, y) give y'= (2, y), then dA=A,(y'— x)dx and y=—A,: Ay are identical equations. Every relation which satisfies Ay=« is a solution, and a singular solution ; except possibly, relations of the form z= const., which must always be examined apart. Also, Ay=o is identical with wa=0. There can exist no solutions whatsoever except those which are contained in A=const., Ay=o, and (possibly) e=const. Again, x,=(logy,),- Of this equation the author has found neither notice nor use: supposing it to have ever been given, he holds it most remarkable that it has not become common as the mode of connecting the two well-known and widely used tests of singular solution. It easily shows that y,=© contains all extraneous solu- tions, and all intraneous solutions which (as often happens) can be also obtained by making a a function of x. It also easily gives a conclusion arrived at by the author in his last paper, namely, that when x,=© is satisfied and not y'=x, it follows that y,+ ,x is infinite. 7. The author gives his own version of the demonstration of a theorem of M. Cauchy, for distinguishing extraneous and intraneous solutions. If y=P, P being a given function of 2, satisfy y'= (2, y), that is, if P’ and x(a, P) be identical, then y=P is an extraneous or 138 intraneous solution of y'= (x, y), according as i ae dy Pp x(%,¥)—x(#, P) (x being constant) is finite or infinite for small values of 8. This theorem has attracted little notice in this country: the author believes it to be fully demonstrated, and considers it one of the most remarkable accessions of this century to the theory of differential equations. | 8. It is observed that the validity of the extraneous solution may depend upon the interpretation of the sign of equality by which A=B is held satisfied when both sides are O, or both infinite, even though A: B=1 is not satisfied. Thus y'=2/y or y=(x+a)*, has the extraneous solution y=0, which, however, is not a solution if by y'=2 »/y we understand in all cases y!: /y=2. 9. The common mode of obtaining the singular solution of a bior- dinal (by combining ¢(2, y, a, b)=0, da, 06=0, da, o62;y=0) though sufficiently general, is never shown to be so. Let y=7(2, a, 6), combined with y'=ya,, give a=A(a, y, y'), 6=B(a, y,y'), from either of which follows y= y(z#,y,y'). The most general primordinal is f(A, B)=0, f being arbitrary. Any given curve, y=m@z, may be made to solve this for some form of f; but, generally speaking, this solution will be extraneous. For A and B are so related that every intraneous solution makes A and B con- stant. And any primordinal equation whatever may in an infinite number of ways be thrown into the form f(A, B)=0, so that the intraneous solutions shall make A and B constant. (Given y=@za, required a key to all the primordinals of which it is a singular solution. Take any equation y=w(z, a, b), eliminate x between a=A(a, wx, a'x) and b=B(a, wa, w'x), and write A(z, y, y’) and B(a, y, y') for a and 6 in the result.) The equations d(A=A,(y’—x)dz, dB=B, (y”—x)d@ are identi- cally true. And Ay=o, or any relation which satisfies it, is a singular primordinal of y”=y, whenever it is a primordinal at all ; that is, when y' appears in it. When A,/=oo is satisfied by a rela- tion void of y’, that relation is not necessarily a solution. The ordinary solutions of Ay»=o are solutions of y!/= ; but not (neces- sarily) the singular solutions. The singular solutions of a relation which makes Ay =o may make A,» finite. Comparing A and B with w, we have ss Ws hee Wa 7 Chapengramtin. «Wire So Maney trail cf aVee—WPibar —* Waie— Debus Xy = {log(pabou— Pear) } From these are obtained results in complete analogy with those for primordinal equations. But when WaWse—wWitae=0, the usual criterion of singular solution, is made valid by ya=0, W,=0, a sin- gular primitive of the singular primordinal may be obtained, which does not necessarily satisfy y”=y. 139 10. Similar forms are given for triordinal equations. In noticing - the manner in which the equations of the general theory may be easily expressed by what are called determinants, Mr. De Morgan expresses his admiration of the system, and his sense of the important services rendered by those who have laid its foundations. But he refuses to employ the word determinant in the sense proposed, on account of its not expressing any distinctive property of these func- ‘tions. Until those who have a better right to give a name provide themselves with a distinctive one, he intends to call them eliminants. The forms connected with y=W(«, a,b) may be easily translated into others derived from ¢(2,y,4,6)=0. But the formula which connects x,’ with ¢ is as follows :— aT te | \ 4 $abiy—Prbay | Xs { ( ? zy Pahdaly— doPariys ‘ where by U,, is meant U,+ U,y’, even when U is a function of y’. Thus (zy'—y)ay , as here used, is Q. 11. The following idea of reciprocal polarity has been presented by M. Druckenmiiller (as cited from Crelle’s Journal by Mr. Boole), and, independently, by Professor Boole: it occurred to the author of this paper before he had seen the researches of either. If there be equations involving m+ variables, and if, determining a point by fixing m of the variables, a curve be defermined by giving all possible values to the remaining (point and curve being here merely names of objects determined), we may say that the (m)-point is the pole of the (n)-curve. Similarly, we may make each (z)-point the pole of an-(m)-curve. And all the points of any curve have polar curves which contain the pole of that curve. If the two sets of variables be severally made primordinally compensative, the general properties which arise are easy extensions of the well-known theory of reci- procal polars. Let (x,y) and (a,b) be two points: the polar pro- perty of 2?+y*=az‘+ by contains the direct and converse property of the angle in a semicircle. If 9(2, y, a,b) be the modular equation, and if x, y and a,b be compensative, any element (2, y,y') of any (z, y)-curve to the pole (a, 6) determines an element (a, 4, 6’) of an (a, b)-curve to the pole (z,y). These curves are reciprocal polars. In the common system, the modular equation is linear with respect to both pairs of coordinates, and the locus of those poles which lie in their polar straight lines is a conic section, to which the polars are tangents. 12. The method of transforming differential equations, given by the author in his last paper, is precisely the reference of the curves sought to their reciprocal polars, the modular equation being taken at pleasure. Mr. De Morgan now proposes to call it the method of polar transformation. Let $(2,y, a, b)=0 be the modular equation, and let ¢,+9,y'=0, 9,+¢,6'=0, 0’ being db: da. Hence a=A(2,y, y"), B=B(2,y.y'); 2=X(a, 0,0"), y=V(a, 0,0!) b'=By +Ay y' =Yy +Xy; the biordinal factors, y’—y(2,y, y'), 6’—a(a, 6, b'), disappearing 140 from 6! and y'. Hence 6! depends on x,y, y’. Similarly, 6” depends on a,y,y',y", &c., and similarly for y',y’,&c. If inf(w,y, y',y”,&c.)=0 we substitute for 2, y, y', &c. in terms of a, b, 6’, &c., the two equa- tions belong to polar reciprocals. If either can be integrated, the integration of the other depends on elimination: thus if the equa- tion in a, b, &c. can be integrated, the solution of the equation in x, y is obtained by eliminating a and b between the integral obtained and r=X, y=Y. i 13. There are two reciprocal biordinal equations belonging to the modular equation (a, y, a, b)=0; y”=y when a and 0 are constant, b’=a when « and y are constant# The two have the same condition _of singular solution; for Ayg,=Xyy. Let this be o(2, y, a, b)=0, when cleared of y' or b'. The following table exhibits the relations of the double system :— p(a,y,a,b)=0 . ris, cnnaned ii =0 3 : pat Po ee, o(2,y,0,)=0 y'=a(2y) o=A(ay,y') b=Beayy’) e=X(a,b,b!) y=V(d,0') b'=X(a,b) y=T(2, C) y"=x(2,y,y') b!'=a(a,b, b=A(a,Z). Eliminate a and b between ¢=0, o=0, oa, =0, and we have '=q@, y=, the singular primordinal and primitive of y’= ; those. of b’=a are obtained by éliminating x and y from ¢=0, o=0, $x\y =O. There is a relation involved between C and Z, the con- stants of integration. For each value of C, y=II is the zy-curve which touches all in ¢(a,y, a, A)=0, for the corresponding value of Z and all values of a. Thesame of Z, b-=A, and ¢(a, Il, a, 6)=0. The contacts are of the second order, and y=II, b=A, are polar reciprocals for corresponding values of C and Z. But the singular primitives of y'=@ and b/=) are not necessarily reciprocals: when this does happen, their contacts with primitives are of the third order. 14. When a surface is described by one set of curves, as in the surface obtained by eliminating a from ¢(a,y,2z,a)=0, V(a,y,z,a)=0, it is proposed to call it a shaded surface, and the curves lines of sha- ding. ‘The equation f(a, y,z, y',z')=0, y and z being functions of x, cannot, generally, belong to any family of surfaces in an unre- stricted sense; that is, it cannot be always true of a point moving in any way upon a surface. Such a supposition would be equivalent to imagining a surface every point of which has the primordinal] character of the vertex of a cone. But it may belong to any surface, properly shaded, or to any mode of shading, if the proper surface be chosen. 15. Two equations of the form y= (a, a, b,c) z= (a, a, 6, c), give one, and only one, primordinal of the form f(a, y, z,y', z')=0. Assume any surface w(2, y,z)=0; by this, and compensative rela- tions between a, 6, c, another pair of primitives may befound. But ‘the primitives obtained from w=0 do not shade this surface, except in cases determined by two relations between the constants. Again, making a, 6, c compensative, without any assumed surface, we find 141 one equation of the form (a, 8, c, a', b!)=0, any primitives of which lead to other primitive forms for f=0. Each of the second primi- ' tives has contact of the first order with one family of curves from among the original primitives ; and all ordinary primitives are found, in an infinite number of ways, among the connecting curves of others. There is a singular solution, a curve of contact to all primitives, when ©, =0, ¥,=0, &c. can all be satisfied at once. Since y=®, z=¥, give a primordinal equation independent of constants, the polar reciprocal properties of curves in space are of a restricted form. Every surface dictates another surface, and a mode of shading both, so that each line of shading on either surface is the polar reciprocal of a line on the other. 16. The conversion of constants into compensative variables may give restricted solutions, as in the ordinary case of two variables, and every other in which the constants are converted into separately self-compensating variables. When these variables are made collect- ively compensating, and the equations permit elimination of the original variables, ordinary differential equations may be produced, the integration of which may, after substitution, give primitives of the same form as those from which they came. But when the ori- ginal variables cannot be eliminated, arbitrary relations may be required, in number enough to eliminate the differentials of the new variables: in this case arbitrary functions enter the primitives finally deduced, Of this last case one instance is Lagrange’s transition from a primitive of a primordinal partial equation having two con- stants to the complete primitive of that equation. 17. A biordinal partial equation may be produced from U(a, y, z, a, b, c, e, h)=0 by eliminating the five constants between U=0 and the five results of primordinal and biordinal differentiation. But it is not true that every form of U=0 leads to one biordinal equation only: many forms lead to an infinite number. Two attempts to procure other primitives by making a, b, &c. compensative variables, end in two different forms of result. First, when all the resulting equations are required to be integrable, by introduction of a proper factor, the success of the process requires the integral of two partial equations, one primordinal and one biordinal, between four variables. Secondly, when no such condition is required, the result is another form in- volving five constants. 18. A primordinal partial equation belongs to a family of surfaces of which one is determined by any given curve through which it is to be drawn. A biordinal equation belongs to an infinite number of families ; and a distinct conception of the conditions which select an individual surface is best formed by an extension of the following kind. A curve on a surface is analogous to a point on a curve: two curves being drawn on a surface, the analogue of the chord joining two points on a curve is the developable surface (or surfaces) drawn through the two curves. The developable surface which touches the given surface in a curve (and not the tangent plane) is the ana- 142 logue of the line which touches a curve in a point. A biordinal equation being given, one surface satisfying it is selected by a curve through which that surface is to pass, and a developable ‘surface passing through that curve which the surface is to touch. 19, 20. The restrictions under which two arbitrary forms must enter, in order that a biordinal partial equation may exist indepen- dent of these functions, are wholly unknown. The case which is fully analogous to a biordinal of two variables, is of the most limited character. Ampére has noticed this: Mr. De Morgan was led to it by an examination of the polar properties of 9(a, y, z, a, 6, c)=0. This equation leads to a=A, 5=B, c=C, where A, B, C are func- tions of x, y, z, p,q. The primordinal f(A, B, C)=0 is satisfied by @=0, subject to ¢(a, 6, c)=0, and leads to a biordinal, independent of f, of the form Q+Rr+Ss+Tt+ U(s?—rt)=0, in which Q, R, &c. are not wholly independent of each other. If the pole (a, 6, c) move along a certain curve, the polar surface must touch a certain surface in one of the lines of a certain shading. That is, every abc-curve has a shaded surface, which is its polar reciprocal: and every line of shading of that surface has another surface for its polar reciprocal, shaded by lines of which the original abc-curve is one. And every surface has a reciprocal surface such that for each point on one there is a point on the other; and the point on one surface being taken, the polar surface of that point touches the other surface in the other point. The singular solutions of the two biordinals derived from o(x, y, 2, a, b, c)=0 by means of 2, y, z and of a, b, c, are connected by relations analo- gous to those already seen in the case of two variables. In fact, there is perfect coincidence and coextension between the properties of the general equation y”= (a, y, y') and a particular species of the equation Q+Rr+Ss+Tt+U(s?—rt)=0. It is proposed to call this species the polar biordinal. 21. The general method of transforming partial equations, given in the last paper, is the investigation of the class of surfaces con- tained under a given equation by reference of them to their polar reciprocals, any convenient modular equation ¢(#, y, z, a, 6, c) being made the means of transformation. ; 22. The following notation is proposed for eliminants. The com- ponents being A,, A,, &c., B,, &c., the eliminants are (A,), (AB, 9)» (ABCpqr), &c. ; the components being A, A’, &c. B, &c., the elimi- nants are (A°), (AB°’), (ABC°'”), &c. Thus (Ap=Ay (ABpq)=Ap (By )—By (Ay) (ABCpqr) = Ap (BC gr) + By (CAgr) + Cp (ABgr) (ABCD pqrs)=Ap(BCD gr) —Bp(CDAgrs) +C,(DABgrs) —Dp(ABCyrs), andso on. Some slight investigation of properties is made, to ex- hibit the notation. 143 The following rule is suggested to determine, in any complicated case, whether the number of contiguous interchanges by which one arrangement of letters is converted into another shall be odd or even. This is an important matter in the theory of eliminants, though very complicated instances may seldom occur in practice. Write down one arrangement under the other, and, beginning at one letter in one line, mark the companion letter in the other line, pass on to that companion in the first line, mark its companion, and so on, until we arrive at a letter already marked. Call this sequence a chain, each mark being one dink. Having formed one chain, begin at a letter not yet used, and form another; and so on until every letter has been used. Then, according as the number of chains with even links is odd or even, the number of interchanges of contiguous letters required is odd or even. For example, the two arrangements being ABCDEFGHIJKLMNOPQ HMOGQBKLJPFCINADE 1-21°2°322122212 4128. Under A is H, under H is L, under L is C, under C is O, under O is A, already taken: the first chain has five links, the second is found to have nine, the third two, the fourth one. The number having even links is one, an odd number; hence an odd number of contiguous interchanges converts the first arrangement into the second. 23. The following is the method of ascertaining whether the bior- dinal equation Q+Rr+S8s+Ti+U(st—rt)=0 . . . . (1) possesses a primordinal of the form /(2, y, z, p, g)=0, containing an arbitrary function. Considering 2, y, z, p, g as five independent variables, integrate, by common methods, the equations dv dv dv k dv U(EtE)t Sie a dv dv dv 1 dv os peasph parceled any | 8 bo atte) * dg 1+k dp k being one of the roots of AS°=(1+4)*(RT+QU). Ifa common solution v=A can be found, then A=const. is a primordinal of (1). If two common solutions, A and B, can be found, then B=@A is a primordinal, w being arbitrary. But though in this case A=const. and B=const. are solutions, they cannot coezist, unless the values of k-be equal, or unless S*=4(RT+QU). This last equation is one condition of polarity; and if, when satisfied, we find three (and there cannot he more) common solutions, A, B, C, inexpressible in terms of each other, then f(A, B, C)=0 is the most general primordinal, any two forms of it may coexist, or even any three, which amount to A=const., B=const., C=const. Elimination of p and g between these last equations gives ¢(2, y, z, a, 6, c)=0, the modular equa- 144 tion. And the general solution of (1) is found by assuming 6 and c in terms of a, and then making a a self-compensating variable. 24. The paper is concluded by some remarks on notation. In an appendix to the preceding paper, read to the Society on the Ist of May, 1854, Mr. De Morgan points out an error committed by M. Cauchy in a very remarkable theorem, of which his enuncia- tion is as follows. Let gx be a function which can be developed in integer powers of x. Let r(cos0+sin@./—1), 7 being positive, be any one of the roots of ¢r=o or of g'v=a. Then the development of oz is convergent from 2=0 up to x = the least value of r. M. Cauchy stipulates that the function shall be continuous ; but he defines a function to be continuous so long as it remains finite, and receives only infinitely small increments from infinitely small accessions to the variable. It is then obviously impossible that the above theorem should be universally true. Were it so, it would follow that the development of (1 +2)? is convergent for all finite values of x, whereas it is well known that this development becomes divergent when z is greater than unity. The error of M. Cauchy’s demonstration (which contains a valuable method for establishing a large class of definite integrals) i is the assumption that if an infinite number of- convergent series of the form a+br+cx?+..., all with one value of x but different values of a, 6, c,..., be added together, the sum divided by the number of series is also a convergent series. This assumption is not universally true. Mr. De Morgan takes a totally different line of dementia and establishes the following theorems. If r(cos@+sin@. /—1), r being positive, represent a root of any one of the equations gr=0, glva=a, g"v=a . then the development of gz in powers of z is always convergent from 2=0 up to z= the least value of 7, and divergent for all greater values of 2. If the development have all its coefficients positive, or if all beyond an assignable coefficient be positive, the least value of 7 is obtained from a real and positive root. If the signs of the development be, or finally become, recurring cycles, with / in each cycle, the least value of 7 is obtained from a root in which cos #+sin@. “—1 is one of the /th roots of unity. If no such cycle be finally established, cos 04+sin 0. /—1 may have a value of 6 which is incommensurable with the right angle. M. Cauchy has established from his own theorem (the want of sufficient statement of conditions not affecting this particular case) the necessity of the observed fact, that the developments produced by Lagrange’s theorem for the development of implied functions always give, when convergent, the least of the real values which are implied. 145 May 1, 1854. A paper was read by Professor De Morgan on the Convergency of Maclaurin’s Series, being an Appendix to a paper on some Points in the theory of differential equations. See the abstract of the former paper, Phil. Mag. vol. vii. p. 450. Mr. Kingsley made an oral communication on the Chemical Nature of Photographic Processes. - May 15, 1854. A paper was read by Mr. Warburton on Self-repeating Series. In computing Bernoulli numbers by the formula of Laplace*, the author of this paper was led to notice, that in the fraction whose development isa series of the form gt gt pg Lae —&c., the numerator of that fraction is a recurrent function of ¢. This led him to investigate the question, what are the conditions which the denominator of the generating fraction, and the terms of the series generated, must satisfy, in order that the numerator of such a frac- tion may be a recurrent function of ¢. The paper contains the result of that investigation. The author calls those series “ self-repeating,” which, when ex- tended without limit in opposite directions, admit of separation into two similar arms, each arm beginning with a finite term of the same magnitude. Between this pair of finite terms, either no zero-term, or one or more zero-terms, may intervene. One arm repeats, and contains arranged in reverse order, the terms of the other arm, either all, or none, of the terms having their signschanged. ‘The different positive integer powers of the natural numbers, of the odd numbers, and of the figurate numbers of the several orders, present familiar examples of self-repeating recurring series. The author demonstrates the following three theorems respecting self-repeating recurring series :— I. If the series arising from the development of a proper fraction is the right arm of a self-repeating recurring series, and if the deno- minator of such a fraction is a recurrent function of ¢, then the nu- merator also is a recurrent function of ¢. II. Other things remaining the same, if the numerator of the fraction is a recurrent function of ¢, then the denominator also is a recurrent function of ¢. III. If the numerator and the denominator ofa proper fraction are each a recurrent function of ¢, then the series, arising from the deve- lopment of the fraction according to the positive integer powers of f, will be the right arm of a self-repeating recurring series. By way of example, the author applies his first theorem to the summation of the infinite series 19—2°+3°—&c., and compares his process with the corresponding processes of Laplace and of Sir John Herschel. The sum in question is given by Sir John Herschel (see Jameson’s Journal, January 1820) in terms of the differences of the powers of 0, extending from A'0° to A90°, In the author’s process, * See Memoirs of the Academy of Sciences, 1777. 146 the requisite differences extend from A'0° only to A*0°, and the nu- merical coefficients of these are of diminished magnitude, and of very easy determination. The author makes other applications of his theorems; but on these we forbear to enter. A paper was read by Professor Challis on the Determination of the Longitude of the Cambridge Observatory by Galvanic Signals, The experiment of which this paper contains the details, was made at the suggestion of the Astronomer Royal, and conducted according to a scheme arranged by him for giving and receiving the signals. A galvanic connexion having been established between the Greenwich Observatory and the Cambridge Telegraph Office, by means of the Londan central station of the Electric Telegraph Company, signals were sent on the nights of May 17 and 18, 1853, between 115 and 125 mean time. The signals were made by causing two needles, one at Greenwich, the other at Cam- bridge, to start by completing the galvanic circuit at either place of observation. The times of starting were noted at both places, and reduced to the sidereal times of the respective observatories, to serve by comparison for determinations of the difference of their longi- tudes. On each night the signals were made alternately for a quarter of an hour at one station, in batches containing an arbitrary number of signals not exceeding nine, and then for a quarter of an hour at the other station in a similar manner. On the first night the total number of signals was 151, and on the second night 139. The two observers, Mr. Dunkin of the Greenwich Observatory, and Mr. Todd of the Cambridge Observatory, changed places in the interval between the two nights’ observations; Mr. Todd observing at Greenwich, and Mr. Dunkin at Cambridge, cn the second night. Also it was arranged that the two observers should observe identical stars on the two nights, as well as the stars ordinarily used for clock errors, and that the same apparent right ascensions of the stars should be employed for reducing the signal-times at both observatories. ‘The Cambridge Observatory time was conveyed with the greatest care to the Tele- graph Office at the Cambridge Railway Station by the transfer of three chronometers. By a first calculation, the longitude of the Cambridge Observatory was found to be 23%:03 east of Greenwich. Professor Challis subsequently made another calculation, taking into account the effect on the times of meridian transits of stars produced by the forms of the transit-pivots, according to a method which he has described in the Memoirs of the Royal Astronomical Society (vol. xix. p. 103). The errors arising from the deviation of the pivots from the cylindrical form being eliminated, the longitude is found to be 22*-70 east of Greenwich, which is less by 05-84 than the value hitherto adopted. May 29, 1854. A paper was read by Professor Fisher, entitled « Additional Ob- servations on the Development of the Vertebral System.”’ PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. ——_—_—_—____ November 13, 1854. A paper, by R. L. Ellis, Esq., was read, entitled ‘‘ Remarks on the Fundamental Principle of the Theory of Probabilities.” Also, ‘‘ On the Purbeck Strata of Dorsetshire.” By the Rev. O. Fisher. The object of this paper was to describe the beds from which a series of insect remains and other fossils had been collected by the author, and presented to the Woodwardian Museum. The connexion of the Purbeck beds with the Oolitic rather than _ with the Wealden series was maintained, while both were shown to be unconformable in this district to the cretaceous system. Reasons were given for thinking that the materials, of which both the Wealden and Purbeck were composed, had travelled from west to east; and the beds of the New Red Sandstone, as they occur in Devonshire, were pointed out as affording a mass of strata which would furnish a detritus of the character of a large portion of the Hastings sands of Hampshire and Dorsetshire. Indescribing the Purbeck beds,the author followed the system of the late Professor E. Forbes, dividing them into upper, middle, and lower; and entered into some detail of the alternations of salt and freshwater conditions that prevailed during their deposition. The aspects under which the same beds appear at different points of the district under examination were particularized, and it was attempted to be shown that these were in conformity with the theory of a current setting from the west towards the east. The mode of occurrence of the remains of insects in the middle and lower Purbecks was somewhat minutely described, and it was suggested that some interesting chronological speculations might be grounded upon it. The paper concluded with an attempt to explain the singular frac- tured condition of about thirty feet of the lower Purbeck strata throughout the eastern part of the county. It was supposed that this might have been caused by the deposition of sediment upon the remains of the Portland forest before the mass of the trees had been removed by decomposition; the sediment, after it had become con- solidated, settling unequally as the carbonaceous matter was gra- dually removed. No. XI.—Proceepines or THE CAMBRIDGE Puit. Soc. 148 2 November 27, 1854. Prof. Willis gave an account of a new form of Atwood’s Machine. December 11, 1854. A communication was made by Dr. Paget on a case of involuntary tendency to fall forwards, February 19, 1855. Mr. Hopkins gave a lecture on certain changes of Terrestrial Temperature, and the causes to which they may be attributed. March. 5, 1855. Dr. Clark gave an account of some recent discoveries respecting the origin, migrations, and metamorphoses of Entozoa, and their bearing on the notion of spontaneous generation, April 23, 1855. A paper was read by the Master of Trinity, on Plato’s Survey of the Sciences, contained in the seventh book of the Republic. Plato, like Francis Bacon, took a review of the sciences of his time ; and like him, complained how little attention was given to the phi- losophy which they involved. The sciences which Plato enumerates are arithmetic and plane geometry, treated as callections of abstract and permanent truths; solid geometry, which he ‘notes as deficient”’ in his time, although, in fact, he and his school were in possession of the doctrine of the ‘‘ five regular solids ;” astronomy, in which he demands a science which should be elevated above the mere knows ledge of phenomena. The visible appearances of the heavens only suggest the problems with which true astronomy deals; as beautiful geometrical diagrams do not prove, but only suggest geometrical propositions. Finally, Plato notices the subject of harmonics, in which he requires a science which shall deal with truths more exact than the ear can establish, as in astronomy he requires truths more exact than the eye can assure us of. It was remarked also, that such requirements had led to the progress of science in general, and to such inquiries and discoveries as those of Kepler in particular. 149 May 7, 1855. The Master of Trinity read a paper on Plato’s notion of Dialectic. _ At the end of the survey of the sciences contained in the seventh book of the Republic, which was the subject of a paper at the last meeting, Plato speaks of Dialectic as a still higher element of a philosophical education, fitted to lead men to the knowledge of real existences and of the Supreme Good. Here he describes Dia- lectic by its objects and purpose. In other places Dialectic is spoken of as a method or process of analysis; as in the ‘ Phedrus,’ where Socrates describes a good dialectician as one who can divide a subject according to its natural members, and not miss the joint, like a bad carver. Another Dialogue, in which there are examples given of dividing a subject, is the Sophistes, where many examples of dichoto- mous or bifurcate division are given. But this appears from the Dialogue to have been a practice of the Eleatic rather than of the Platonic school. Aristotle proposed a division of subjects according to his ten Categories, which he and others since have extensively used. Xenophon says that Socrates derived Dialectic from a term implying to divide a subject into parts, which Mr. Grote thinks unsatisfactory as an etymology, but which has indicated a practical connexion in the Socratic school. The result seems to be, that Plato did not establish any method of analysis of a subject as his Dialectic ; but he conceived that the analytical habits formed by the comprehensive study of the exact sciences, and sharpened by the practice of dia- logue, would lead his students to the knowledge of first principles. Also, Mr. Maxwell gave an account of some experiments on the mixture of colours. May 21, 1855. A paper was read by Mr. Hopkins on the External Temperature of the Earth and the other Planets of the Solar System. We have not sufficient data to determine the superficial tempera- ture of any planet besides our own. We know, however, that it must mainly depend on the temperature of the planetary space, and on the heat which the nearer planets at least receive directly from the sun, but modified, and possibly in a far greater degree than has been generally supposed, by the particular circumstances by which each planet may be characterized. The modifying circum- stances more particularly referred to in this paper, are the existence of atmospheres surrounding the planets, the positions of their axes of rotation, and the conductivity and specific heat of the substances forming the outer crust of each planetary body of our system. No astronomer, judging from the appearances which Mars and Jupiter present to us, would entertain any serious doubt as to the existence of atmospheres surrounding those planets, and the probability would seem to be almost equally strong of Saturn being likewise enveloped 150 in a similar manner. ‘The obliquity of the axis of rotation is known with considerable accuracy in the cases of Mars and Jupiter; and also in that of Saturn, if it coincide with the axis of rotation of his ring. Venus presents great difficulties to the observer, but it ap- pears now to be pretty satisfactorily determined, that the period of rotation about her own axis is nearly the same as that of the Earth, and that the obliquity of her axis is large, amounting to as much as about 75°. This must produce an extraordinary difference between the changes of aznual temperature in that planet and those which we experience. The author has endeavoured, in this paper, to estimate numerically the effect of this anomalous obliquity. Practical astro- nomers have entertained the opinion that Venus likewise has an atmosphere. Of Mercury we know too little by direct observation to form any opinion on those points founded on observed facts, and the same remark will apply to the remoter planets beyond Saturn ; but most astronomers probably feel much the same conviction that Mercury, Uranus, and Neptune have atmospheres of greater or less extent, as that they revolve round their own axes with greater or less angular velocity. It is not the author’s object, however, to adjust the balance of pro- babilities for particular hypotheses in favour of planetary atmospheres or against them; but assuming their existence, to estimate their effects on the planetary temperatures. And in like manner he points out the influence which must be exercised by a greater or less conductivity, and specific heat in the superficial matter of a planet, without professing to discuss the probability of such properties being materially different in the different planets. The Earth’s atmosphere is known to be almost completely diathermanous for heat radiating directly from the Sun ; and it is assumed to be equally so for the heat which proceeds directly from the fixed stars, and to which the general temperature of space is due. This radiating heat therefore has little or no effect in heat- ing the atmosphere during its transmission to the Earth’s surface ; but after falling upon and heating terrestrial objects, it loses the power of radiating completely through the atmosphere, and is trans- mitted back into space through the atmosphere by conduction, con- vection, and partial radiation to limited distances. But for any of these modes of transmission, it is essential that the temperature of the atmosphere should be greater in its lower than in its upper por- tions, and in a degree greater as the quantity of heat to be trans- mitted is greater. The temperature (r,) of the upper portion must be determined by the condition, that, in a given time, a quantity of heat must radiate from it into surrounding space equal to that which falls upon it from external sources in the same time, and is trans- mitted back after reaching the surface of the earth or objects near » to it. Consequently r, must be independent of the height of the Earth’s atmosphere. At lower points the temperature will increase till we reach the surface of the Earth ; and if we denote the tempe- rature there by 7,, it is manifest that 7, will be greater, the greater the height of the Earth’s atmosphere. It must here be particularly observed, that 7, is the proper tem- 151 perature of the component particles of the atmosphere, and is pro- bably widely different from the temperature which would be indi- cated by a thermometer placed at the upper extremity of the atmo- sphere, since the instrument would not only be affected by the ex- change of heat between its bulb and the atmospheric particles, but also by the heat radiating upon its bulb from every source of heat in surrounding space; while the atmosphere, on account of its diather- mancy, would remain unaffected by this radiating heat. Conceive now a thermometer to be placed at a point sufficiently above the earth’s atmosphere. If the bulb were sheltered from the direct influence of the solar rays, the thermometer would indicate the temperature of that point of space, independent of the effect of radiation from the central luminary of the solar system, but depen- dent on the radiation from all other sources of heat in the universe. -If the instrument thus sheltered were sufficiently remote from the sun and every planet, it would indicate very nearly the same tem- perature at every point within the solar system, assuming the absence of all unknown centres of heat within that system or near to it. This is what may be understood by the general temperature of pla- netary space. Let it be denoted by T. We shall then have T greater than 7,; and therefore if we now conceive the thermometer to be transported to the upper limit of the atmosphere, it will be affected by the lower temperature there, and will indicate a temperature ih- termediate to T and r,. If the instrument be brought still lower within the atmosphere, it .will indicate a still lower temperature, from its being entirely surrounded by a portion of the atmosphere more dense than that at the extreme boundary, till this tendency to lower the indications of the thermometer is counteracted by the greater temperature of the atmospheric particles as we descend towards the Earth’s surface. At some point, consequently, within the Earth’s atmosphere the indication of the thermometer would attain its minimum ; after which, in descending continuously towards the Earth, the temperature indicated would constantly increase, omitting variations due to temporary or local causes. Thus it fol- lows that the ¢xistence of an atmosphere like that of the Earth, en- veloping a planet, may, according to its extent, either elevate the superficial temperature of the planet above, or depress it below that of surrounding space independently of the direct solar radiation. With respect to our own globe, we are entirely ignorant of the height to which the thermometer, in ascending, would continue to indicate a decreasing temperature, but we are sure that such height is great. This is important with reference to the ultimate object of this paper ; for if the height of a planet’s atmosphere were too small to allow a thermometer descending in it to attain its minimum indication, it is manifest that an increase of atmosphere would cause a decrease in the planet’s superficial temperature ; whereas if the height of the atmo- sphere were great enough to allow the thermometer to attain the mini- mum,any increase of atmosphere would necessarily cause an increase in the superficial temperature of the planet. Inthe Earth’s atmosphere, we are sure (as just remarked) that the indications of a thermometer 152 would constantly increase in its descent from a very high point above the Earth’s surface; and therefore it follows, that if a planet be en- veloped in an atmosphere similar to that of the Earth, but of greater height, the superficial temperature of that planet will be higher than that of the Earth, supposing both to exist in the planetary space un- affected by the heat which radiates from the Sun; while, on the contrary, the superficial temperature of the planet would necessarily be less, under the same conditions, than that of the Earth, if its atmosphere were smaller, unless it should be so small as not to allow a thermometer descending in it to reach its minimum indication. If the planet were entirely without atmosphere, its superficial tempe- rature (in the assumed absence of solar radiation) would be that of surrounding space; but we have no means of determining what rela- tion that temperature bears to existing terrestrial temperature, or to what this latter temperature would become in the absence of solar radiation. | Nu} ; The author has calculated from Poisson’s formule the increase of temperature in the superficiul crust of the Earth, due to the amount of heat received by direct radiation from the Sun, in different lati- tudes, above that temperature which would be common to all parts of the Earth’s surface in the absence of solar radiation, and with a uniformity of intensity of stellar radiation in all directions upon our globe. But-this increased temperature must produce an augmentation of temperature in the atmosphere, which must react on the terrestrial temperature till equilibrium of temperature be established. The author has endeavoured to estimate the amount of this indirect effect of solar radiation by means of the data furnished by M. Dove’s work on terrestrial temperatures, combined with calculations based on Poisson’s formule. He concludes that the whole effect of solar heat at any proposed place is very nearly double that due to the im- mediate and direct effect of solar radiation. Having thus ascertained this entire effect, he finds the temperature which would pervade the whole surface of the earth if the solar heat were extinguished. He estimates this temperature at —39°'5 C. The annual variation of temperature in any latitude is found to be nearly the same in amount for the terrestrial surface and for the part of the atmosphere resting upon it. This must be understood as applying to those places at which the temperature is not materially affected by the horizontal transference of heat by marine or aérial currents, or any local causes, which disturb the dependence of tem- perature on latitude alone. The author also points out the depend- ence of the annual inequalities of the terrestrial temperature (and consequently of those also of the atmosphere) on the conductivity and specific heat of the matter which constitutes the Earth’s crust: If these were much greater, the annual changes of temperature would be much less. Before applying these results to other planets, the author states that he does not admit the notion, that the remoter planets may derive a considerable superficial temperature from the remains‘ of that in- ternal heat which they probably possessed in the earlier stages of 153 their existence. It is a well-established conclusion, that the super- ficial temperature of our own globe has arrived at that point below which it can never descend by more than the small fraction of a degree, so long as all external conditions remain the same as at - present; and the superficial temperature of the remoter planets will in all probability be reduced to the corresponding limit. To these external conditions, therefore, and not to their primitive heat, must the existing temperatures on the surfaces of these planets be attri- buted, assuming always that they are not of less antiquity than our own globe. Hence the superficial temperature of the Earth, with its present atmosphere, placed-at the distance of Neptune, Uranus, or Saturn, would be very nearly —39°°5 C., since the effect of solar radiation at those distances would be nearly insensible. But if the extent of the atmosphere were increased, the superficial temperature would be augmented in a corresponding degree. Judging by the decrements of temperature observed by Mr. Welsh, the author con- cludes that an increase in the height of the Earth’s atmosphere of 35,000 or 40,000 feet, would elevate her superficial temperature, if placed in the remoter planetary regions, to nearly the mean tempe- rature of our present temperate zone. The same conclusion will hold with respect to the three planets above mentioned, if we sup- pose them to have atmospheres similar to that of the Earth, and of suffiqent extent. Their temperatures must be sensibly uniform over the whole of their surfaces, not being subject to any appreciable annual variation. The same conclusions will apply to Jupiter, except that there will be a small augmentation of temperature arising from solar radiation, which the author calculates might amount to about 23° C. at his equator. Hence the author concludes that those views which assign a necessarily low temperature to the above-mentioned planets in con- sequence of their distance from the Sun, are altogether untenable. The conditions under which Mars is placed approximate more nearly to those of the Earth than for any other planet. The author calculates, that with an atmosphere similar to that of the Earth, and exceeding it in height by about 15,000 or 20,000 feet, the equatorial temperature of Mars may be about 60° F., or 15°°5 C., and his polar temperature about —10°C. The extent of the annual variations would be about half those on our own planet in corresponding lati- tudes, supposing the conductivity, specific heat, and radiating power of the matter composing his superficial crust to be the same as for the Earth. Again, if the Earth, with her present atmosphere and obliquity, were placed in the orbit of Venus, the mean equatorial temperature would be upwards of 90° C., subject to the reduction, which would doubtless in this case be great, due to the horizontal transference of heat. The mean polar temperature would be about 16°C. A diminution in the atmosphere would reduce these temperatures in any assigned degree. But the obliquity of Venus, though not satis- factorily determined, is considered to be much greater than that of 154 the Earth, amounting, according to the estimate of some astronomers, to as much as 75°, as heretofore stated. This would of course render the character of her seasons entirely different from those of the Earth. The greatest mean annual temperature would be at the pole. Independently of the horizontal transference of heat by aérial currents or other causes, taking the extreme obliquity of 75°, and sup- posing the atmosphere of Venus to be exactly like that of the Earth, her mean temperature at the equator would be about 56° C., and at the pole 95° C. This latter would probably be much lowered by currents; but if the height of the atmosphere of Venus be less than that of the Earth’s atmosphere by about 25,000 feet, the author con- siders that the mean temperature of Venus in her equatorial regions would not exceed that of the temperate regions of the Earth; while the mean polar temperature would probably be about 40° C., or about 12° or 18°C. higher than the Earth’s equatorial temperature. The heat of sunshine may be moderated by an atmosphere more laden with vapour than that of the Earth. Supposing the atmosphere of Venus like that of the Earth in its nature and its magnitude, the temperature at her poles, with the supposed obliquity, must be subject to an enormous annual inequality, amounting to between 70° and 80° C. above or below the mean tem- perature, liable, however, to a great reduction by horizontal trans- ference of heat. It may also be considerably reduced by the nature of the matter which constitutes her outer crust. A reduction, like- wise, in the extent of her atmosphere, like that above supposed, would probably diminish the amount of this inequality, as well as the mean temperature, though not in the same degree. It is easy to conceive that the coefficient of the inequality may be thus reduced to some 40° C.; and supposing the mean temperature then, as above estimated, at about 40° C., the annual polar temperature will oscil- late between 0° C, and 80° C. At the equator, the semi-annual in- equality might amount, under the above suppositions, to about 10° or 12° C., in which case the equatorial temperature might oscillate between something below zero (C.) and some 25° C. It should be recollected also, that a much greater reduction of the mean tempera- ture would result from a greater reduction in the extent of this planet’s atmosphere than above supposed with reference to the height of our own atmosphere. This would not, the author conceives, be inconsistent with the existence of a large quantity of vapour in the atmosphere, affording shelter from the heat and glare of sunshine. The Moon is under the peculiar circumstances of the absence of a sensible atmosphere, and her long period of rotation about her axis. Assuming her to have no atmosphere at all, the mean tem- perature of her outer crust, in the absence of the Sun, would be the general temperature of that portion of planetary space in which the solar system is situated. How much this might differ from the superficial temperature which the Earth would have with the like absence of the Sun, and which the author estimates at —39°-5 C., as above stated, it is impossible to determine; but whatever it may be, the influence of the Sun’s heat would be to increase it by about 155 40° C. at the Moon’s equator, and by a small amount only at her poles. This must be attended by an enormous monthly inequality, amounting to nearly 60° C., supposing the matter of which her su- perficial crust is composed to have the same conductivity, specific heat, and radiating power as the crust of the Earth. If these be much greater for the Moon, this inequality might be considerably diminished. At the poles it must be comparatively small. The lunar temperatures here spoken of are those of the matter forming her external crust. The temperature which would be indi- cated by a thermometer placed in her immediate vicinity would be affected by the Moon (in the assumed absence of an atmosphere) only by her direct radiation. We have not the means of determining what this temperature may be. Also a paper was read “On the singular Points of Curves.” By Professor De Morgan. Mr. De Morgan defines a curve as the collection of ail points whose co-ordinates satisfy a given equation; and contends for this definition as necessary in geometrical algebra, whatever limitation may be imposed in algebraic geometry. He divides singular points into points of singular position and points of singular curvature; the character of the former depending on the axes, but not that of the latter. Both species are defined as possessing a notable property, and such as no arc of the curve, however small, can have at all its oints. The form first considered is that of which the case usually taken is an algebraic curve. Let ¢(#, y) be a function which for all real and finite values of z and y is real, finite, and univocal; let the curve be ¢(z, y)=0, considered as an individual of the family ¢(z, y) =const. The two curves d¢ : dr=0, dp: dy=0, or ¢2=0, dy=0, are the sub- ordinates of this system, on which the singular points of all depend. When ¢ is not reducible to another function of the same kind by extraction of a root, it divides the plane of co-ordinates into regions in which, severally, it is always positive or always negative. By this consideration it is easily shown (independently of y’, y”, &c.), that if («+dz, y+dy) be a point on the tangent at (7, y), o(a+dz, y+ dy) has the sign of $,,d2*+ 2¢,,dxdy + ¢y,,dy*.. Hence, immediately after leaving the curve, ¢ agrees with or differs from —¢,y” at the point left, according as the curve is left on the convex or the concave side. Hence easily follow the criteria of flexure, and also the following relation between any two points whatsoever of the curve. Let two points be called similar when a line drawn from one to the other cuts the curve an even number of times (0 included) with the same abutments (on convexity or on concavity), or an odd number of times with different abutments. Let other points be called dissimilar. ‘These points are similar or dissimilar, according as their values of ¢, .y” agree or differ in sign. An 4 priori proof is given that multiple points, cusps, and isolated points, must be determined by ¢,=0, ¢,=0, or can only take place when both subordinates meet the curve. It is shown that, in the 156 system ¢(2, y)= const., the cusp of ¢(z, y)=0 must be an evanes- cent loop, and the isolated point an evanescent oval, or bounded portion. Some discussion of the meaning of y'=a+b/—1 at an isolated point is given. There have been two methods of treating the singular points: The first has recourse to the theory of equations, using differentia- tion, if at all, only to supply coefficients. ‘The second attempts canonical forms derived from differential coefficients, and examines, in succession, the meaning and bearing of the successive orders of differential coefficients. Mr: De Morgan affirms that this second method cannot be what it pretends to be; and, by treating it gene- rally, shows that its questions are ultimately dependent upon the theory of equations. An equation of the form ZAy'*=0, when it has no equal roots, decides the character of a singular point defini- tively ; and reduces it to a number of intersecting branches without contact, a number of coinciding isolated points without real tangents, or some of one and some of the other. When the equation has some real roots, each set furnishes either multiple branches with contact, or cusps, or conjugate points with real tangents. All this is easily illustrated by examining the curve in which 9(#, y) is an infinitely small constant, near to the singular point of ¢(v, y)=0. A theorem given by Lagrange, and strongly indicated in the writings of Newton, Taylor, Stirling, Cramer, and John Stewart, but apparently nearly forgotten, solves the question of finding the higher or lower degrees of all the roots of SAy*=0, where A is a function of x of the degree a; that is, where A=2%(a+A’), and A! vanishes when z= o or when #=0. By this theorem (which is also given in the first* Number of the Quarterly Journal of Mathe- matics), y being x”(u+ U), all the values of r, and their corresponding values of uw, are very easily found; and repetition of the process upon a transformed equation gives U=a”(u,+U,), and so on. It obviously follows, that when the origin is removed to any singular point of a curve, the discussion of the branches which pass through that point, and of their contacts with the tangent and each other, is made very easy. In proof of this, the author takes the following instance,— xl? + elt 4 glly— g8y2 4+ Qr7y3 —gty? + y6 — Bays + 21413 =0, and discusses its infinite branches, and the sextuple point at the origin (which turns out to be a couple of isolated points, and a cusp of similar flexures), with very much less space and trouble than ordi- nary methods would demand from a much less complicated instance. It is also shown that the lower form of Lagrange’s theorem solves the following question:—Given an equation with a certain number of equal roots, what effect will be produced upon these roots by given infinitesimal alterations in the coefficients, how many will remain real, and how many will become imaginary ? Newton has given the foundation and the chief step of a geome- * There attributed to Mr. Minding, by a mistake caused by M. Serret, ge = eae it with a theorem of Mr. Minding, without any notice of its author. 157 trical method (Newton’s parallelogram) which has passed into oblivion, though it occurs in the celebrated second letter to Oldenburg, has been fully described by Stirling, used by Taylor and De Gua, and forms the main method of Cramer’s work on curves. Mr. De Mor- gan proposes to call it the method of co-ordinated exponents. He proceeds to describe and enlarge this method ; observing that, of the polygon which represents an equation, Newton and his fol- lowers are in full possession of the connexion of the sides with the solutions, and fail only in not grasping the connexion of the whole polygon with the whole equation. Both Newton’s method and Lagrange’s, the second of which is an arithmetical version of the first, may be applied to irrational equations, but it will be convenient to confine the description to the form Zax"y"=0, where m and 2 are integers. F - In avy”, let n be an abscissa, and m an ordinate, and let (m, n) be called the exponent point of the term az”y”._ Take some paper ruled in squares (or ruled both ways in any manner, for any equal rectangles will do) to facilitate the process when z and m are always integers, and lay down all the exponent points in Zaxz”™y"=0. Through some of these points draw a convex polygon including all the rest, which can only be done in one way. Should the points be so many and so scattered that some method must be applied, the geometrical method is a translation of the main arithmetical method of Lagrange’s theorem. The points which end on, or otherwise fall in, the sides of the polygon show the essential terms of the equa- tion: no others are wanted to determine g and uw in y=a2"(u+U). The upper contour of the polygon shows how all the solutions com- mence in descending powers of x; the under contour does the same for ascending powers. Take any side of either contour, its projec- tion on the axis of nm shows the number of roots it represents, the tangent of the angle it makes with the negative side of the axis of n shows the value of r. It will not be needful to abstract the developments given in the paper: we shall only notice the inverse method. The following example is taken, and the construction of the equation is even easier (under Cramer’s form) than the direct treatment of it. The example chosen by the author is the following :—Required ¢(z, y)=0, of the twelfth dimension in terms of y, such that the twelve roots of y, with reference to lower degrees, shall be as follows: two roots of the degree 1, four of 3, two of 0, one of —1, two of —3, one of —2. But with reference to higher degrees, there are to be one root of the degree 3, two of 4, three of 0, three of —1, two of —1, one of —3. On examination these conditions are found compatible, and the most general equation which satisfies the conditions is found. The paper is terminated by a discussion on the pointed branch, for the admission of which, as a branch altogether composed of sin- gular points, the author contends. 158 November 12, 1855. A paper was read by the Master of Trinity on the Intellectual Powers according to Plato. ; Also, Prof. Sedgwick gave a lecture on the Classification and Nomenclature of the Paleozoic Rocks. November 26, 1855. A paper was read on the Earthquake in Switzerland in July last, by the Rev. O. Fisher. The 25th of July, 1855, on which the first and most severe shock was felt, was a very wet, close day, and the little wind stirring came from the S.W. In the Miinster Thal the earthquake began by a rumbling vibra- tion like that caused by a carriage run under an archway, gradually increased for about four seconds, and then suddenly ceased. The oscillation seemed to be from E.S.E. to W.N.W., but would be affected by the build of the house. In the church at Bienne two stones fell from the groining thirty or forty feet into the organ pipes, to a point between 2 and 23 feet N. by E. of the point vertically beneath their first position; and allowing for the direction of the building, this would give the motion of the earth about from N.E. to S.W. This wave may have been a reflexion caused by the wave entering the Jura from the valley. Another shock was felt at Bienne, at 10 a.m., on the 26th. The great shock was felt at Strasburg, slightly at Lyons in a direction from E. to W.; likewise at Chambery, Alessandria, and Genoa. The account given by Plana in the Times does not seem very intelligible, but as far as can be made out from the stopping of the clocks, it gives the direction of the shock at Turin about 30° W. of S. Chiavenna, the western shore of the Lake of Constance, and Schaffhausen seem to fix the limit to which it was felt towards the east. The area shaken was therefore an oval, having its largest dimension about 300 miles N. and S., and its shortest 250 miles E. and W. At Geneva the shock appeared to be directed toE.N.E. At Thun it appeared to come from Frutigen. At Kandersteg, at the north foot of the Gemmi, the shock was N. and SS. At Interlaken the shocks were more severe; and at Ormont, Canton Vaud, the oscil- lation came from W. to E., preceded by a noise which lasted for an instant only, and the roof of a house fell in. It seems that nearer the centre of the oval the intensity of the shock was greater. At the baths of Leuk a chimney was thrown down and the walls cracked ; but on ascending the valley of the Rhone the evidence of disturbance became rapidly more marked up to Visp, where only seven houses remained habitable. At the little inn, the “Soleil,” the flag pave- 159 ment was burst upwards as if by a blow from beneath: a continual succession of shocks have occurred there at variable intervals up to the present time: Passing on towards Brieg, the evidence of the violence of the shock rapidly diminished. ‘The valley of Zermatt showed the chief disturbance; the bridle road was continually fis- sured, and in some places slipped down into the valley. At Stalden there was much destruction, but at St. Nicholas the havoc was very great indeed. Higher up the traces of the shock were less and less, until at Tesch, Randa, and Zermatt, there was no mischief done. The other branch of the valley by Saas did not suffer so much. Drawing lines through the different places in the direction in which the wave proceeded, it will be found that they converge very nearly to Visp, showing that to be nearly the centre of disturbance. Mr. Croker of Caius College was walking between Stalden and Visp when the great shock occurred, which appeared to him to be a blow from beneath like the springing of a mine under him, and he observed that the path sunk several inches from the solid rock; a lofty isolated rock on the opposite side of the valley vibrated, and blocks of stone came tumbling down on all sides. The quivering lasted about thirty seconds. He did not observe any sound prece- ding the shock, though this was heard at Visp; but a crashing sound accompanied the great shock, and a fainter sound continued afterwards beyond the motion. He felt continued shocks from one o’clock till four, when he proceeded towards Sion. At Zermatt the same shock was felt very much less violently, and no sound preceded it; and after attaining its maximum, it ceased somewhat suddenly. It was felt less strongly on the Riffelberg; and on the 27th another, felt at Zermatt as strongly as before, was not felt on the little Mont A sound seems in general to have preceded the earthquake at places near the centre of disturbance: at Visp likened to the echo of an avalanche, but at a distance there was only a sound simul- taneous with the shock. The sound may have arisen from the grind- ing of the walls of the fissure, or whatever violent action may have occurred at the origin, and the sound-waves travelling more rapidly than the earthquake-wave. This is opposed to Mr. Mallet’s view, though he gives a table in which the least rate given for sound tra- velling through any kind of stone is 3640 feet per second, while the rate of motion of the earthquake of Lisbon was 1750 feet. If the view stated be correct, the disturbance must have been deep in the earth, which would also explain the upward blow felt by Mr. Croker. At greater distances the sound-wave would be expended sooner than the earthquake-wave, and the accompanying sound be due to local action. Chimneys and such like structures appear to have fallen away from the centre of disturbance, being thrown down by the return stroke of the wave; the forward stroke having to move them only from a state of rest, whereas the return stroke would have to over- come the momentum generated by the former. Near the centre the shock was sudden, passing away gradually. 160 At a distance it began with slight quivering, gradually attained a maximum, and then suddenly ceased. Now if the disturbance oc- curred along a large fissure, perhaps several miles in length, and of unknown depth, the waves from different portions would reach any given point in succession, and at intervals the combined effect of many waves would be felt, producing a result analogous to the rolling of thunder due to the varying distance of the source of sound, while the sudden concussion at a nearer point is like the detonation heard when the lightning is near the auditor. — j The shocks were less severe in the mountains than in the valleys. As far as the wave progressing horizontally is concerned, it would, on entering a mountain, at first be nearly bounded by a horizontal plane continuous with that of the valley, just a8 light is propagated in straight lines; but there would also be a diversion of a part (ana- logous to the diffraction of light at a screen) into the mountain, so that where the wave passed for some distance into a range it would finally be felt at the summit. It is observable that the shock on the 25th was less severe on the Riffelberg than at Zermatt, yet it travelled through the mountain and was felt at Turin. The period of elevation of the Alps seems about contemporaneous with the older Pliocene of Sir C. Lyell. The country is broken up with faults, which probably there, as elsewhere, follow the lines of valleys. The valley of Visp lies in the axis of two ranges which have all the appearance of a mighty valley of elevation. ‘The shock may have arisen from a shifting of the beds on this line of ancient disturbance, and very probably the somewhat rectangular corner between the valleys at Visp suffered the principal displacement. Earthquakes in non-volcanic regions probably arise from a failure of support. At the period of the elevation of the Alps, the more heated lower parts of the earth’s crust must have come nearer to the surface than their normal position, and contractions and failure of support must occur while cooling; and the comparatively recent elevation of the Alps may give reason for thinking this to be still going on. ~ December 10, 1855. A paper was read by Mr. Maxwell on Faraday’s Lines of Force. The method pursued in this paper is a-modification of that mode of viewing electrical phenomena in relation to the theory of the uni- form conduction of heat, which was first pointed out by Professor W. ‘Thomson in the Cambridge and Dublin Mathematical Journal, vol. iii. Instead of using the analogy of heat, a fluid, the properties of which are entirely at our disposal, is assumed as the vehicle of mathematical reasoning. A method is given by which two series of surfaces may be drawn in the fluid so as to define its motion completely. The uniform motion of an imponderable and incompressible fluid permea- ting a medium, whose resistance is directly as the velocity, is then discussed, and it is shown how a system of surfaces of equal pressure 161 tnay be drawn, which, with the two former systems of surfaces, divides the medium into cells, in each of which the same amount of work is done in overcoming resistance. It is then shown that if the fluid be supposed to emanate from certain centres, and to be absorbed at others, the position of these centres can be found when the pres- sure at any point is known; and that when the centres are known, the distribution of pressures may be found. Methods are then given by which the motion of the fluid out of one medium into another, the resistance of which is different, may be conceived and calculated, and the theory of motion in a medium in which the resistance is different in different directions is stated. The mathematical ideas obtained from the fluid are then applied to various parts of electrical science. It is shown that the expres- sion for the electrical potertial at any point is identical with that of the pressure in the fluid, provided that “sources” of fluid are put instead of positive electrical ‘‘ matter,” and centres of absorption or ‘sinks ” for negative ‘‘ matter.” The theory of Faraday with respect to the effect of dielectrics in modifying electric induction, is illustrated by the case of different media having different conducting power; and it is shown, that, in order to calculate the effects by the ordinary formule of attractions, we must alter in a certain proportion the quantities of electricity within the dielectric, and conceive an imaginary distribution of elec- tricity over the surface which separates it from the surrounding medium. _ The theory of magnets and of the phenomena of paramagnetic and diamagnetic bodies is expressed with reference to the “lines of inductive magnetic action ;” and elementary proofs of the tendency of paramagnetic bodies toward places of stronger magnetic action, and of diamagnetic bodies toward places of weaker action, are given. This distinction of paramagnetic and diamagnetic is not here used absolutely, but indicates a greater or less conductivity for the lines of inductive action than that of the surrounding medium. The magnetic phenomena of crystals are then examined, and referred to unequal magnetic conductivity in different directions ; and the case of a crystalline sphere in a uniform field of force is worked out. The laws of electric conduction, as laid down by Ohm, are shown to agree with those of the imaginary fluid, and definitions of quan- tity and intensity are given, which will apply to magnetism as well as galvanism. The theory of the attractions of closed circuits, as established by Ampére, is shown to lead to the following results :— 1. The total intensity of the magnetizing force estimated along any closed curve embracing the circuit is a measure of the quantity of the current. 2. The quantity of the current, multiplied by the quantity of in- ductive magnetic action, from whatever source, which passes through it, gives what may be called the potential of the circuit. The ten- dency of the resultant forces is to increase this potential. 162 The theory of Faraday with respect to the induction of currents in closed circuits takes the following form :— When the quantity of inductive magnetic action which passes through a given circuit changes in any way, an electromotive force proportional to the rate of change acts in the circuit, and a current is produced whose quantity is the electromotive force divided by the total resistance of the circuit. The mathematical discussion of the electro-magnetic laws is re- served for another communication. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. > February 11, 1856. A paper was read by Mr. Maxwell of Trinity College, ‘‘ On Fara- day’s Lines of Force,” in continuation of a former paper (Proceedings of the Society, Dec. 10, 1855). This paper was chiefly occupied with the extension of the method of lines of force to the phenomena of electro-magnetism, by means of a mathematical method founded on Faraday’s idea of an “< electro- tonic state.” In order to obtain a clear view of the phenomena to be explained, we must begin with some general definitions of quantity and inten- sity as applicable to electric currents and to magnetic induction. It was shown in the first part of this paper, that electrical and mag- netic phenomena present a mathematical analogy to the case of a fluid whose steady motion is affected by certain moving forces and resistances. [The purely imaginary nature of this fluid has been already insisted on.] Now the amount of fluid passing through any area in unit of time measures the quantity of action over this area ; and the moving forces which act on any element in order to over- come the resistance, represent the total intensity of action within the element. In electric currents, the quantity of the current in any given direc- tion is measured by the quantity of electricity which crosses a unit area perpendicularly to this direction; and the intensity, by the resolved part of the whole electromotive forces acting in that direc- tion. In a closed circuit, whose length =/, coefficient of resistance =k, and section =C®, if F be the whole electromotive force round the circuit, and I the whole quantity of the current, I C*F G k=F, I= 7 The laws of Ohm with respect to electric currents were then ap- plied to cases in which the conducting power of the medium is dif- ferent in different directions. The general equations were given and several cases solved. : 3 In magnetic phenomena, the distinction of quantity and intensity is less obvious, though equally necessary. It is found, that what Faraday calls the quantity of inductive magnetic action over any No. XII.—Proceepines or THE CamBripGE Putt, Soc. 164 surface, depends only on the number of lines of magnetic force which pass through it, and that the total electromotive effect on a conduct- ing wire will always be the same, provided it moves across the same system of lines, in whatever manner it does so. But though the quantity of magnetic action over a given area depends only on the number of lines which cross it, the intensity depends on the force required to keep up the magnetism at that part of the medium; and this will be measured by the product of the quantity of magnetiza- tion, multiplied by the coefficient of resistance to magnetic induction in that direction. : The equations which connect magnetic quantity and intensity are similar in form to those which were given for electric currents, and from them the laws of diamagnetic and magnecrystallic induction may be deduced and reduced to calculation. We have next to consider the mutual action of magnets and elec- tric currents. It follows from the laws of Ampére, that when a magnetic pole is in presence of a closed electric circuit, their mutual action will be the same as if a magnetized shell of given intensity had been in the place of the circuit and been bounded by it. From this it may be proved, that (1) the potential of a magnetized body on an electric circuit is measured by the number of lines of magnetic force due to the magnet which pass through the circuit. (2) That the total amount of work done on a unit magnetic pole during its passage round a closed curve embracing the circuit depends only on the quantity of the current, and not on the form of the path of the pole, or the nature or form of the conducting wire. The first of these laws enables us to find the forces acting on an electric circuit in the magnetic field. Give the circuit any displace- ment, either of translation, rotation, or disfigurement, then the dif- ference of potential before and after displacement will represent the force urging the conductor in the direction of displacement. The force acting on any element of a conductor will be perpendicular to the plane of the current and the lines of magnetic force, and will be measured by the product of the quantities of electric and magnetic action into the sine of the angle between the direction of the electric and magnetic lines of force. The second law enables us to determine the quantity and direction of the electric currents in any given magnetic field; for, in order to discover the quantity of electricity flowing through any closed curve, we have only to estimate the work done on a magnetic pole in passing round it. This leads to the following relations between a, 6, , the components of magnetic intensity, and a, 6, c,, the resolved parts of the electric current at any point, ' = dp, ay b= dy, __ da, a= da, ? > fark: paar aOs Yn J eS dz dy dx dz dy In this way the electric currents, if any exist; may be found when we know the magnetic state of the field: When a,de+,,dy+9,dz is a perfect differential, there will be no electric currents, Since it is the intensity of the magnetic action which is immedi- a, _ Bi dx 165 ately connected with the quantity of electric currents, it follows that the presence of paramagnetic bodies, like iron, will, by diminishing the total resistance to magnetic induction while the total intensity is constant, increase its quantity. Hence the increase of external Sti due to the introduction of a core of soft iron into an electric elix. From the researches of Faraday into the induction of electric cur- rents by changes in the magnetic field, it appears that a conductor, in cutting the lines of magnetic force, experiences an electromotive force, tending to produce a current perpendicular to the lines of motion and of magnetic force, and depending on the number of lines cut by the conductor in its motion. It follows that the total electromotive force in a closed circuit is measured by the rate of change of the number of lines of magnetic force which pass through it; and it is indifferent whether this change arises from a motion of this circuit, or from any change in the mag- netic field itself, due to changes of intensity or position of magnets or electric currents. This law, though it is sufficiently simple and general to render intelligible all the phenomena of induction in closed circuits, con- tains the somewhat artificial conception of the number of lines pass- ing through the circuit, exerting a physical influence on it. It would be better if we could avoid, in the enunciation of the law, making the electromotive force in a conductor depend upon lines of force external to the conductor. Now the expressions which we obtained for the connexion between magnetism and electric currents supply us with the means of making the law of induced currents depend on the state of the conductor itself. We have seen that from certain expressions for magnetic intensity we could deduce those for the quantity of currents, so that the cur- rents which pass through a given closed curve may be measured by the total magnetic intensity round that curve. Here we have an integration round the curve itself instead of one over the enclosed sur- face. In the same way, if we assume the mathematical existence of a state, bearing the same relation to magnetic quantity that mag- netic intensity bears to electric quantity, we shall have an expression for the quantity of magnetic induction passing through a closed cir- cuit in terms of quantities depending on the circuit itself, and not on the enclosed space. Let us therefore assume three functions of x y 2, & Bo Yo, such that a, b, c, being the resolved parts of magnetic quantity, 4B, dy nee dy, da, oz da, dio. q=——-———, 4=—-—-—, dz dy dz dz ; da, dBy ad then it will appear that if we assume 2, _ ; re as the expres- sions for the electromotive forces at any point in the conductor, the total electromotive force in any circuit will be the same as that ex- pressed by Faraday’s law. Now as we know nothing of these in- ductive effects except in closed circuits, these expressions, which are 166 true for closed currents, cannot be inconsistent with known pheeno- mena, and may possibly be the symbolic representative of a real law of nature. Such a law was suspected by Faraday from the first, although, for want of direct experimental evidence, he abandoned his first conjecture of the existence of a new state or condition of matter. As, however, we have now shown that this state, as described by him (Exp. Res. (60.)), has at least a mathematical significance, we shall use it in mathematical investigations, and we shall call the three functions a, Bo, Yo, the electrotonic functions (see Faraday’s Exp. Res. 60. 231. 242. 1114. 1661. 1729. 3172. 3269.). That these functions are otherwise important may be shown from the fact, that we can express the potential of any closed current by the integral d d d ((« ay M2 + 8, + cay ds, and generally that of any system of currents in a conducting mass by the integral ( i) (24% + Bb +7 90a) da dy dz. The method of employing these functions is exemplified in the case of a hollow conducting sphere revolving in a uniform magnetic field (see Faraday’s Exp. Res. (160.)), and in that of a closed wire in the neighbourhood of another in which a variable current is kept up, and several general theorems relating to these functions are proved. February 25, 1856. A paper was read, “‘ On a direct method of estimating Velocities, Accelerations, and all similar magnitudes with respect to Axes move- able in any manner in Space, with applications.”” By Mr. Hayward, of St. John’s College. The frequent recurrence, in many different investigations of kine- matics and dynamics, of exactly corresponding equations, suggests the inquiry whether they do not result from some common principle, from which they may be deduced once for all. An investigation based on this idea forms the first part of this paper, and the result is the method mentioned in the title. This calculus shows how the variations of any magnitude, capable of representation by a straight line of definite length in a definite direction, and subject to the parallelogrammic law of combination, may be simply and directly determined relatively to any axes what- ever. If such a magnitude (u) be estimated in a given direction, its intensity in that direction will be represented by the projection on it of the line which represents u. If this given direction be not fixed, but move according to a given law, the projection of w upon it will change by the alteration of its inclination to the direction of u; and the rate of that change is easily calculated, whence an expres- 167 sion for the acceleration of the resolved part of u along a given axis as due to the motion of that axis. If u itself be variable, its variations may be conceived to be due to an acceleration f in a definite direc- tion, which in the time dt produces a quantity fdt in the direction of f to be combined with u by the parallelogrammic law; hence result expressions for the changes in intensity and direction of wu. If, u being variable, the variations in its intensity estimated along a given moveable direction be sought, it will consist of two parts: one, that due to the resolved part of f in the given direction; the other, that due to the motion of the axis, which is the same as if f had not existed, or u had been constant: hence expressions for the’ total acceleration of the resolved part of u along the given moveable axis. If u be resolved along three rectangular axes, these expressions take the forms of familiar kinematical and dynamical equations. These results furnish immediately expressions for the relative velocities of a point with respect to moving axes when its absolute velocities in their directions are given, and vice versd. They also furnish very ready means of estimating accelerations in variable di- rections; as, for instance, the radial and transversal accelerations of @ point moving in a plane or in space, or the tangential and normal accelerations in the same case. ‘These are some kinematical appli- cations of the calculus. The dynamical applications form the second part of the paper. Here the general problem of the motion of a system, so far as it is due to external forces, is divided into two steps: one from force to momentum, the other from momentum to velocity. If the momenta of the particles of a system be reduced like a system of forces, they produce a single linear momentum and a single angular momentum, just as a system of forces produces a single force anda single couple. The linear momentum is (in our received language) the momentum of the mass of the system collected at its centre of gravity; the an- gular momentum is a magnitude the constancy of whose intensity in a given axis is equivalent to the assertion of the principle of the conservation of areas for that axis, and the constancy of whose direc- tions determines the “ invariable plane” as a plane perpendicular to it. The momentum, whether linear or angular, is a magnitude to which the previous calculus applies, and the resultant force and resultant couple are respectively the accelerators of the two kinds of momentum: hence the equations obtained in the first part, inter- preted with respect to these magnitudes, furnish equations in any required form for the determination of the momenta at any instant. The step from force to momentum is independent of the nature of the system, that from momentum to velocity requires the system to be particularized. In the paper the case of an invariable system only is considered, and in particular its motion of rotation about its centre of gravity. The axis of rotation or angular velocity is related in direction to that of angular momentum, as the radius of the central ellipsoid with which it coincides to the normal at its extremity. Hence an angular momentum constant in intensity and direction, in general gives rise to an angular velocity variable in both respects, 168 and vice versd. The question then becomes, to determine the acce- leration of angular velocity due to the motion of the system. This is obtained by determining the acceleration of angular momentum for a line fixed in the body, which is then shown to be a maximum for the normal to the plane containing the axes of angular momentum and velocity ; then the acceleration along this line is the total acce- leration of angular momentum due to the motion, and the accelera- tion of angular velocity determined from it (just as the angular velo- city is determined from the momentum) is that due to the motion of the system. Also the acceleration of angular velocity due to the forces is related to the resultant couple and its axis, just as the an- gular velocity to the angular momentum. Thus the accelerations of angular velocity due both to the motion and to the forces being determined, the intensity and direction of the angular velocity at any time is to be found by combining these effects by integration. The problem is worked out in the case of the axis of the resultant couple being coincident with that of angular momentum, so that this remains fixed. The paper concludes with a simple solution of the problems of Foucault’s gyroscope as applied to show the effects of the earth’s rotation, the simplicity arising from the method of this paper enabling us at once to refer the motion to those axes (neither fixed in the body nor in space) whose motion it is desired to determine. March 10, 1856. A paper was read by L. Barrett, Esq., ‘‘ On the Distribution of the Mollusca on the Coast of Norway.” (Vide Annals of Nat. Hist. May 1856.) In this the author observed, that when the fauna of the coast of Nor- way is compared with that of the other side of the North Atlantic, a great difference will at once be perceived, not only in the number of species, but also in the different distribution of northern and south- ern types; the Mollusca of Greenland being peculiarly arctic, those of Scandinavia a mixture of southern and northern species. In the southern part of Norway we find the species living on our coasts abundant; but they become rarer as we go north, their place being supplied by arctic forms. Many of the northern species have a great geographical range, at which we need not be surprised when we consider their great antiquity, many of them having existed since the pliocene period; and, in the author’s opinion, whenever we find a species with a great geographical range, we may at once infer that it has continued to live from a remote period. It is extremely diffi- cult, according to the present state of the currents in the northern seas, to account for the wide distribution of arctic shells on this side of the North Atlantic; but when we consider that at not a very distant period the temperature and other conditions of this area were totally different, that a cold climate prevailed, certainly accom- panied by a current setting from the north (as is fully proved by the 169 fact that boulders are always found nearly south of the moun- tain ranges from which they have been originally transported), and that many of the shells are found fossil in the Sicilian tertiaries, this wide distribution may be fully accounted for. As these frigid conditions were gradually altering to more genial ones, those species requiring a lower temperature would gradually die out, and only con- tinue to exist in higher latitudes. The littoral and shallow water species would be most affected by such an alteration of climate; and while the fauna of the littoral and laminarian zones would be entirely changed,—the shells composing that fauna replaced by other species,—those living in the deep sea would continue to exist, per- haps at a greater depth, mingled with the species brought in with the new physical conditions of the area. This we know to be the case; for while the northern littoral shells, such as Mya truncata, &c., are found only fossil in Sicily, many of the deep-sea arctic species that existed there when those fossils were alive are still found living in the deeper parts of the Mediterranean. The same thing occurs on our coasts, where the arctic littoral or shallow-water shells, as Astarte arctica, Tellina proxima, Natica helicoides, &c., which are found in shallow water on the Scandi- navian or Greenland coasts, are now rare as deep-sea shells, and that in the same area in which they were formerly abundant as shallow- water species. Some species are capable of enduring great differ- ences of climate, the Mytilus edulis being found as abundantly on the coast of Greenland as on our own shores. It is not difficult to account for the presence of the southern species on the coast of Norway, as the Gulf-stream sets directly along the coast, warming its waters, and rendering them habitable for species requiring a moderately high temperature. ‘The great abundance and wide distribution of these species show that the present order of things has continued for a great length of time. The gradual ex- tinction of northern shells on our coasts is still going on; the number of living specimens of Pecten danicus is very small, while dead shells are very abundant, and fresh dead specimens of Pecten islandicus are frequently dredged, though a living specimen has not yet occurred. It is probable that this species has died out very recently. On the eastern shores of Davis’s Straits the Mollusca are about half as numerous as on the coast of Norway. The fauna differs in the prevalence of arctic types and the total absence of southern. At a former period the fauna was of a mixed character; species now in- habiting more southern latitudes are found fossil in the raised beaches at Disco Island, which species are no longer found living on the coasts. April 28, 1856. A paper was read ‘‘ On the Theory of Heat,” by Mr. A. A. Har- rison, of Trinity College. : ‘ The object of this paper was to show thaf there is considerable 170 reason for supposing that radiant heat is identical with light, and that they both consist of vibrations of the ultimate particles of matter. There is a strong presumption of this from the facts, that every body heated to a certain temperature, dependent only on the nature of the surface, emits light as well as heat; and that ‘‘ whenever light manifests itself, heat appears along with it” (Kelland) : the difference between radiant heat and ordinary heat is, that radiant heat is due to vibrations in planes normal to its direction of propagation, and that ordinary heat consists of vibrations in all three dimensions. The author endeavoured to show, in the first place, that the mo- tions of the particles of matter, which must be caused by friction, or in the union of two gases in combustion, is sufficient of itself to account for the following phenomena of heat :—. sind I. That a body once heated continues of the same temperature, with the exception of heat lost by radiation, conduction, &c. This follows immediately from the principle, that in any system of par- ticles held together by mutual attractions and repulsions, the vis viva is independent of the time, and depends merely on the position of the particles. . II. That bodies expand by heat. Before proceeding to this, the author argued that in gases the repulsive force varies inversely as the cube, and not, as usually stated, the simple power of the distance; that it is not true, without some limitation, that the force varies as the inverse first power, was urged from the fact that such a force would decrease more slowly than one varying as the inverse square, and consequently would be the force observed in astronomical phenomena; and even the oxygen of the ocean would repel that of the air instead of attracting it. That the force varies as the inverse cube was deduced from the law of elas- ticity, that the density varies as the pressure ; for if a particle repels other particles with a force varying as the inverse cube, it repels a fixed plane of them with a force varying as the inverse first power. That this is the case may be seen, by considering that though the particle repels particles similarly situated with a force varying as the inverse cube, yet the number of such particles varies directly as the square of the distance, and therefore the whole effect upon the plane varies inversely as the first power. And if this is true for a plane, it is also true for the solid side of the containing vessel; for any solid may be considered as made up of a succession of planes. The law being the inverse cube, it follows that in any position the sum of the forces exerted by any particle on two particles, one on each side of it, is least when that particle is half-way between them, and increases the further the particle is removed from the middle oint. This is seen directly, c fi p y, for the value of (aa) + (a—zy is least when 2=0, and increases until =a. And therefore, in order to produce the same force, it would be necessary that the mean distance should be increased ; and hence if the particles of any aéri- form body be in motion, the force exerted by them would be greater than when at rest; that is, if the pressure to be supported be con- 171 stant, the average distance of the particles must increase, and the body must expand. Ill. That every aériform body not in contact with a liquid expands in the same proportion. This was accounted for by the circumstance, that the increase of pressure depends only upon the ratio of the dis- turbance to the original distance, and not at all upon the absolute distance. IV. That air and elastic fluids give out heat oncompression. By compression the absolute distance of the particles from one another is diminished ; but the absolute motion remaining the same, the rela- tive motion is increased. V. That the same amount of heat is generated in two gases sub- jected to the same pressure ; for the absolute distance of the particle in both being diminished in the same proportion, and the absolute _ Motion remaining unaltered, the relative motion is increased in the same proportion in both. VI. The specific heats are inversely as the atomic weights. Here it was necessary to show that mass is not necessarily proportional to the quantity of matter, as usually stated; or rather, that a body may have a different mass when considered with regard to the molecular force from what it has with respect to the force of gravity. With regard to elasticity of gases, the weight of any single particle is so small as not to affect the result. The question remains, whether we know anything of the masses of different particles relatively to this repulsive force. To determine their masses we have these data. In several different gases equivalent volumes under the same pressure occupy the same space, that is (assuming the Daltonian theory, that equivalent volumes contain the same number of particles), that each particle of the two different gases exerts an equal pressure on the adjacent particles: and hence, with reference to this law, the mass of a single particle in each of these two different gases is the same, and therefore the ‘ vis viva” of equivalent weight or volumes subject to the same motion is the same for both; that is, the quantity of heat of an equivalent of each is the same, and therefore the specific heat of a given weight is inversely as its equivalent number or atomic weight. With reference to the phenomena of radiation, it may be shown from theoretical considerations that the inverse cube is the law required. ‘The inverse first is impossible, for then there could be no vibrations. For the same reason the inverse second is impossible (Camb. Phil. Trans. vol. vii. p. 98). The inverse fourth is also im- possible, for then there could be no vibrations, and the velocity would be infinite (vol. vi. p. 325). It has also been shown that neither the second nor the fourth would satisfy the conditions of the equations (vol. vii. p. 419). Hence, from the theory of radiation, it is supposed that the luminiferous ether consists of solid particles, attracting one another with a force varying as the inverse square (vol. vii. p. 110), and repelling with a force varying as the inverse cube. Now from the Daltonian theory, and the law of elastic fluids, it 172 has been shown that the ultimate particles of our atmosphere com- pose such an xther. But if our atmosphere is the luminiferous ether, we must next inquire whether it does pervade space. Omitting variations of temperature, and merely considering the atmosphere as subject to the two forces of elasticity and gravity, we have for the equation of a column of air on a unit of surface, dp=— 2% paz, or P= 19", where k= 8. 4 p dz e* pP Integrating this, we find that p and p, though they become extremely small, never vanish; and therefore, if these laws are absolutely true, our atmosphere does pervade space. It may be well to obviate the objection, that black substances radiate heat best, and white substances light. ‘This arises from employing the same word radiation to denote two different things : by radiated heat is meant heat given out from a heated body; by radiated light is meant the secondary radiation from the surface of a body exposed to light. If Sir J. Leslie’s experimental calculation of the heat lost from the sun be correct, there is no need of any theories to account for its generation. From the foregoing arguments and facts, it was urged that mo- tions and forces, which certainly exist in cases of combustion, would produce phenomena exactly similar to those of heat, and therefore that part of the phenomena usually attributed to heat are due to this motion; and if part of them, probably the whole. And further, that if the phenomena of radiation of heat are explained by this motion of the particles of matter, light is simply radiated heat of considerable intensity ; and that imponderable substances, whether under the names of ether, caloric, or phlogiston, are equally ima- ginary. Also, a paper was read ‘‘ On the Question— What is the Solution of a Differential Equation?” By Professor De Morgan. This paper is a short supplement to § 3 of a paper on some points of the integral calculus (Camb. Trans. vol. ix. part 2). It discusses the principles on which such an equation as y’*=a%, giving (y—azx+b)(y+ar+c)=0, is generally affirmed to be completely solved when b=c. It dwells on the distinction between a relation and an equation, which may express the alternative of one or more relations; it points out several cases in which conclusions applicable to the simple relation only are affirmed of any equation; and, with reference to the question asked in the title, discusses the manner in which the answer depends on the cross-question, what degree of discontinuity is allowed to be im- plied in the word solution? 173 May 12, 1856. } A paper was read by Mr. Warburton “ On Self-repeating Series,” in continuation of a former paper. The author showed in his former paper on self-repeating series, printed in vol. ix. part 4. of the ‘ Transactions’ of the Society, that in the fraction which generates a series of either of the following forms, 1°42” .t4+3™" 2+ &..., or \ ial Sy adie : ea 2+ &e. . : the numerator of such fraction is a recurrent function of ¢. Healso - then determined the coefficients of the several powers of ¢ in such numerator to be given linear functions of the differences (as the case may be) of 0”, or of 0°"*?. In his present paper, from the z pairs of equal coefficients which the recurrent numerator contains, the author obtains linear equations between the 2n differences concerned; and selecting any 2 of these differences, he concludes that each of them can be expressed in terms of the other n differences not so selected ; and consequently that no formula, expressed in terms of the differences of 0°” or 0?”*!, need contain more than of those differences. He gives the equations requisite for obtaining A”*”(0”") in terms of (A”, A”, ... A%, A')0? ; and A”*!t?(0"""'), in terms of (A”*?, a”, .. . A’, A2)0”"*!; and he applies these and other of his equations to the elimination of particular differences of zero from sundry formulas. Also, Mr. Bashforth exhibited models illustrating the Moon’s motion. Also, a paper was read by Mr. Maxwell ‘‘ On the Elementary Theory of Optical Instruments.” The object of this communication was to show how the magnitude and position of the image of any object seen through an optical in- strument could be ascertained without knowing the construction of the instrument, by means of data derived from two experiments on the instrument. Optical questions are generally treated of with respect to the pencils of rays which pass through the instrument. A pencil is a collection of rays which have passed through one point, and may again do so, by some optical contrivance. Now if we sup- pose all the points of a plane luminous, each will give out a pencil of rays, and that collection of pencils which passes through the in- strument may be treated as a beam of light. In a pencil only one ray passes through any point of space, unless that point be the focus. in a beam, an infinite number of rays, corresponding each to some point in the luminous plane, passes through any point; and we may, 174 if we choose, treat this collection of rays as a pencil proceeding from that point. Hence the same beam of light may be decomposed into pencils in an infinite variety of ways; and yet, since we regardjit as the same collection of rays, we may study its properties as a beam independently of the particular way in which we conceive it analysed into pencils. Now in any instrument the incident and emergent beams are com- posed of the same light, and therefore every ray in the incident beam has a corresponding ray in the emergent beam. We do not know their path within the instrument, but before incidence and after emergence they are straight lines, and therefore any two points serve to determine the direction of each. Let us suppose the instrument such that it forms an accurate image of a plane object in a given position. Then every ray which passes through a given point of the object before incidence passes through the corresponding point of the image after emergence, and this determines one point of the emergent ray. If at any other distance from the instrument a plane object has an accurate image, then there will be two other corresponding points given in the inci- dent and emergent rays. Hence if we know the points in which an incident ray meets the planes of the two objects, we may find the incident ray by joining the points of the two images corresponding to them. ~ It was then shown, that if the image of a plane object be distinct, flat, and similar to the object for two different distances of the object, the image of any other plane object perpendicular to the axis will be distinct, flat, and similar to the object. When the object is at an infinite distance, the plane of its image is the principal focal plane, and the point where it cuts the axis is the principal focus. ‘The line joining any point in the object to the corresponding point of the image cuts the axis at a fixed point called the focal centre. The distance of the principal focus from the focal centre is called the principal focal length, or simply the focal length. There are two principal foci, &c. formed by incident parallel rays passing in opposite directions through the instrument. If we sup- pose light always to pass in the same direction through the instru- ment, then the focus of incident rays when the emergent rays are -parallel is the first principal focus, and the focus of emergent rays when the incident rays are parallel is the second principal focus. Corresponding to these we have first and second focal centres and focal lengths. Now let Q, be the focus of incident rays, P, the foot of the per- pendicular from Q, on the axis, Q, the focus of emergent rays, P, the foot of the corresponding perpendicular, F,F, the first and second principal foci, A,A, the first and second focal centres, then PF, _P,Q,_F,P, A,F, P,Q, F,Ag lines being positive when measured in the direction of the light. 175 Therefore the position and magnitude of the image of any object is found by a simple proportion. In one important class of instruments there are no principal foci or focal centres. A telescope in which parallel rays emerge parallel is an instance. In such instruments, if m be the angular magnifying power, the linear dimensions of the image are — of the object, and m the distance of the image of the object from the image of the object- glass is =, of the distance of the object from the object-glass. Rules were then laid down for the composition of instruments, and sug- gestions for the adaptation of this method to second approximations, and the method itself was considered with reference to the labours of Cotes, Smith, Euler, Lagrange, and Gauss on the same subject, November 6, 1856. A paper was read by Dr. Donaldson “ On the Structure of the Athenian Trireme, considered with reference to certain difficulties of Interpretation.” The author’s intention was to show in this paper that the arrange- ments for seating the three tiers of rowers in the trireme, which Dr. Arnold has called ‘‘an indiscoverable problem,” may be adequately explained by an examination of the terms which are used to discri- minate the rowers, and of other words referring to the different parts of the war-galley. The name of the zygite, or rowers of the middle tier, implies that they sat on the {vyd, or transverse planks connect- ing the opposite sides of the vessel, also called céAyara, and in earlier times xAnides. The thalamite, or rowers of the lowest tier, must, in accordance with their name, have had their seats attached to the ribs of the vessel in the @adAapos, or hold. And the thranite, or rowers of the highest tier, sat on Op4vves, or benches like low stools, extending for seven feet along the alternate vya. The epi- bate, or marines, whether as working the supernumerary oars, or as fighting, occupied platforms running along the bulwarks. This view of the matter explained the fact that there was a gangway from the stern to the prow for the passage of the officers, &c. along the céApara or fvyd, between the ends of the stools on which the thranite sat. This gangway was called the cedis, and the same name was given to the passages leading down to the orchestra from the upper part of the theatre between the rows of seats occupied by the spectators. Hence was derived a philological explanation of the words in Aristoph. Equites, 546 :— aipest’ avrg odd 7d pdtoy, raparéupar’ ég’ Evdexa kwwats O6ovBov xypnoroy Anvatrny for there were eleven tiers of seats between each diazoma of the theatre, which were divided again by the selis ; so that the spectators 176 would represent eleven banks of oars, seated, as in the trireme, with the lower rows in advance. In the same way, the use of the selis in a trireme, as the gangway for the officers, &c., explains the lines in the Agamemnon of AXschylus, 1588-9 :— ov Tavra gwreis veprépg Tpoohpevos kom, kparovytwy trav éxt Cvy@ dopds ; for if the zygite had been intended, they must have been described as ray émt fuywv. The same view of the oéApuara, as the proper place for the officers, was used to explain another passage in the same play (v. 1413), where Agamemnon’s companion is described as vaurtkov cedparwy tororpAys. And the risk of passing along these planks, with intervals between them, was considered to explain the proverbial warning that we must take care not to miss our footing and fall into the hold (Eurip. Heracl. 168). Other points were in- cidentally noticed. November 10, 1856. The Master of Trinity read a paper “On the Platonic Theory of Ideas.” . In this, he first stated the Platonic theory of ideas as given by the late Professor Butler of Dublin, in his ‘Lectures’ (vol. ii. p.117); he then remarked that this theory had evidently, for one of its objects, to explain the possibility of necessary, and therefore eternal truths; and thus ‘was an attempt to solve the.problem, often debated in modern times, of the grounds of mathematical truth; an attempt especially called out by the Heraclitean skepticism of Plato’s time. The doctrine of ideas which belong to the intelligible, not to the visible world, and which are the basis of demonstration, did really answer its purpose, and account for the existence of real and eternal truths; and at the same time, by the tenet that sensible things par- ticipate in those ideas, accounted for the securing of truth respecting the sensible world. But when Plato goes on to speak of ideas of tables and chairs and the like, he gives an extension of the theory which solves no difficulty, and for which no valid reason is rendered. The arguments against this extension of the theory are given with great force in the Dialogue entitled Parmenides, and are not answered ‘there, nor in a satisfactory way, in any part of Plato’s writings. Moreover, throughout this Dialogue, Parmenides is represented as having, in his conversation with Socrates, vastly the superiority, not only in argument, but in temper and manner; and Socrates and his friends, after a little show of resistance, assent submissively to all that Parmenides says. On this ground the writer maintains that the Dialogue is not Plato’s, but anti-Platonic, written probably by an - admirer of Parmenides, and tending to represent Socrates and his disciples as poor philosophers, conceited talkers, and feeble dis- putants. 177 This view was further confirmed by arguments drawn from the external circumstances of the Dialogue. November 24, 1856. An account was given by Professor Miller, of the restoration of the Standard of Weight (vide Proceedings of Royal Society, vol. viii. Nos. 21, 22). December 8, 1856. Mr. Humphry read a paper “ On the relations of the Vertebrate Skeleton to the Nervous System.” He pointed out that the central parts, both of the skeleton and of the nervous system, are composed of segments placed in front of, or above one another: those of the former being called *‘vertebre,” those of the latter “ganglia;”’ that the vertebre correspond with the ganglia, each vertebra having its appropriate ganglion; and further, that the processes, or nerves, emanating from the central ganglionic portion of the nervous system corre- spond with, and accompany the processes, or bones, appended to the central portion of the skeleton, so that the bones appended to any particular vertebra are generally accompanied by the nerves ema- nating from the ganglion connected with that vertebra. Hence, where a difficulty is found in referring a bone to its vertebra, assist- ance may often be derived from a reference to the nerve or nerves which accompany that bone. Following the guide thus indicated, Mr. Humphry would refer the upper extremity, not to one—the occipital—vertebra, according to the plan of Professor Owen, but to several cervical vertebra, forasmuch as it derives its nerves from a considerable tract of the cervical portion of the cord. For the like reason, the lower extremity may be regarded as appertaining to several lumbar and sacral vertebra. The relations of the bones of the face to their respective cranial vertebree were pointed out in ac- cordance with the distribution of the cerebral nerves. It was shown, that although the size and shape of the skull are proportioned to the size and shape of the brain, yet that, as a general rule, the thickness and weight of the skull are in an inverse ratio to the size of the brain. A comparison of the different nations of mankind proves, moreover, that the size of the whole skeleton, as well as that of the skull, is usually proportioned to the size of the brain; a well-deve- loped physique being the natural associate of an ample cerebrum, PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. —_—<>____ February 9, 1857. Prof. Challis gave an account of his Observation of the Occulta- tion of Jupiter on Jan. 2, 1857. February 23, 1857. A paper was read “On the Theory of Polarized Fasciculi, com- monly known as Haidinger’s Brushes.” By the Rev. J. Power, M.A., Librarian of the University. In this paper the view taken of the subject is similar to that which had occurred to M. Jamin, and which will be found in Pog- gendorff's Annalen, 1849, p. 145, and in the Comptes Rendus, tome xxvi. p. 197. The author arrived, however, at the present theory quite independently in the course of last summer, and before he had acquainted himself with the literature of the subject. M. Jamin had taken as an essai de calcul the particular semi-visual angle 20°, which lies far beyond the limits within which the phenomenon is visible ; and he has not attempted to give the general law for small- angles, which was the real problem to be solved. This is what the author has attempted in the present communica- tion, availing himself of the experimental researches of Chossat given in the Bulletin de la Soc. Philomatique, 1818, p. 94. The subject was rendered more complicated by the circumstance that the formule for the intensities of the refracted pencils are given differently by Neumann, Airy, and the author of this paper. Instead of taking any one set of formule, the author managed to take them all into consideration by previously showing that Airy’s formule result from Neumann’s by multiplying them by ae , which is equi- valent to : eee ; while his own result from the same by multi-. / No. XIII.—Procrrpines or THE CAMBRIDGE Part. Soc. 180 plying them by 28 6 and 0, being the angles of incidence and ‘ refraction, It follows from thence that, whichever set of formule we may prefer, we shall have after refraction, in all cases, the following sim- ple relation for two oppositely polarized incident pencils of equal in- tensity ; namely, =cos* (0—0,). ff |e This fraction decreases as the deviation 0—0, increases; it is therefore less for the violet rays than the red, for the indigo than for the yellow; and this serves to explain in a sonore way the dingy ellow stripe in the plane of polarization, and the bright violet stripe in the plane at right angles. The author has also considered the effect in a plane making an angle » with the plane of greatest polarization, and arrives at the following result; that, provided we attribute to the distribution of the optic nerve such a variation of sensibility as, taken in conjunc- tion with the action of the iris, shall produce a field of view uni- formly bright from the centre outwards when common daylight is viewed (a condition which the author believes is common to all eyes with his own), we shall have for the brightness at any point of the field of view the following expression, M .(1 ey cs 24), where M is the central brightness, e the degree of polarization (bein 0 for common daylight, and 1 for completely polarized light), an y='07309 for rays of mean refrangibility. The last expression gives us without difficulty the form of the curves of equal brightness, Assuming this constant brightness to be ¢ M, and putting v==0 008 , y=0sin ¢, we find for the equation sought 2(1 —0) ey The curves are therefore equilateral hyperbolas having the lines in octants for their common et ba 0 which confound themselves with the curves themselves when c=0, the case of mean brightness. The yellow fasciculi have their vertices in the plane of polariza- tion, and the violet fasciculi have their vertices in the plane at right sy hee t will be seen that for a given value of 0, the brightness, for rays gt — ytcs 181 of all degrees of refrangibility, that is for all values of , is least in the of greatest polarization and greatest in the plane at right trary to the idea of Moigno, who, for insufficient rea- sons, imagined that the maximum occurred in the plane of polariza- tion and the minimum in the plane at right angles. The yellow tint in the position of minimum intensity, and the vio- fectly accounted for by the consideration that y is greater for the violet and indigo rays than for the red and yellow. The paper further contains some observations respecting a subjec- tive centre of the eye, distinct from the usual objective centre, which may be read with interest, as they remove some difficulties con- nected with the theory of vision, which had often occurred to the author, and may have occurred to others. March 9, 1857. Mr. Hopkins gave an account of some experiments on the conduc- tivity of various substances, and pointed out the bearing of the results on theories of terrestrial heat. : April 27, 1857. Mr: Humphry read a paper “ On the Proportions of the Human Frame.” May 11, 1857. A paper was read by Professor Stokes, ‘‘ On the Discontinuity of Arbitrary Constants which appear in Divergent Developments.” — In a paper “‘ On the Numerical Calculation of a class of Definite Integrals and Infinite Series” printed in the ninth volume of the ‘Cambridge Philosophical Transactions,’ the author succeeded in put- - ting the integral E cos 5 (w® —mw)dw under a form which admits of 0 receiving every numerical calculation when m is large, whether po- sitive or negative. ‘The integral is obtained in the first instance under the form ogee functions for m positive, or an exponential for m negative, multiplied by series according to descending powers of m. ‘These series, which are at first convergent, though ultimately divergent, have arbitrary constants as coefficients, the determination of which is all that remains to complete the process. _ From the nature of the series, which are applicable only when m is large, or when it is an imaginary quantity with a large modulus, the passage from a large positive to a large negative value of m cannot be made 182 through zero, but only by making m imaginary and altering its am- plitude by z. The author succeeded in determining directly the arbitrary constants for m positive, but not for m negative. It was found that if; in the analytical expression applicable in the case of m positive, —m were written for m, the result would become correct on throwing away the part involving an exponential with a positive index. There was nothing however to show @ priori that this pro- cess was legitimate, nor, if it were, at what value of the amplitude of m a change in the analytical expression ought to be made, although the occurrence of radicals in the descending and ultimately divergent series, which did not occur in ascending convergent series by which the function might always be expressed, showed that some change analogous to the change of sign of a radical ought to be made in pass- ing through some values of the amplitude of the variable m. The method which the author applied to this function is of very general application, but is subject throughout to the same difficulty. In the present paper the author has resumed the subject, and has pointed out the character by which the liability to discontinuity in the arbitrary constants may be ascertained, which consists in this, that the terms of an associated divergent series come to be regularly positive. It is thus found that, notwithstanding the discontinuity, the complete integrals, by means of divergent series, of the differen- tial equations which the functions treated of satisfy, are expressed in such a manner as to involve only as many unknown constants as correspond to the degree of the equation. Divergent series are usually divided into two classes, according as the terms are regularly positive, or alternately positive and negative. But according to the view here taken, series of the former kind ap- pear as singularities of the general case of divergent series proceeding according to powers of an imaginary variable, as indeterminate forms in passing through which a discontinuity of analytical expression takes place, analogous to a change of sign of a radical. A communication was likewise made by the Rev. W. T, Kingsley, ‘On the application of Photography to Wood Engraving.” May 25, 1857. Mr. Bashforth made a communication ‘‘ On some Calculations and Experiments undertaken for the purpose of testing the Theories of Capillary Action.” Also Mr. Candy exhibited a Physiological Alphabet. The principle of this alphabet is to make the form of the letter indicate the manner in which the sound is produced, by showing the position of the organs of speech concerned. CANDY’S PHYSIOLOGICAL ALPHABET. Gee eee L. BN | BR Ga us eee ae ef XN YF l ee GEES ae, en a 4 ee ee ae Sey a oh ae eee | 7) NL 2 ai 5 gS SE ial ad tke 2 ‘tee meee a TN Tt ee ea coped = S ae Lb e;e 4 4 183 The open mouth, ‘affording an uninterrupted, unimpeded passage to the breath, is represented by two horizontal straight lines (—_). The sudden, complete, closing of the lips, which gives the sound of the letter P, is represented by drawing a line joining these two on the left (/— ); the mouth looking in the same direction as the Queen’s head on the coins and stamps. All the letters in the first column of the Table have a line on the left, and are therefore labials. The second column, having a line inclining to the left, are deatals. The third column have a vertical line in the centre of the mouth, and are the linguals. The fourth column incline to the right, and are the palato-linguals or weakened linguals. The fifth column, with a line curving over to the right, are the Sanscrit cerebrals, in which the tongue is curled up, so that the lower side of it comes against the roof of the mouth. The sixth column, in which the line leans from the right to the centre, are the palatals. The seventh column, with a vertical line on the right, are the gut- turals, at the back of the mouth. The eighth column, with a line on the right, leaning to the right, are the faucals, still further back than the gutturals. The first row of letters in the Table, with a thin line from top to bottom, are the mute explodents, in which there is a sudden and complete stoppage of the breath. The second row, with a thick line, are the corresponding sonant explodents. The third row, in which there is an interval between the top line and the connecting line, are the whispered continuants, in which there is a passage of breath, producing a hissing. The fourth row, with a short thick line, are the corresponding voiced continuants, or buzzes. The fifth row contains additional hissing letters, and the sixth row corresponding additional buzzing ones. In the letters of the seventh row, there is represented an opening upwards at the back of the mouth, leading intu the nose. ‘These, therefore, are the nasal letters. The double curl in the eighth and ninth rows indicates a vibration of both sides of the tongue, while the tip is fixed. This is the case in L, which is represented in Sanscrit by a similar form. The eighth row, with thin lines, are the whispered Ls. The tenth and eleventh rows, with a single curl, are the Rs or trills. The curl indicates vibration. The twelfth row are the Caffre clicks. The short line darting out from an angle, shows that the tongue is placed in a certain position, and then suddenly jerked away. The thirteenth row are the breathings. The dots and accents (*’‘) in the last row are the vowel-points. The last figure shows the applicability of this alphabet to mono- grams, being a combination of F, DZh=J, and K=C, the initials of the inventor, F. J. Candy. 184 The analysis of sound exhibited in this alphabet is that in ‘ Universal Writing and Printing,’ by Mr. A. J. Ellis, a Fellow of this Society. ‘The following is Mr. Ellis’s Universal Digraphic Alphabet arranged in the order of this alphabet, to facilitate comparison. Pro sik he bs) Bie ok ale bdd@a@qadawqygqg ph 88 8 a. gk. eR OR bh 22 z2 ww zh jh gh € wh f th sh | gh: w ove dh zh sy m mn ny a ngj ng lih th ee oe l shi: wie ¢ brh r 5. ‘srh . & 8. ork CO, 6) eh Gas ¢ € 2 Oe ae a Key to the Consonants and Breathings. [See the Table for the forms of the Physiological consonants. ] rp P, 7, B with dagesh. tt Arabic dental ¢, Hebrew {). t 'T, r, F\ with dagesh. Row | #7 like ty. 1. < *\ ¢ Sanscrit cerebral ¢. kj French qu, Italian chi. kK, x, 3 with dagesh. q Hebrew /. (b B, 3 with dagesh. dd Arabic dental d. d D,“} with dagesh. 2.< d Hungarian gy. d Sanscrit cerebral d. gi Old Engl. guard, French gueux. .g G, 3 with dagesh. cph >, D without dagesh. ss Arabic dental s, Hebrew Y. s 8,o, D. 3. 4 sj Polish s, Hebrew {. sh Sanscrit cerebral sh. ch German palatal ch in “ ich.” .kh German guttural ch in “ach!” 5..< 6. 2 y 185 rbh German w, without dagesh. zz Arabic dental z. z Z,¢,%., zj Polish 2. 4.2% zh buzz of the hiss sh. ce German palatal 9 in “ teig.” gh German guttural g in “tag.” -€ Hebrew y, Arabic e. (wh whispered w, English wh, Saxon hw. fF: in which the lower lip touches the upper teeth. th 0, D) without dagesh. sh &, English sh, French ch, German sch. yh whispered y in “hue.” L@ Spanish z or /. fw English w, Hebrew }. v English v. dh th in “the,” “¥ without dagesh. zh French /j. _y English y, German j, Hebrew ’. [‘m M, B> i. nn Arabic dental n. n N,v», 3. mj Fr. and Ital. gn, Span. %, Port. nh. n Sanscrit cerebral n. ¢ ngj back palatal n. Lng ng in “sing,” y before gutturals. 8 ‘ib whispered Polish /. lh whispered /, Welsh Ji. Wl Polish barred /. tb A, D> YG Ital. gli, Span. WZ, Pert. lh. Zl Sanscrit cerebral é. ry Sanscrit cerebral 7. 10 i whispered r, Welsh rh. (brh German lip-trill. r initial R, p, “V. rj palatal r. 11. < zrh Polish rz. “ English r in “ fir,” “ her.” rR English r in “fur,” ‘ poor. rhArabic <. 9 é 2” 186 cc dental click, African q. ce flat click, African ec. 12. < ck side click, African z. cj palatal click, African-ge. c cerebral click. —y, Hebrew &, Greek ’. =; Arabic Hamza. athe sh. ae eB — sh strong h, Hebrew fF. | a Sanscrit anuswara, mark of nasalization. Vowels. There are three series of vowels; the upper, middle, and lower; the points representing which are placed respectively above, between, and below the two horizontal lines of the letter. There are also three positions of vowels ; labial and dental (which may be called front), palatal, and guttural. The points are placed respectively at the left hand, middle, and right of the letter. In some cases there are three vowels of the same series in the same position. These are expressed respectively by - /‘ in the ap- propriate place. The number of vowels which may be expressed is thus twenty- seven; but there are only twenty distinct vowels recognized, of which ten occur in English. There are no front upper vowels. Each of these twenty vowels may be long or short. The long vowels may be expressed by doubling, or thickening, the sign for the short vowel; or perhaps better by combining with it the mark (—), giving —, Z or <, A or> for the long sounds of (-) (”) (‘) merpecavely: Scale of Vowels. Position :— Front. Palatal. Guttural. Short. Series. ‘ i: N/- \ 7: Upper. Pryce } 2 WI \ : Middle. Alphabet. , - Lower. Ellis’s iiae ea aea Upper. Pipi ue uh ih eh oh eo oe a Middle. Alphabet u ua oOo 0a ao aa Lower. Long. Candy’s he<- >= >< Upper. Physiol. >= > — Middle. Alphabet. ~ > — Lower. Ellis’s ii iia ee eea aae aa Upper. Daerah} uue uuh ith eeh ooh eeo 00e aa Middle. Alphabet.) xy xua oo 00a aao —aaa-_—s(“Lower. 187 Key to the Vowels. Short. Long. Ph. Digr. Phys. Digr. — a, Ital. and Germ. short a. aa, a in “ father.” — ae, English short a. —aae, Provincial English. — ea, English short e. = cea, ea in “ bear.” —e, French é. =e, English ay, French ée. — ia, English short i. —< iia, the long sound of > =i, French short i. it, Engl. ee, Fr. 7, Germ. ie. ea — a, Engl.&Dutchw,Sanscr.a. —= 2a, the long sound of —= — oe, Fr. eu, Ger. oe or 6 short. —> ove, Fr. e#, Germ. oe or é long. = eo, French mute e. = eeo, the long sound of =~ é —~ oh, Gaelic ao in “ laogh.” ooh, the long sound of / eh, Polish y short. eeh, the long sound of =— ih, Welsh y (tongue between =~ ith, Welsh y long. Z— uh, Swedish u short. ee uuh, Swedish u long. — ue, Sc. ui, Ger. ue or i, Sw. y =— uue, German we or é long. =, aa, French a short. = aaa, French d. —, ao, English o short. aao, English a in ‘‘ water.” ihe — 0a, Italian o aperto, short. | —= 00a, Ital. o aperto, long. =—’o, English o in “ omit.” = 00, Eng]. long o, Gr. w. “sh ma, Ital. o chiuso, Sw. o short > uua, Ital. o chiuso, Sw. o long. — 4, Engl.oo short; Germ. &c. = uu, Engl. oo long, Germ uh. u short. November 9, 1857. Professor Sedgwick gave a description of a series of dislocations which have moved the Cambrian and Silurian rocks between Leven Sands and Duddon Sands, several miles out of their normal posi- tion in the Geological Map of the Lake Mountains. To make the subsequent descriptions clear, the author first gave a normal or typical section of the older Paleozoic rocks, by enume- rating (in an ascending order) the great groups which have been well established, as follows:—1. Skiddaw Slate. 2. Chloritic slate, porphyry, trappean shales, &c. 3. Coniston Limestone, calcareous 188 slate, Coniston Flagstone, &c. (All the above three groups are called Cambrian.) 4. Coniston Grit,—a good physical group, which how- ever, from the paucity of its fossils, may be thought of ambiguous relations: it appears to represent the May Hill Sandstone, and there- fore to be the base of those rocks which, in the north of England, represent the Wenlock and Ludlow groups of Siluria. 5. A coarse slate, often much contorted. 6. A bed of impure limestone, which may be traced from Tottlebank Fell towards the north-east for about two miles, after which it thins out and does not again appear further northward. 7. A coarse and often contorted state. When No. 6 is wanting, No. 5 and No. 7 may be considered as one group (now called Banisdale Slate), which is widely spread, and gradually passes into more coarse and gritty masses, that link themselves to the next superior group. 8. Grit, slate, and tilestone, expanded between Kendal and Kirkby Lonsdale. ‘This is followed by unconformable masses of Old Red Sandstone and carboniferous limestone. Of the above groups, Nos. 5, 6, and 7 approximately represent the Wenlock series, and No. 8 abounds in characteristic Ludlow fossils. In the range of these groups from Shap Fell to Tottlebank Fell (which is about two miles south-west of Coniston Waterfoot) there seems to be no ambiguity ; but to the south of Tottlebank Fell the groups have, through the intervention of great faults, been thrown into such abnormal positions that their relations have often been misunderstood. Thus, (in 1822) when the author first attempted to map this part of Lancashire, he was led, by the line of strike as well as by the whole physical characters of the country, to identify the Tottlebank Limestone with a calcareous band a few miles further south, which ranges (on the east side of the Duddon estuary) from the hills above Bank House, through Meer Beck, towards the vil- lage of Ireleth, and which from thence by an enormous fault (upcast towards the south) is thrown into the ridge of High Haume, near Dalton, from which it is continued nearly in the same strike till it is covered by the Old Red Sandstone and Mountain Limestone. This identification was however erroneous; for the limestone-beds above mentioned, ranging on the east side of Duddon Sands, are altogether unconnected with the Tottlebank Limestone, and are in mineral type and fossils absolutely identical with that portion of the Coniston Limestone which appears, as is well known, on the other side of Duddon Sands in the south-western extremity of Cumberland. The above mistake (made in like manner by several subsequent observers) was partially corrected by the author in 1845; when, on good physical and fossil evidence, he placed the High Haume Lime- stone on the same parallel with that of Coniston. Not having any fossils from the limestone quarries north of the village of Ireleth, and not having found a single characteristic fossil from the slate-rocks between Ulverston Sands and Duddon Sands, he was in 1845 unable to carry his correction any farther; but even then he remarked again and again that the calcareous beds north of the village of Ireleth in structure resembled the Coniston Limestone, which is seen on the north-west side of the Duddon, much more nearly than they resem- bled the calcareous beds of Tottlebank. 189 _ Inthe autumn of 1856 the author, accompanied by his friends (Mr. Gough of Reston Hall, and Mr. John Ruthven of Kendal), saw for the first time the excellent local collection formed during the labours of many past years by Mr. Bolton of Ulverston. It was evident almost at a glance that the fossils he had collected from the slate-rocks between Ulverston and Duddon Sands belonged to the upper part of the Coniston group (No. 3). He kindly pointed out to them some of his best localities, and they left the country con- vinced that nearly all the older rocks between Ulverston and the Duddon estuary belonged to the Coniston group, and consequently that these rocks were not superior, but inferior to the Coniston grits, though the prevailing dip and the geographical position of the groups might seem to indicate the very contrary. In 1857 they again visited the district, re-examined Mr. Bolton’s unrivalled local collection, and again during two days made traverses under his guidance. They then devoted a few days to the approxi- mate determination of the vast breaks and faults, which have so much disturbed the normal position of the physical groups in a part of Furness, and made the colours of the Geological Map to appear almost incredibly anomalous. The author then described, by help of plans and sections, the Faults above alluded to. 1. Black Coomb, protruded as it is at the south-west end of Cumberland, seems to have been a kind of centre of disturbance. The chloritic slates and Coniston group which skirt the south-east side of Black Coomb have been ripped up by a north and south fault which at one cast throws the Coniston lime- stone about three miles to the south of its previous range. 2. Similar enormous up-casts towards the south-east cause the re- petition of the Coniston limestone and flagstone on the other side of the Duddon Sands. This repetition is not produced by undula- tions, but by great up-cast faults. 3. A great east and west fault descends near the rivulet of Beck Side with a down-cast to the north, which brings the Ireleth slates down to the level of the sea at Sandside. 4. A complicated fault, or system of faults, with a very great up- cast to the south-east, runs from Kirkby Hall, skirting the brow of the hill under the great Ireleth slate quarries. The slates of Ireleth cannot be separated from the Coniston flag. They do not overlie the Coniston grits (as the author and other observers had long sup- posed), but abut against them. This conclusion seems inevitable, though the sections are broken and difficult of interpretation. 5. By a complication of faults the Coniston grits are widely ex- panded in the hills immediately north of the great Ireleth slate quar- ries; but all the above mentioned groups are by an east and west fault (or series of faults), with an enormous up-cast towards the south, cut off from the normal groups (viz. Nos. 4, 5, 6, and 7 of the typical section), which range towards Coniston water and thence into Westmoreland. This east and west fault runs down into the valley of the Crake, not far from Lowick Bridge. 6. Another great fault appears to descend from Coniston water- 190 head down the Crake, producing an up-cast on its south-east side. The facts given in the above abstract necessitate a partial change of nomenclature. The Ireleth slates can no longer be appealed to as groups suverior to the Coniston grits and on the parallel of the Wenlock shale. But in Banisdale there are old slate quarries in the group which, without sectional difficulty or ambiguity, does over- lie the Coniston grits. For the future, therefore, the author pro- poses to use the term Banisdale slates for the lower part of the group which overlies the Coniston grits. Having approximately laid down the faults above mentioned, there was still something wanting to complete the evidence; for the slate rocks between Duddon Sands and Leven Sands, in spite of their contortions, form an ascending section of great thickness. If, therefore, these slates be a repetition of the Coniston flags, the Coniston grits (typical section No. 4) might be looked for somewhere towards the south-east. To put this to the test, the author (accom- panied by his two friends) made a complete traverse from Broughton to the upper part of Leven Sands; and they found, as they were finishing their traverse, that the ridges which skirt the estuary of the Leven, below Penny Bridge, were composed of the Coniston grits in their characteristic form. The evidence was then complete, and they next day left the country. Also a paper was read by Professor De Morgan, ‘‘ On the Beats of Imperfect Consonances.”’ This subject has been left in great obscurity by Dr. Smith, and subsequent writers have either neglected it, or misunderstood it, or obtained results by methods which miss the principal simplification of which the theory appears susceptible. Omitting historical matter, Mr. De Morgan’s method may be described as follows :— The grave harmonic of ‘Tartini, formed by sounding two notes of which the vibrations take z and m equal parts of time (m : n being in its lowest terms), has a vibration which Jasts through mn of those times. This is called Tartini’s beat, whether it produce a sound, or whether it only produce what Dr. Smith calls a fluttering. This beat is most perfect when the consonance is in perfect tune. If the consonance be a little out of tune, Tartini’s beats are not destroyed, but do not succeed each other with perfect reiteration of circum- stances, owing to the gradual advance or regression of the position in one vibration of the commencement of the other. A cycle of disturbances is the result, which cycle is repeated, or repeated guam proxime ; and the ear recognizes this recurrence in Smith’s beats, which are entirely due to the imperfection of the consonance. The connexion has a close resemblance to that of the instantaneous ellipse of a planet and its disturbed orbit. The simplest connexion of beats and vibrations is as follows :— The smaller of the two numbers, n and m, being 2, every vibration by which the upper note is tuned wrong gives m beats per second. Thus, the consonance being a fifth (2:3), every vibration by 191 which the upper note is too flat or too sharp gives two beats per second. In an appendix, Mr. De Morgan gives some tables of beats, repeats some theorems on temperament from the Penny Cyclopedia, and recommends and argues in favour of tuning being performed by a whole octave of tuning forks, adjusted by beats to the system em- ployed. November 23, 1857. A paper was read by Professor Thompson, “On the Sophista of Plato.” In this paper the genuineness of the Sophista was defended, and some of its philosophical bearings pointed out. In answer to the doubts expressed by the Master of Trinity in a previous communi- cation, it was shown that the Sophista, as well as the Politicus, which is a continuation of it, are repeatedly referred to in the works of Aristotle. In particular, Arist. Metaph. v. ii. 93 was appealed to as evidence that Aristotle had not only read the dialogue called Sophista, but believed it to have been written by his master. The Dialogue was analysed, and shown to be a critique of the negative or Eristic systems of logic, derived from the Eleatics, which were taught by Euclides and Antisthenes, the founders of the Socratic sects of the Megarics and Cynics respectively. ' Many allusions, personal and otherwise, to Antisthenes were pointed out, as existing both in this dialogue and in the Theetetus, of which it is a professed continuation. The Theeietus was regarded as a critique of the contemporary psychology, and the Sophista as a confutation of the prevailing schemes of logic; and both were shown to contain exem- plifications of the twin processes of Induction or Collection, and Division or Classification, which constitute, according to Plato in the Phedrus, p. 265 E., the science or art of Dialectic. It was also argued, in opposition to Schleiermacher, that the Materialistic doctrines confuted in Sophista, p. 246, represent those of Antisthenes, rather than the atomic theory of Democritus, or the empirical system of Aristippus. The analysis of the simple Proposition (Soph. p. 262) was shown, by the testimony of Plutarch and others, to be Platonic, and the imperfect Idealists refuted in p. 276 were identified with the Megarics, and distinguished from the Platonists. — Passages were also quoted from the Politicus, showing the dis- ciplinary and educational uses to which the method of Division was made subservient in the teaching of the Academy ; and this teaching was further illustrated by a quotation of considerable length from a Comic Poet ap. Athen. lib. ii. Incidentally, Porphyry and Abelard were appealed to in evidence that Plato’s Method of Division was known to the Neo-Platonists and the Schoolmen, and recognized by them as characteristic of his Dialectic. 192 December 7, 1857. Professor Miller made a communication on the Planimeters of Wetli, Decher, and Amsler, and communicated the following simple proof of the principle of Amsler’s, due to Mr. Adams. P Let O be the fixed point, P the tracer, M Q the hinge, w W the centre of wheel, Q M the middle point of P Q, OQ=a, PQ=s, MW=c. oO The area of any closed figure whose boundary is traced out by P, is the algebraical sum of the elementary areas swept out by the broken line O Q P in its successive positions. Let @ and i be the angles which OQ, QP at any time make respect- ively with their initial positions. s the arc which the wheel has turned through at the same time. If now OQP take up a consecutive position, and 9, W, s receive the small increments d¢, dp, ds, we see that ds = motion of W in direction perpendicular to PQ. Hence motion of M in the same direction =ds+cdy, and there- fore the elementary area traced out by QP=4(ds+cdy). Also ele- mentary area traced out by OQ= jaro. Hence the whole area swept out by OQP in moving from its initial to any other position is « Za°p + be + bs. If OQP returns to its initial position without performing a com- plete revolution about O, the limits of @ and w are 0, and the area of the figure traced out by P is ds. If OQP has performed a complete revolution, the limits of ¢ and p are 27, and the area traced out is a(a?+ 2bc) + bs. A paper was also read by the Astronomer Royal, “On the sub- stitution of Methods founded on Ordinary Geometry for Methods based on the General Doctrine of Proportions, in the treatment of some Geometrical Problems.” The doctrine of proportions laid down in the fifth book of Euclid is the only one applicable to all cases without exception, but it is cumbrous and difficult to remember. It is therefore natural to attempt, in special applications of the doctrine, to introduce the facilities which are special to each case. This has been done long 193 since in the case of numbers, and this the author of this paper attempts in some cases in which geometrical lines only are the subject of consideration, by a new treatment of a theorem equivalent to Euclid’s simple ex equali and of the doctrine of similar triangles, referring to nothing more advanced than Euclid, Book II. The author proves,— 1. If the rectangle contained under the sides a, B be equal to the rectangle contained under the sides 5, A; and if these rectangles be so applied together that the sides a and 6 shall be in a straight line and that the side B shall meet the side A, the two rectangles will be the complements of the rectangles on the diameter of a rectangle. 2. If the rectangle contained under the lines a, B is equal to the rectangle contained under the lines 4, A; and if the rectangle under the lines 5, C is equal to the rectangle contained under the lines _¢,B; then will the rectangle contained under the lines a, C be equal to the rectangle contained under the lines ¢, A. (This is equivalent to the ordinary ex equali theorem. If a:6::A:B and 6:¢::B:C, then will @:¢::A:C.) 3. If two right-angled triangles are equiangular, and if a, A are their hypothenuses, and 4, B homonymous sides, the rectangle con- tained under the lines a, B is equal to the rectangle contained under the lines 5, A. _ (The equivalent theorem in proportions is a:b::A:B.) 4. If a, eand A, C are homonymous sides of equiangular triangles, the rectangle contained under a,C€ will be equal to the rectangle contained under c, A. 5. If b, ¢ and B, C are homonymous sides including the right angles of two equiangular right-angled triangles, the rectangle con- tained under 6, C will be equal to the rectangle contained under c, B. 6. If the rectangle contained under the lines a, B is equal to the rectangle contained under the lines 6, A; the parallelogram con- tained under the lines a, B will be equal to the equiangular paral- lelogram contained under the lines 6, A. (This is equivalent to the proposition, If a:b::A:B then a:b::A cos a:B cos a.) These propositions will suffice for the treatment of the first thirteen propositions of Euclid’s sixth book (Prop. I. excepted), and of all the theorems and problems apparently involving proportions of straight lines (not of areas, &c.) which usually present themselves. The author then proceeds, as an instance of their application, to prove by means of them the following theorem :— 194 If pairs of tangents are drawn externally to each couple of three unequal circles, the three intersections of the tangents of each pair will be in one straight line. Also a paper was read by Professor De Morgan, ‘‘ On a Proof of the existence of a Root in every Algebraic Equation: with an examination and extension of Cauchy’s Theorem of Imaginary Roots; and remarks on the proofs of the existence of Roots given by Argand and by Mourey.” The extension of Cauchy’s theorem is very easily found, when the proof is the first of those given by Sturm in Liouville’s Journal. The extended theorem is as follows :— Let oz be any function of z, and let z=a+y/—1. Let (2, y) be a point on any circuit which does not cut itself. Let this point describe the circuit in the positive direction of revolution; and, o(a+y/—1) being P+ Q. /—1, let 5 change sign & times as in +0—, and / times as in —O+. Let (a, y) be called a radical point when ¢(a+y/”—1) =0, or=a. Let there be m radical points of the first kind within, and m! upon, the circuit: let there be n radical points of the second kind within, and x! upon, the circuit. Then k—l=2m+m!—(2n+7'). Sturm’s demonstration of the case where m!/=0,2=0, n'=0, which is Cauchy’s theorem, assumes the existence of the roots of an algebraical expression. Mr. De Morgan’s proof of the existence of these roots is as follows:—He shows, @ priori, that in the se- quence of signs which Cauchy’s theorem requires to be examined, k—/ never undergoes any alteration except after 0 and @ have coincided, that is, where P=0, Q=0O, simultaneously. It is then easily proved that change in k—/ happens in every algebraical equation. The proofs given by Argand and Mourey were intended as illus- trations of the power of the extension which is now called double algebra. Stript of this interpretation, they are purely algebraical, and Argand’s proof is really that which was afterwards found by Cauchy. Argand’s proof is more simple in form than Cauchy’s. February 8, 1858. A paper was read by the Rev. O. Fisher, ‘‘On the probable origin of numerous Deep Pits on some Heaths in Dorsetshire.” Also a paper was read by Professor De Morgan, ‘On the Syllogism, No. III., and on Logic in general.” This paper is divided into two sections, the first of which is de- scriptive and controversial, the second is an abstract of the system. ° 195 Between the opinion of Kant that logic cannot be improved, and that of some recent writers, who hold it perverted, and not always correct, the truth is held to lie in this,—that existing logic, in its quod semper, quod ubique, quod ab omnibus, is true and accurate; but that it is only a beginning, and that the low estimation in which it has been held is a consequence of its incompleteness. The modern definition of logic, the form of thought, relates to a distinction which is more familiar to mathematicians than to logicians, but is rather in the common use of the mathematician than in his clear apprehension. Aristotle, who first implicitly made the distinction of form and matter, was a mathematician ; and so also was Kant, who first explicitly introduced this distinction into the definition of logic. The only two nations who had a logic taking character from the distinction, the Greeks and Hindus, are precisely the two nations to whom we owe the rudiments of our mathematics. It is affirmed by the author, that, in our time, the distinction is more in the theory of the logician than in his practice, more in the practice of the mathematician than in his theory. Various illustrations are given of the manner in which recent logical writers have, according to Mr. De Morgan, misconceived the distinction of formal and material. In another part of the paper he suggests that this distinction has been confounded with the distinction which he designates as onymatic and non-onymatic. By onymatic he means what arises out of the use and meaning of nomenclature : thus the relation of containing and contained is an important relation of names to each other as names, or an onymatic relation. The modern logic, by the simplicity of its final examples, is pre- vented from being of much use as a mental gymnastic. Instances are given of a proposition and a syllogism which are more worthy of being propounded as ezercises than the instances which are found in works on logic. The objections to symbols are discussed. Every science which has thriven has organized symbols of its own: and logic, the only science which confessedly has made no progress for many centuries, is also the only science which has grown no symbols. The logicians have confined themselves hitherto within what Mr. De Morgan calls the logico-mathematical field: they now begin to contend for the inclusion of what he calls the logico-metaphysical. This distinction they take as that of extension and comprehension. The author contends for a distinction of extension and intension in both the sides of logic, the mathematical and the metaphysical; though undoubtedly extension predominates in the mathematical side, and intension in the metaphysical. These distinctions are onymatic. If the name C contain all that is in A or in B, or in both, symbolized by C=(A, B), then A and B are in the extension of C. But if C be contained both in A and in B, symbolized by C=A-B, or AB, then A and B are in the intension of C. A name is used in four senses. It is the name of an object, or of a quality inhering in an object, and distinguishing a class : these twa B 196 uses are objective. It is also the name of a class, or of an attribute by which the mind thinks of a class: these two uses are subjective. The subjective uses are reductions of plurality to unity, a description for the truth and reality of which the author contends. It is affirmed that the logicians have not only confined themselves within the mathematical side of logic, but that even the recent attempts to introduce the metaphysical appear like attempts to create a second mathematical branch. This is evidenced by the manner in which unity of attribution has been discarded in favour of plurality of qualification. Thus it has been said, in obedience to the theory of quantification of the predicate, that the humanity of Newton is a different thing from the humanity of Leibnitz. That this view, though true, belongs to the mathematical side of logic, is contended for and enforced at length. Aristotle made the distinction which the logicians now recognize as that of extension and comprehension, and which Mr. De Morgan distinguishes as that of mathematical and metaphysical reading, as follows:—In one sense the species is in the genus: in another the genus is in the species. Thatis, all the species are aggregants of the genus: the whole genus is a component of the notion of the species. Recent English logicians of high name have misconceived this distinction to the extent of imagining that by changing ‘ Every A is some B’ into ‘Some B is every A,’ they make the change alluded to by Aristotle. Mr. De Morgan restores the old distinction, and completely incorporates what was only partially introduced, the change of quantity which takes place in passing from the mathe- matical to the metaphysical reading. Thus ‘ Every A is B’ is in the first reading ‘'The whole class A is one aggregant of the class B’; and in the second, ‘ The whole attribute B is one component of the attribute A.’ The limitation of the universe of a proposition, made throughout the author’s preceding writings, is again contended for. A proposition is the assertion or denial of a relation between two notions. Relations which are of necessity involved in nomenclature, are called onymatic; and these must be first studied. Mr. De Morgan believes that the logicians have described, under the di- stinction of formal and material, no more than the distinction of onymatic and non-onymatic. ‘The mathematical notion of class, and the metaphysical notion of attribute, give four different readings of a proposition :—1. Logico-mathematical, class aggregate of class ; man contained in animal. 2. Logico-physical, attribute predicated of class; animality attribute of the class man. 3. Logico-meta- physical, attribute component of attribute; animality a component of humanity. 4. Logico contraphysical, attribute subjected to class ; humanity only predicable within the class animal. The logicians confined logical predication to the idea of class contained in class, species in genus. The genus in species, attribute component of attribute, they relegated to metaphysics. Hence their distinction of the logical and metaphysical whole. ‘The class composed of individuals they called the mathematical whole: Mr, 197 De Morgan calls it the arithmetical whole, transferring the word mathematical to what was called the logical whole. The common mode of expression, as ‘Every A is B,’ &c., he considers as speaking the language of the arithmetical whole, though the speaker may attach the idea of either of the other wholes. Extension predominates in the mathematical whole ; intension in the metaphysical. - ‘Ihe most usual mode of speech is the physical : man is educated a mathematician as to the subject of his proposition, a metaphysician as to the predicate. The most remarkable point at issue between Mr. De Morgan and the logicians, is in his opposition to their notion of the whole attribute being the sum of its components. The difference between aggregation and composition is one of the turning-points of his whole system. The distinctions above drawn require differences of language to express the relations which enter : the logicians have nothing but the copulais. At the outset, however, we have the distinction which is expressed by speaking of relations of terminal ambiguity and relations of terminal precision :—the first seen in ‘ A is contained in B,’ where it is left unsaid whether or no A fills B; the second seen in the case in which it is implied that A is part only of B. By. speaking in the arithmetical whole, the logicians have made a system of syllogism from which the numerical syllogism cannot be _ excluded. The propositions ‘Some As are Bs,’ and ‘50 As are Bs,’ are of the same kind: they are both referred to the arithmetical whole. This whole is subordinate to both the mathematical and metaphysical wholes; though more prominent in the first than in the second. When inclusion and exclusion are opposed to one another, and combined with assertion and denial, the ordinary proposition takes a form in which quantity is but an emergent incident, and not a fundamental mode of discrimination. Thus the propositions A and O are the assertion and denial of the inclusion of class in class ; E and I are the assertion and denial of the exclusion of class from class. The opposition of the two kinds of quantity, extensive and inten- sive, is not easy and natural, when the word quantity is used in metaphysical reading. Mr. De Morgan proposes the word force to express quantity in the second case. He finds this word in use. Thus it is sometimes said that a term is or is not used in its com- plete force, when the meaning is, that all the attributes of which the term is compounded are or are not involved in the use made of the term. ‘This is, according to Mr. De Morgan, one of the cases ‘in which the logical system of the world at large has got beyond that of the logicians. The spicular notation of the former papers is extended: the signs ) and } being used to distinguish mathematical and metaphysical reading. Then X))Y signifies that the whole class X is contained in the class Y; X]]Y signifies that the whole attribute Y is a com- ponent of the attribute X. a The syllogism is the deduction of a relation from the combination of two others. By distinction of the mathematical and metaphysical, 198 of the terminally ambiguous and terminally precise, four modes of combination are obtained. Logicians have but the copula is for all cases. Mr. De Morgan proposes to use a complete system of schetical terms, by which the combination of relations shall be ex- hibited. Leaving out the cases of terminal precision, which are more complex and less usual, the two kinds of reading under which the common syllogism is included are as follows :— Terminal Ambiguity. Mathematical reading. Relation of Class X to Class Y. The class x is the contrary of X, or contains all the rest of the universe. Proposition. X—of Y. Y—of X. Notation. Assertion of X containedin Y | Species Genus X))Y Denial of X contained in Y Exient Deficient X(.(Y Assertion of X excluded from Y | Coexternal | Coexternal | X).(Y Denial of X excluded from Y | Copartient | Copartient | X()Y Assertion of x containedin Y | Complement | Complement | X(.)Y Denial of x contained in Y Coinadequate | Coinadequate| X)(Y Assertion of x excluded from Y | Genus Species X((Y Denial of x excluded from Y _| Deficient Exient X).)Y Terminal Ambiguity. Metaphysical reading. Relation of attribute Y to attribute X. Y—of X. X—of Y. Notation. Assertion of Y acomponent of X_ | Essential Dependent |X]]Y Denial of Y a component of X Non-essential| Independent} X[.[Y Assertionof Yincompatible with X/Repugnant |Repugnant | X].[Y Denial of Y incompatible with X |Jrrepugnant |Irrepugnant | X{]Y Assertion of Yacomponent ofx Alternative | Alternative | X[.]Y Denial of Y a component of x Inalternative | Inalternative| X][Y Assertion of Y incompatible with x} Dependent /Essential |X[[Y Denial of Y incompatible with x | Independent | Inessential | X].]Y The extension of the four forms to eight, the notation, &c., are treated in the second paper on syllogism. ‘The two sets contain the same propositions, differently read; and the quantities in the two are different. In the first reading X) and (X denote X taken uni- versally in extension; X( and )X denote X taken particularly. In the second reading ]X and X[ are universals, X] and [X are parti- culars. ‘I‘hus, when we say that the classes X and Y are copartient, or in common language ‘some Xs are Ys,’ denoted by X()Y, both X and Y have particular quantity in extension. In saying this we also say that X and Y, as attributes, are irrepugnant, or not incompatible, denoted by X[]Y. But the intensive force of both X and Y is uni- versal; no one attribute of X is repugnant to any one attribute of Y. The syllogism denoted by X ))Y)(Z contains the assertions that 'X is a genus of Y and Y a coinadequate of Z, (Y and Z not together filling the universe), The conclusion is X)(Z, X is a coinadequate of 199 Z, and the combination of relations is seen in—Every species of a - coinadequate is a coinadequate. In metaphysical reading, we have X))Y)[Z, X is a dependent of Y, Y aninalternative of Z. The con- clusion is X}][Z, X is an inalternative of Z, and the combination of relations is seen in—The dependent of an inalternative is an inalter- native. When the terms become as familiar as genus and species, the axiomatic character of the combination is as clearly manifest as in—Species of species is species. Mr. De Morgan gives the following instance of a good inference which would probably not be seen with ease in its present form, though the phrases are not technical: ‘‘We must not say that either bodily strength or meanness is a necessary alternative, for courage and meanness are incompatible, while courage does not depend on bodily strength.” And he maintains that the educated world has made considerable advance in the use of rela- tions of attributes, though the logician has nothing but what he calls the arithmetical abacus on which to exhibit the process. Some modern logicians have so completely fallen into the mathema- tical view of quantity, that there is a school which treats all thought — as relation of more and less. Mr. De Morgan opposes this view. The second part of this paper, being a non-controversial summary of Mr. De Morgan’s system, so far as onymatic relations are concerned, hardly admits of abstract. Its principal points have been touched on. In a postscript, such notice is taken of the late Sir W. Hamilton’s criticism on Mr. De Morgan’s second paper as circumstances re- quire and will allow. - February 22, 1858. Dr. Donaldson, of Trinity College, read a paper ‘‘ On the Statue of Solon mentioned by A%schines and Demosthenes.” The object of the author of this paper was to fix the age and subject of a beautiful statue in the Museo Borbonico at Naples, which was recovered from the ruins of the theatre at Herculaneum. This statue has generally been regarded as representing Aristides the Just, the son of Lysimachus ; and one attempt has been made to show that it is a portrait of A®lius Aristides, the rhetorician, who was born 38 years after the destruction of Herculaneum in a.p. 79. Amore plausible hypothesis, supported by great names, considers the statue as a portrait of Aischines. But this rests on a palpable misconception. After refuting these theories, the author undertook to show that the statue was probably a copy of that erected in honour of Solon in the agora at Salamis, and mentioned in a striking manner by A®schines (c. Timarch. p. 4) and Demo- sthenes (De Fals. Leg. p. 420). This was argued from the peculiar and distinctive attitude; from the fact that the treatment of the drapery accorded with that belonging to the school of Scopos, and the costume corresponded to that of the epoch (about fifty years before s.c. 343) assigned to the statue of Solon by Demosthenes ; and from the suitableness of a statue of Solon, who was an elegiac 200 poet as well as a legislator, to the place where the statue was found, namely, the theatre at Herculaneum. Attention was also directed to the improbability of a later appropriation of a statue in such a peculiar posture, and Dion Chrysostomus was cited to show that even the Rhodians, who had adopted the practice of altering the in- scriptions of honorary statues, abstained from interfering with those which were defined, not only by the name, but by the characteristics of the person represented. March 8, 1858. i A paper was read by Dr. Paget, ‘‘ On some Instances of remark- able Defects in the Voluntary Muscles.” Four original cases, in which large and important muscles; such as the pectorals, were wholly absent or in a state of extreme tenuity; the defects either congenital, or existing from early in- fancy; limited to certain groups of muscles, and unaccompanied with any defect or deformity of the bones. In three of the cases, the effects symmetrical; in the fourth confined to one side of the chest. Enormous development of the calves in one of the cases. Also a paper was read “On Organic Polarity,” by H. F. Baxter, Esq. The object of the paper is to show the intimate connexion that exists between organic force and the ordinary polar forces, such as chemical force, for example. The principal experiments, showing that organic action, viz. secre- tion, is accompanied with the manifestation of current force, have already appeared in the Royal Society’s ‘ Transactions’ for the years 1848 and 1852; but in the present communication the author enters more minutely into the resemblance between the actions which take place in the voltaic circle and those that occur during secretion than could be prudently attempted in his previous papers. But whatever view may be entertained in regard to the origin of the power in the voltaic circle, whether by mere contact or by chemical action, the decision of this point is of no importance to the question under consideration; since the manifestation of current force during voltaic action is allowed both by the chemical theorist as well as the contact theorist; and if we admit the manifestation of this force (current force) to be evidence of polar action. in one class of cases, viz. during voltaic action, we are certainly justified in logically concluding that it may be adduced as evidence of polar action in other cases also, viz. during organic action as in secretion. Reference is made to Prof. Graham’s researches on osmose. Accord- ing to Prof. Graham, osmose would appear to be dependent upon chemical action, and consequently, should we be disposed to class the phenomena of secretion with those of osmose,~we should be thus compelled to acknowledge that the act of secretion must be polar in its nature. a ¥ peas: aby PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. April 26, 1858. Professor Challis made a communication ‘‘ On the Annular Eclipse of the Sun, March 15, 1858.” The Master of Trinity read a paper “‘ On Barrow, and his Acade- mical Times.” May 10, 1858. Professor Miller made communications—(1) ‘‘ On an improved Method of finding the position of any face of a Crystal belonging to the Anorthic System.” (2) ‘‘ On the direction of the Axis of a Zone.” (3) On a substitute for the Reflective Goniometer.”’ Mr. Godfray read a paper “ On a Chart and Diagram for facilita- ting Great-Circle Sailing.” It has long been known that in many cases a very great saving of distance would be effected by guiding the ship along the arc of a great circle instead of following the rhumb ; but the tedious calcula- tions which great-circle sailing requires, and the difficulty of tracing the track on the chart, have hitherto stood in the way of its adop- tion. The great advantage which Mercator’s sailing offers in these respects sufficiently explains the preference given to it, even by those who are fully aware of the longer route it obliges them to follow. The object of this communication was to show how the same ad- vantages are secured to great-circle sailing by the adoption of a chart on the central or gnomonic projection, which, with the addition of - a diagram, solves the problem of this sailing with the same facility as Mercator’s chart does for sailing on a rhumb. The great-circle track becomes a straight line joining the two places; and this being drawn, the various courses and the distances to be run upon each are obtained, as also the distance from the ship to her destination, by a mere inspection of the diagram and without calculation. No. XIV.—Proceepines oF THE CAMBRIDGE PuiL. Soc. 204 The great-circle track is not always practicable, on account of its taking the ship into too high a latitude, where the ice would render it dangerous or impossible to penetrate. When this is the case, the same chart and diagram will, with just as much facility as before, point out that which, under the circumstances, is the shortest route. Some parallel of latitude is fixed upon for the maximum, and the track to be followed will then consist of a portion of that parallel and of the portions of two great circles which are tangents to it,— one passing through the ship, the other through the destination. On this great-circle chart the track will be the two straight lines drawn from the two places, so as to touch the circle of highest lati- tude and the part of this circle between the points of contact. The paper explains the construction of the chart and diagram, and illustrates their use by two examples: the first, from the south- ern extremity of Africa to Perth in Australia, which shows a gain of 204 miles; the other, from 30° S. lat., 18° W. long. to Melbourne, gives a gain of 1130 miles, without going into a higher latitude than 55° south. . A chart and diagram on this principle have been engraved by the Hydrographic Office, Admiralty. May 24, 1858. The Master of Trinity read the conclusion of his paper ‘‘ On Bar- row, and his Academical Times.” November 8, 1858. Professor Challis made a communication ‘‘On Donati’s Comet.” November 22, 1858. The Public Orator gave a lecture on ‘‘ The Battle of the Trebbia,” the object of which was to compare the different authorities, and to illustrate them by a description of the neighbourhood of Pia- cenza, which he had recently visited. He showed that Polybius’s narrative was the most valuable, as being in all probability the ori- ginal source of all subsequent accounts, and as deriving an especial interest from the author’s personal knowledge of the localities. Livy in the main followed Polybius, amplifying and varying the details rather with a view to an effective and ornamented style than to the actual truth of his statements. Some particulars, however, appeared, in this part of his work as elsewhere, to have been borrowed from L. Cincius Alimentus, or some other historian of the Second Punic 205 War. Cornelius Nepos, from his brevity, and Appian, from his in- accuracy, were not worthy to be taken into account. No modern historian appeared to have visited the place. The principal points to be determined were (1) the situation of Scipio’s camp after he had abandoned his position in the immediate vicinity of Placentia, and (2) the identification of the deep watercourse where Hannibal placed Mago with two thousand men in ambush on the morning of the battle. Scipio’s camp was, beyond doubt, at or near Rivalta, a castle and hamlet situated on a ‘high bank’ (as the name imports) on the further side of the Trebbia, about nine miles south-west of Piacenza. The ambuscade was placed in the watercourse called the Trebbiola, a small stream, of which the banks were from 6 to 8 feet high, about six miles from Piacenza, above the place called ‘ La Volta di Settima.’ The passages of Polybius, to which reference was made in the lecture, are in Book III. chapters 66 sqgq.; those of Livy, in Book XXI. chapters 47, 48, 52-56. The rude plan given below may make this abstract more intel- ligible. December 6, 1858. “ Suggestion of a proof of the Theorem that every Algebraic Equa- tion has a Root.” By G. B. Airy, Esq., Astronomer Royal. In this paper the equation to be discussed is expressed under the form dary + bere +... +m, =0, where a,=a(cos «+ Vv —lsina), re=r(cos 0+ VY —I sin 0), 2. vor, 206 as it may be written, P+ 7 —1Q=0; and the object is to show that there will be at least one value of r between 0 and positive in- finity, and one value of @ between Q and 27, which, used in combi- nation, will make both P and Q =0. This is effected by constructing two curves whose common ab- scissa is 0, and whose ordinates are respectively the corresponding values of P and Q, produced by substituting in their expressions the same value of 7, and observing the change which takes place in the form of these curves, and in the position of their points of intersec- tion, as r successively assumes all values from 0 to positive infinity. The existence of a root will be indicated by a point of intersection of these curves (the P-curve and Q-curve, as they may be called) falling on the axis of abscisse. When r=0, each of these curves will be a straight line parallel to the axis of abscisse. When r=, the cor- responding values of P and Q will generally be indefinitely great ; but by reducing their values in the same proportion, which will not affect the validity of the demonstration, the Q-curve will become a line of sines, and the P-curve a line of cosines, ‘or a line of sines drawn back through 7 On constructing these curves, which we may call respectively P(0), Q(0), P( ), (Q o ), it will be remarked—. (1) That P(O) and Q(0) do not intersect. (2) That P(o ) and Q( ) intersect in two points. (3) That one of these points of intersection is above the line of abscissze, and the other is below it. On considering the change in the forms of the P-curve and Q-curve as r increases from 0 to infinity, it will be seen that the P-curve must have intruded on the Q-curve, at first by simple contact; and that, as the intrusion advances, the simple contact is changed into two in- tersections, which will at first be on the same side of the line of abscissee. But as, where r is indefinitely increased, any two conse- cutive intersections necessarily lie on opposite sides of the line of abscisse, it may be shown, by considering the various ways in which the intrusion may take place, that in all cases one at least of the intersections must have crossed the line of abscisse during the in- crease of 7; and a root is thus determined. A communication was also made by Professor Miller ‘‘ On the contrivances employed by M. Porro in the construction of instru- ments used in Surveying and Astronomy.” February 14, 1859. Mr. Humphry made a communicatisn ‘“‘On the Limbs of Verte- brate Animals.” He gave a brief description of the fore and hind limbs in the 207 several vertebrate classes, directing attention to the tripartite divi- sion of their distal segments, and to the uniformity of plan upon which they are constructed. He argued that this uniformity has relation to the mode in which their development proceeds, and to _ the similarity of their functions rather than any adhesion to an **ideal archtypal pattern ;”’ and expressed his belief that the exist- ence of such an ideal in the minds of anatomists proves in some measure a hindrance to the full study of the laws which regulate the formation of animal bodies. The differences between the fore and hind limbs were shown to depend,—/first, upon the fact that the hind limbs are required to pro- pel as well as to support the trunk, and less variety of movement is needed in them; hence they are larger and firmer, and the same bone of the leg forms a main constituent both of the ankle and of the knee-joint; secondly, in walking and running the fore limb is ex- tended in front of the trunk, and draws the latter after it during its flexion, whereas the hind limb is bent up beneath the trunk and drives the latter on before it during its extension. This antagonism in the mode of their action leads to an antagonism in construction of the two limbs,—of their upper segments at least, the posterior aspect of the one corresponding with the anterior aspect of the other. The antagonistic relations are brought about by a partial rotation in the long axis of the two limbs which takes place in opposite directions ’ during development; and coincident with the rotation of the proxi- mal segments of the fore limb in one direction is a rotation of its distai segment in an opposite direction, so as to turn the palm towards the ground. Reasons were given for regarding the scapular and pelvic arches as formed by mpdifications of the hemal, and not of the pleural, parts of the vertebre; and for believing that the scapular arch be- longs, not to the occipital bone, but to the vertebra of the fore part of the chest or of the hinder part of the neck. February 28, 1859. Dr. Donaldson read a paper “On Plato’s Cosmical System as - exhibited in the tenth book of ‘ The Republic.’ ” The author first gave a translation of the whole passage (Plato, Resp. x. 616 B, 617 E), accompanied by a critical and philological examination of the Greek text. He then undertook to show the connexion between the fanciful picture of the universe which Plato has here given, with his other speculations on the origin of things, and especially with the occult philosophy of numbers. And he argued, finally, that the tradition preserved by Clement of Alexandria, which identifies Er, the son of Armenius, with Zoroaster, rests upon a foundation in fact; and that while there is good reason to believe that the doctrines of Heracleitus and Zoroaster agreed in many essen- tial particulars, and that Plato was well acquainted with the specu- lations of the Ephesian philosopher, there are certain particulars in 208 the cosmical myth of ‘The Republic’ which agree exactly with the theories known to have been common to Zoroaster and Heracleitus. March 14, 1859. A paper was read ‘‘ On the general principles of which the Compo- sition or aggregation of Forces is a consequence.”’ By Prof. De Morgan. This paper examines the fundamental grounds of the composition or, as Mr. De Morgan calls it, aggregation of forces. By a tendency is meant anything which has both magnitude and application: by ap- plication is meant any notion which, not presenting the idea of mag- nitude, presents the idea of opposition. ‘Two tendencies have a third tendency for their aggregate, to which they are jointly equivalent : and equivalence is any notion which, given that things equivalent to the same are equivalent to one another, satisfies the following postulates, which are the grounds of every method of aggregation known in mechanics. 1. Any two tendencies have one aggregate (0, the aggregate of counteraction being included among possible cases), and one only. 2. The magnitude of the aggregate, and its application relatively to the applications of the aggregants, depend only on the relative, and uot on the absolute, applications of the aggregants. 3. The order in which tendencies are aggregated produces no effect either on the magnitude or application of the aggregate. 4, Tendencies of the same or opposite applications are aggregated by the law of algebraical additions. From these postulates follow the following theorems :— 5. In any aggregate, the result of partial aggregation may take the place of its own aggregants. 6. Two tendencies cannot counteract one another unless they have equal magnitudes and opposite applications. 7. An aggregate has not more than one pair of aggregants, when the applications of the aggregants are given, and are different. 8. If the aggregants be altered in any ratio, without change of application, the aggregate is altered in the same ratio, also without change of application, 9. Any tendency may be disaggregated into two of any two dif- ferent applications, neither of which is its own. From the preceding it is proved,— 1. When by application is meant direction, the law of aggregation must be the well-known law of the aggregation of forces meeting at a point. 2. When by application is meant choice of a point through which a given direction is to be drawn, the law of aggregation must be the well-known law of aggregation of parallel forces. In the case of translations and rotations, the postulates are all laws of thought; in the case of pressures, whether divergent or parallel, whether equilibrating or producing motion, all the postu- 209 lates contain results of experience. Accordingly, the multifarious proofs of the laws of aggregation, in the case of pressures, are not the mathematical playthings which they are often supposed to be from their grounds being insufficiently stated. If to the postulates necessary to make the law of aggregation a consequence, be added the following, ‘‘ the velocity due to the aggregate of pressures in one given direction is the aggregate of the velocities due to the pressures taken separately,” it follows, as a mathematical consequence, that the pressure varies as the velocity created by it in a given time. This great law of dynamics, therefore, is not fundamentally distinct from the laws of aggregation, but follows from them with the addition of a postulate more simple than itself. _A proof was also given by Professor Stokes, of the theorem that. “ Every Equation has a Root.” May 2, 1859. Professor Miller made a communication “On the employment of the Gnomonic Projection of the Sphere in Crystallography.” May 16, 1859. Mr. Hopkins gave a lecture “ On Glacial Theories.” De Saussure attributed the motion of glaciers to their sliding over the bottoms of the valleys in which they exist, but did not appear to have made any observations on the change of form to which the glacial masses may be subject during their motion. No advance was made in our knowledge of glaciers for many years after De Saussure’s death, till certain Swiss observers, Charpentier and others, rather more than twenty years ago, added much to our knowledge of the subject. Among these glacial observers, M. Agassiz, with characteristic zeal and activity, soon afterwards took a prominent ition. He and those who were associated with him, or had pre- ceded him, brought forward incontestible evidence of the former ex- tension of glaciers in the Alps, and of their efficiency in the trans- port of enormous angular fragments of rock from their original sites to other localities, not only in the same Alpine valley, but even on the flanks of the Jura on the opposite side of the great central valley of Switzerland. These were the great facts which bore upon specu- lative geology. Glaciers were still engaged in the work of trans- port, and they had been so on a much larger scale at some former epoch; the masses of ice of which they were composed had been then of much larger dimensions, and consequently the mean climate of Western Europe must have been at that period considerably lower than at present. No one was so active as M. Agassiz in pressing these facts on the notice of geologists, or msisted more strongly on their geological importance; and by the energy of his own character, his great reputation, and extensive personal acquaintance with men 210 of science, he had undoubtedly made the first great steps in giving currency to the glacial theories of geology—theories which, though viewed at that time with much distrust, had since, with such modi- fications as enlarged knowledge and sober judgment had imposed on them, been universally recognized by geologists. For his unflinching advocacy of the glacial theory in its broad out- lines, geologists had been unquestionably much indebted to M. Agassiz; but in his physical theories respecting glacial phenomena, he had not shown that caution or acquaintance with physical science which the subject demanded. With respect to the motion of glaciers, it may be sufficient to state that he regarded it as due to the infil- tration and subsequent freezing of water within the glacier, and a consequent expansion of its mass, by which the glacier in general, and especially those portions near its lower extremity, were urged forwards in the direction in which the bed of the glacier descended. Few persons ever received this theory, and it is no longer considered as deserving of serious attention. A few years later Professor Forbes commenced his researches among the Alpine glaciers. His ‘Travels through the Alps’ was published in 1848, and contained a greater amount of well- arranged information respecting glacial phenomena than perhaps all other works together on that subject. But in this lecture, Mr. Hop- kins remarked, he professed to deal with theories, and not with de- scriptive details. M. Agassiz’s second work on glaciers, his Systeme Glaciére, also appeared in 1847. Professor Forbes introduced a new view of the motion of glaciers, which he attributes to a certain fa- cility with which he supposes glacial ice to be capable of changing its form under the pressures to which it is subjected, in a manner similar to that in which a viscous mass would change its form under the same circumstances. Hence it was called the viscous theory. It was founded on the fact (distinctly ascertained, Mr. Hopkins be- lieved, the same year, both by Agassiz on the glacier of the Aar, and by Prof. Forbes on the Mer de Glace) that the central portions of a glacier moved considerably faster than its lateral portions, as a viscous mass would move along a trough inclined at a small angle to the horizon ; and, moreover, it was obvious that the general mass of a glacier did so change its form as to accommodate itself to the changing dimensions of the valley down which it moved. On the other hand, it was contended that a substance so hard and brittle as glacial ice could not be said to have the property of visco- sity, and that the different velocities of the central and lateral por- tions of a glacier, and the changes of form which the general mass might undergo, were more attributable to the formation of crevasses and to discontinuous ruptures of the mass, than to any continuous change of form in each infinitesimal portion of it, like that which takes place in a mass which can be properly termed viscous. ‘That this view was partly true was obvious, since ruptures and crevasses were actually formed by the unequal motions of different portions of the mass. Those who maintained this latter view held that the glacier moved by actually sliding over its bed; while those who sup- 211 ported the viscous theory contended that there was no such sliding motion, or if it existed at all, it constituted but a small part of the whole observed progressive motion of the surface of the glacier. In the warmth of discussion these theories came to be considered as more antagonistic than they really were. It was manifestly possible that the lower surface of the glacier might slide, and thus cause a part of the observed motion of the upper surface; which might also have an additional motion, due to the more rapid pro- gression of the upper portions of the mass as compared with that of its lower portions retarded by friction, as in the case of a semifluid mass. On this point Mr. Hopkins quoted the following passage from one of his letters ‘‘ On the Mechanism of Glacial Motion,” ad- dressed to the editors of the Philosophical Magazine in 1844-45. If observations ‘‘ should concur in showing an approximate equality in the motions of the upper and lower surfaces of a glacier, every candid and impartial mind must admit, I conceive, the sliding in preference to the viscous theory; but if, on the contrary, it should be proved that the velocity of the upper bears a large ratio to that of the lower surface, the claims of the latter theory must be at once admitted.” Since this was written, several observations had been made by dif- ferent persons, which agreed in showing that the upper surface of a glacier does move faster than the lower surface ; but the only obser- vations Mr. Hopkins had met with which enabled us to compare the actual amounts of those motions, had been made by Prof. Forbes himself near the extremity, Mr. Hopkins believed, of one of the gla- ciers at Chamouni. The result was that the upper surface moved about twice as fast as the lower one, thus proving that in this in- stance the motion of the upper surface was due in nearly equal de- grees to the two causes above mentioned, and that both theories had ‘so far equal claims to be admitted. But at present no one probably doubted the fact of the whole motion of a glacier being made up of that motion which it derives from the property hitherto usually designated as the viscosity or plasticity of its mass, and that which consists of a sliding over its bed. Dr. Tyndall had recently observed proofs of this latter motion in various parts of glaciers as well as near their lower extre. mities ; and all the phenomena of polished and striated rocks indi- cate most clearly that such motion must have existed in the ancient glaciers to which such phenomena are referred. But how was it conceivable that a glacier should thus slide over a surface on which there must be many and considerable inequalities, and at inclinations sometimes not exceeding 2° or 3°? And if it did thus slide, how was it that it did not move, as bodies ordinarily move down inclined planes, with an accelerated motion? These questions were frequently dwelt upon formerly. They were completely answered by the expe- riments made by Mr. Hopkins, and described by him in the ‘ Trans- actions’ of this Society in 1844 (vol. viii. part 1), and in his first letter ‘On the Motion of Glaciers,” dated November 19 of that year, and inserted in the Philosophical Magazine. The motion in question was not at all analogous to that of a body descending down 212 an inclined plane and retarded by friction as a constant force; it was due to the fact of the cohesion of the constituent particles of the mass at its lower surface being insufficient to resist the tendency of such an enormous weight of ice to descend down a plane even of very small inclination. A continuous disintegration is thus pro- duced, promoted probably, in a greater or less degree, by a constant but very gradual thawing of the ice at the lower surface. In this manner it is easy to see that the motion must depend on the rate of disintegration, and therefore must be nearly a uniform, and not an accelerated motion. Professor Forbes had an undoubted claim to the credit of being the first to suggest and insist upon the capability of the general mass of a glacier to change its form under existing conditions, as a cause of glacial motion. The above explanation of the sliding of the mass Mr. Hopkins claimed for himself. Still it was felt that further investigation was required respecting the property of glacial ice which had been designated as its viscosity. There was no conclusive evidence that glacial ice would bear any considerable extension without breaking, for numberless crevasses were formed wherever the ice appeared to be subjected to any great extending force. Again, it was equally certain that the contiguous portions of a dislocated glacial mass, though retaining their perfect solidity, did become reunited into one continuous and unbroken mass. These facts were not sufficiently explained by the assertion that glacial ice was viscous. ‘The true explanation appeared to have been afforded by an observation made some time ago by Dr. Faraday, and the more recent experiments of Dr. Tyndall. The former observed that two pieces of ice in perfect contact would freeze together so as to become one perfectly continuous mass, though the surrounding temperature should be much higher than 32°; and the latter gentleman had shown, by a striking form of the experiment, the extreme facility and rapidity with which a piece of common ice, after being crushed and broken into numberless frag- ments, will reunite into one continuous mass of transparent ice. This process had been designated by the term “ regelation;”’ and manifestly some corresponding term was required to designate the property which, in ice, rendered that process so complete. Such terms as viscous and plastic failed to express adequately the property in question. At the same time it should be remarked that, so far as glacial motion depended on the facility with which the glacial mass might change its form, the manner in which that change was effected was of secondary importance, and did not diminish whatever value attached to Professor Forbes’s first recognition of this change as an important cause of glacial motion. In one respect, however, the mechanism of the motion would be in some measure affected. Dr. Tyndall contends, and in a paper recently presented to the Royal Society has collected a considerable amount of evidence to show, that glacial ice would bear no more linear extension, independently of lateral compression, than ordinary specimens of ice would lead us to suppose, and consequently, when acted on by extending forces, it Peart 213 cracked, forming fissures and crevasses to a much greater extent than would seem consistent with any property to which the term viscous could be applied with strict propriety. Professor Forbes was also the first to make known to us, by systematic and well-directed observations, the facts and laws of the veined structure in glacial ice. He also entered into elaborate specu- lations on the causes which produced this structure, both in his ‘Travels in the Alps,’ and in letters written subsequently. Dr. Tyndall has also put forward a theory suggested by an analogy be- tween the veined structure of ice and the lamination of rocks. As there appeared to have been some confusion as to the differences between these two theories, Mr. Hopkins would endeavour, as far as he was able, to explain them. Both these theories depended primarily on the internal tensions and pressures to which the glacial mass might be subjected. The different parts of a glacier, as was well known, move with different velocities, the most general law being that the central move faster than the lateral portions; but whatever may lead to this unequable motion, its manifest result must be a tendency to drag the slower- moving portions of the ice after those which move more quickly. Moreover, it was easy to see that, in certain directions, this drag- ging might be greater on one portion of the ice than on a contiguous portion, and might thus tend to give different motions to contiguous vertical slices of the mass. ‘This difference of motion, or differential motion, was supposed by Professor Forbes to actually exist, and that ruptures or breaches of continuity were absolutely produced between these vertical thin slices of ice by the strain upon them, and that these ruptures gave rise to the veined structure. His first idea ap- peared to have been that water infiltrated into the small fissures thus formed, where it afterwards froze and formed the veins of blue trans- parent ice, while the intermediate vertical laminz contained a suffi- cient quantity of air-bubbles to render them white and opake. This notion of infiltration appeared to have been subsequently given up by the Professor, the conversion of the opake into transparent ice being supposed to take place by pressure and differential motion, indepen- dently, as far as Mr. Hopkins understood, either of infiltration or the melting of any portions of the ice. Both Professor Forbes and Dr. Tyndall had endeavoured to eluci- date the phenomena of glacial motion by means of a semifluid sub- stance descending down a trough inclined to the horizon. For the purpose of ascertaining the direction of greatest extension and com- pression of the substance when thus put in motion, the latter gen- tleman described circles on its surface while still at rest, and ob- served the compressions and extensions of the radii when the mass was in motion. He thus found that the lines of greatest extension were inclined at angles of 45° to the axis of the trough; each such line pointing centrally and downwards, or laterally and upwards, while the lines of greatest compression were perpendicular to them. In the case of a glacier descending a canal-shaped valley, the former of these directions would manifestly be that of greatest tension; 214 and this is precisely the result which Mr. Hopkins had obtained both by exact mechanical reasoning and by experiment fifteen years ago ; and it was thus that he was able to explain (and he was the first to do so) the formation of crevasses making angles of 45° with the axis of the glacier, and directed centrally and upwards, 7. e. at right angles to the lines of greatest tension. There was also another result to which Mr. Hopkins was led by an exact consideration of the problem, but which he had also elucidated by experiments, as described in the letters above alluded to in the Philosophical Maga- zine. ‘There were not only directions of maximum and minimum pressures and tensions at each point of the mass, there were also two other directions inclined at 45° to the former, not recognized by either of the above-mentioned experimenters, in which there is a - maximum tendency to produce the differential motion, to which Prof. Forbes ascribed an actual rupturing of the ice and consequent forma- tion of the veined structure. In the directions of maximum and minimum pressures or tensions, this tendency to produce a differ- ential motion altogether vanished. The truth of these results was just as certain as that of the parallelogram of forces. In more com- plicated cases than that above supposed of a glacier descending down a canal-shaped valley, the absolute directions of these differ- ent lines would vary with the conditions of the glacial mass, and the external pressures to which it was subjected; but the important fact was, that there must in all cases exist at each point of the mass a direction of maximum tension or of minimum pressure, and a direction, perpendicular to it, of minimum tension or maximum pressure, and two other directions inclined to each of the former at 45°, along which there is a maximum tendency to produce the kind of differential motion above described. It was moreover manifest that where crevasses were formed there must be tension; and equally manifest that the directions of such crevasses must at least approximate to perpendicularity with the directions of maximum tension, and therefore to coincidence with those of maximum pressure. Also, if discontinui- ties and differential motions resulted from these internal pressures and tensions, they must be produced in those directions in which there is the maximum tendency to produce them, 7. e. in directions inclined at 45° to those of the crevasses. All these directions might be supposed to be (as they would generally be in a glacier) nearly horizontal. ‘These conclusions, Mr. Hopkins repeated, were as cer- tain as that of the parallelogram of forces, and no theory which con- tradicted them could possibly be true. Professor Forbes was the first to recognize the law which esta- blishes a certain relation between the veined structure and the cre- vasses. He asserts that, as a matter of observation, the crevasses intersect the structure at right angles; consequently the blue veins must be perpendicular to the directions of maximum pressure, and could not coincide (such being the law) with the directions in which differential motion must necessarily take place, if it should take place at all. The law above enunciated exactly accorded with the conclu- sions above stated, and also with Dr. Tyndall’s views, who asserts, 215 from his own observations, that the laminz of blue and white ice (and especially in those places in which the structure originates) conform to the law of perpendicularity to the direction of greatest pressure. So far from there being any tendency to produce ruptures and fissures lying in the planes of the laminz in these positions, they were the only positions entirely free from such tendency. And hence it became so difficult to conceive how the laminated structure could possibly originate in actual discontinuities such as those to which Professor Forbes had ascribed them, whether we suppose the blue laminz to be produced by subsequent infiltration or any other process. According to Dr. T'yndall’s views, the law stated in the preceding ph was an essential consequence of physical causes, to which the production of the laminz was referred. He had shown, experi- mentally, that if a piece of ordinary ice be subjected to direct pres- sure, it will melt along fine lenticular laminz perpendicular to the direction of the pressure. A similar process is supposed to take place in the évé, or in any part of the glacier where the structural laminz originate. Professor W. Thomson had offered an explana- tion of this phenomenon on thermal principles. It had been shown by himself and his brother that the melting temperature of ice is lowered by compression. Now if, in Dr. ‘Tyndall’s experiment, the ice were a perfectly homogeneous substance, every portion would be equally compressed; and if the uncompressed mass were only just above the melting temperature, the whole would melt under the compressing force. But no substance is perfectly homogeneous ; and consequently the internal pressures in the experiment would not be perfectly equable; and those portions of the ice which were subjected to the greatest pressure would melt the soonest, and pro- duce the aqueous laminz above mentioned in the experiment, or the lamin of blue ice in the glacier. In the latter case the laminar portions must first be supposed to melt, the air-bubbles to escape, and the water subsequently to be refrozen to form the blue transpa- rent lamine. Mr. Hopkins did not profess to maintain the entire adequacy of the above explanation of the formation of the laminated structure, though he could not but feel persuaded that it was founded in truth. It seemed to explain very satisfactorily the conversion of névé, or opake white ice, into transparent blue ice; but he did not well understand how transparent consolidated ice could be converted by pressure into white opake ice. But still, at the bottom of an ice-fall, as that of the glacier of the Rhone, the broken fragments were again united into a continuous mass, and the laminated structure was reproduced on a type entirely new, and conformable to the altered conditions of the mass; and, assuming a large portion of the ice descending the fall to be in a sufficiently consolidated state, the process of recon- struction must consist as much in the conversion of blue ice into white, as of white ice into blue. Mr. Hopkins was not aware whether this difficulty had been previously started, or, if so, what answer had been made to it. His object was more especially to 216 _ point out the essential distinctions between Professor Forbes’s and Dr. Tyndall’s theories respecting the laminar structure. A disposi- tion had recently manifested itself to confound the two theories, whereas they were so fundamentally different, that the physical rea- soning essentially involved in the one was totally inapplicable to the other. The differences were such as could not be ignored, if we would hope to arrive at a complete and final view of the subject. ‘The remarks in this lecture, Mr. Hopkins said, had been made with a sincere desire of eliciting the truth, and notin the mere spirit of advocacy of preconceived opinions ; nor would it, he conceived, be inconsistent with this assertion if, in conclusion, he reminded those who were interested in the subject, that though his own investiga- tions nearly fifteen years ago respecting the internal pressures and tensions of glacial masses were little noticed then, and had been little mentioned more recently, no one had ever attempted to refute them ; and now, on the contrary, all those observations and experiments of Dr. Tyndall which related to this part of the subject, and were at present generally received, were entirely confirmative of them. The nature of the reasoning which has now been applied to the subject, whether founded on analogies with certain phenomena of lamina- tion, or on thermal principles, clearly proved the necessity of more accurate conceptions of these internal pressures and tensions than could ever be acquired from merely elucidatory experiments. May 30, 1859. “On the Occultation of Saturn by the Moon on May 8, 1859.” By Professor Challis. In observing this occultation, Professor Challis was prepared to take especial notice of the occurrence of any phenomenon like that witnessed at the occultation of Jupiter on January 2, 1857, on which occasion the disc of the planet at emergence was seen to be traversed by a dark band contiguous to the moon’slimb. No such appearance was visible in this instance. The circumstances of the reappearance of Saturn at the moon’s bright limb on May 8, were very similar to those of the reappearance of Jupiter, excepting that there was no depression of the limb where Saturn reappeared such as that which was noticed at the place of Jupiter’s reappearance. ‘The comparison of the two occultations seems, therefore, to indicate that the phzno- menon seen in the case of Jupiter was in some way connected with the indented form of the moon’s limb. PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. ————————_—___ October 31, 1859. A communication was made by Mr. Hopkins “On the construction of a new Calorimeter for determining the Radiating Power of the Surfaces of Heated Bodies.” November 14, 1859. A communication was made by the Master of Trinity College **On the Mathematical part of Plato’s Meno.” November 28, 1859. The Rey. Dr. Donaldson read a paper ‘“‘ On the Origin and proper value of the word ‘ Argument.’ ” a The author first investigated the etymology and meaning of the Latin verb arguo, and its participle argutus. He showed that arguo was a corruption of argruo = ad gruo; that gruo (in argruo, ingruo, congruo) ought to be compared with xpotw, which means ‘to dash one thing against another,” especially for the purpose of making a shrill, ringing noise; that arguo means “to knock something for the purpose of making it ring, or testing its soundness,” hence “to test, examine, and prove anything ;”’ and that argutus signifies ‘‘ made to ring,” hence ‘ making a distinct, shrill noise,” or ‘* tested and put to the proof.” Accordingly argumentum means id quod arguit, “ that which makes a substance ring, which sounds, examines, tests, and proves it.” It was then shown that these meanings were not only borne out by the classical usage of the word, but also by the technical appli- cation of “argument” as a logical term. For it is not equivalent to “‘argumentation,” or the process of reasoning; it does not even denote a complete syllogism; though Dr. Whately and some other No. XV.—Proceepines oF THE CAMBRIDGE Puit. Soc. 218 writers on logic have fallen into this vague use of the word, and though it was so understood in the disputations of the Cambridge schools. ‘The proper use of the word ‘ argument’’ in logic is to denote ‘‘ the middle term,” 7. e. ‘‘the term used for proof.” Ina sense similar to this the word is employed by mathematicians ; and there can be no doubt that the oldest and best logicians confine the word to this, which is still its most common signification. The author entered at some length into the Aristotelian definition of the enthymeme, which may be rendered approximately by the word “argument.” He also explained how the words “topic” and “‘argument ” came to denote the subject of a discourse or even of a picture. He showed, by a collection of examples from the best English poets, that the established meanings of the word “ argu- ment” are reducible tu three: (1) a proof or means of proving; (2) a process of reasoning or controversy made up of such proofs; (3) the subject matter of any discourse, writing, or picture. And he maintained that the second of these meanings ought to be excluded from scientific language. December 12, 1859. The following paper from the Astronomer Royal was read :—*Sup- plement to the proof of the Theorem that ‘ Every Algebraic Equa- tion has a Root.’ ” The author expressed his want of confidence in every result ob- tained by the use of imaginary symbols, and in this supplement demonstrated that the left-hand member of every algebraic equation of the form ¢(#)=0 admitted of resolution, either into real linear factors, or into real quadratic factors. Professor Miller also made a communication “ On a new portable form of Heliotrope, and on: the employment of Camera Lucida prisms and right-angled prisms in surveying.’ February 13, 1860. The Rev. H. A. J. Munro read a paper “On the Metre of an Inscription copied by Mr. Blakesley, and printed by him in ‘his ‘Four Months in Algeria,’ p. 285.” February 27, 1860. ft A Bae The Rev. Professor Sedgwick made the following communica- tions :— 1. ‘ Anaccount of Mr. Barrett’s progress in the Survey of Jamaica, with some remarks on the Distribution of Gold Veins.” 219 2 “Some account of the Geological Discoveries in the Arctic Regions.” 4 March 12, 1860. The Rey. Professor Challis made a communication “ On the Planet within the orbit of Mercury, discovered by M. Lescarbault.” By a recent comparison of the theory of Mercury’s orbit with observation, M. Leverrier found that the calculated secular motion of the perihelion of that planet requires to be increased by 38", and that this difference between observation and theory cannot be ac~ counted for by the attractions of known bodies of the solar system. In a letter addressed to M. Faye, and published in the Paris Meteo- . rological Bulletins of October 4, 5, and 6, 1859, he suggested that the difference might be due to the attraction of a group of small planets circulating between Mercury and the Sun. On December 22 of the same year, M. Lescarbault, a physician and amateur astro- nomer, residing at Orgéres, about sixty miles south-west of Paris, announced in a letter to M. Leverrier that he had seen on March 26, 1859, a small round spot traversing the sun’s disk, which he con- sidered to be a planet inferior to Mercury. Naturally much inter- ested by this information, M. Leverrier went to Orgéres on Decem- ber 31, and after closely interrogating M. Lescarbault respecting the particulars of the observation, and the instrumental means by which it was made, he returned with the conviction that the observation was trustworthy, and that a new planet had been discovered (Comptes Rendus, January 2, 1860, p. 40). M. Lescarbault had long conceived the idea of detecting inferior planets by watching the sun’s disk for transits, and in 1858 he put his project into execution. He was in possession of a good telescope of 3 inches aperture and 5 feet focal length, mounted with an alti- tude and azimuth movement, and provided with a finder magnifying 6 times. The power of the eyepiece employed in the observations of March 26 was 150. Not being furnished with a position-circle, he adopted the following means of obtaining angular measurements. The eyepiece of the telescope and the eyepiece of the finder each had at its focus two wires crossing at right angles, and the wires of the latter were so adjusted that a star seen at their intersection was seen at the same time at the intersection of the wires of the telescope. There were also in the eyepiece of the finder two wires parallel to, and on opposite sides of, each cross-wire, and distant by about 16’. A circular card about 6 inches in diameter, and graduated to half degrees, was placed concentric with the tube of the eyepiece of the finder, and apparently could be moved both about the tube and, with the tube, about the axis of the finder. A cross-wire of the telescope and a cross-wire of the finder were adjusted vertically by looking at a distant plumb-line, and the diameter of the card containing the zero of its graduation was placed vertically by means of a small plumb- line and eye-hole approximately arranged for that purpose. The 220 mode of using this apparatus for angular measurements will be seen by the following account of the observations. The observer had also a small transit-instrument by which he obtained true time, using for timepiece his watch, which, as it only indicated minutes, required the supplement of a temporary seconds’ pendulum. ° In the account which M. Lescarbault gives of his observations, he says that it had been his practice to examine with the telescope the contour of the sun for a considerable interval on each day in which he had leisure, and that at length, on March 26, 1859, he saw a small round spot near the limb, which he immediately brought to the inter- section of the wires of the telescope. ‘Then, according to his state- ment, he quickly turned the graduated card till two of the wires of the finder were tangents to the sun’s limbs, or equidistant from them. But it is evident that to effect an angular measurement in this way, one of the middle wires of the finder must have been placed tangen tially to the sun’s limb at the point of their intersection, to which point the spot had just been brought. Assuming that this operation was performed, the angular distance of the point from the vertical diameter of the sun might be read off, as the account states that it was, by applying the plumb-line apparatus to the graduated card. This method could only give a rough measure of the angular position of a point very near the sun’s limb; and in fact M. Lescarbault does not appear to have attempted to determine the position of the spot during the interval between the beginning and the end of the transit. He states that the spot had entered a little way on the sun when he first saw it, and that the time and place of entrance were inferred by estimation. The following are the immediate results of the observations :-— The spot entered at 45 5™ 36° mean time of Orgéres at the angular distance of 57° 22! from the north point towards the west, and de- parted at 55 22™ 448, at 85° 45! from the south point towards the west, occupying consequently in its transit 1" 17™ 8%. The length of the chord it described was 9' 14", and its least distance from the sun’s centre 15! 22!'. M. Lescarbault also states that he judged the apparent diameter of the spot to be at most one-fourth of that of Mercury, when seen by him with the same telescope and magnifying power during its transit across the sun on May 8, 1845. The lati- tude of Orgéres is 48° 8’ 55", and longitude west of Paris, 2™ 358. From these data M. Leverrier ascertained, by calculating on the hypothesis of a circular orbit, that the longitude of the ascending node is 12° 59!, the inclination 12° 10’, the mean distance 0°1427, that of the earth being unity, and the periodic time 19°7 days. Also he found that the greatest elongation of the body from the sun is 8°, the inclination of its orbit to that of Mercury 7°, the real ratio of its diameter to Mercury’s 1 to 2°58, and that its volume is one- seventeenth the volume of Mercury on the supposition of equal den- sities. This mass is much too small to account for the perturbation of Mercury’s perihelion. According to these results, the periods at which transits may be expected are eight days before and after April 2 and October 5, the body being between the earth and sun 221 near its descending node at the former period, and near its ascend- ing node at the latter. After the announcement of this singular discovery, it was found that other observations of a like kind had been previously made. Several instances are collected by Professor Wolf in the tenth num- ber of his Mittheilungen iiber die Sonnenflecken, eight of which are quoted in vol. xx. (p. 100) of the Monthly Notices of the Royal Astronomical Society. T'wo of these, the observation of Stark on October 9, 1819, and that of Jenitsch on October 10, 1802, agree sufficiently well with the calculated position of the node of the object seen by Lescarbault. But the spot seen by Stark is stated to have been about the size of Mercury. Capel Lofft saw at Ipswich, on January 6, 1818, at 11 a.m.,a spot of a ‘sub-elliptic form,’ which advanced rapidly on the sun’s disk, and was not visible in the evening of the same day (Monthly Magazine, 1818, part 1, p. 102). Mr. Benjamin Scott, Chamberlain of London, saw about mid- summer of 1847 a large and well-defined round spot, comparable in apparent size with Venus, which had departed at sunrise of the next day (Evening Mail, January 11, 1860). Pastorff of Buchholz records that he saw on October 28 and November 1, 1836, and on February 17, 1837, two round black spots of unequal size, moving across the sun at the respective hourly rates of 14’, 7”, and 28’. Also he announced, January 9, 1835, to the Editor of the Astronomische Nachrichten, that ‘‘ six times in the previous year he had seen two new bodies pass before the sun in dif- ferent directions and with different velocities. The larger was about 3" in diameter, and the smaller from 1! to 1/25. Both appeared perfectly round. Sometimes the smaller preceded, and at other times the larger. The greatest observed interval between them was 1’ 16"; at times they were very near each other. Their passage occupied a few hours. Both appeared as black as Mercury on the sun, and had a sharp round form, which, however, especially in the smaller, was difficult to distinguish.” Schumacher considered it his duty as editor to insert the communication, but evidently did not give credit to it (Astron. Nachr. No. 273). In vol. ii. of the Correspondence between Olbers and Bessel, mention is made in p. 162 of an observation at Vienna by Steinhiibel, of a dark and well-defined spot uf circular form which passed over the sun’s diameter in five hours. Olbers, from these data, estimates the distance from the sun to be 0°19, and the periodic time thirty days. It is remarkable that Stark saw about noon of the same day a singular and well-defined circular spot, which was not visible in the evening. This is one of the instances in vol. xx. of the Monthly Notices of the Astronomical Society. These accounts appear to prove that transits of dark round objects across the sun are real phenomena; but it would perhaps be prema- ture to conclude that they are planetary bodies. If the object ob- served by Lescarbault be a planet, it is certainly very surprising that it has not been often seen. Schwabe, after observations of the sun’s 222 face continued through thirty-three years, has recorded no instance of such a transit. It is probable that now attention has been espe- cially drawn to the subject, future observations, accompanied by measures (of which Lescarbault’s are the first instance), may throw light on the nature of these phenomena, April 23, 1860. Professor De Morgan read a paper ‘‘ On the Syllogism, No. IV., and on the Logic of Relations.” In the third paper were presented the elements of a system in which only onymatic relations were considered; that is, relations which arise out of the mere notion of nomenclature—relations of name to name, as names. The present paper considers relation in general. It would hardly be possible to abstract the part of it which relates to relation itself, or to the author’s controversy with the logicians, who declare all relations material except those which are onymatic, to which alone they give the name of formal. Mr. De Morgan denies that there is any purely formal proposition except ‘there is the probability a that X is in the relation L to Y;” and he maintains that the notion ‘ material’ non suscipit magis et minus ; so that the relating copula is as much materialized when for L we read identical as when for L we read grandfather. Let X..LY signify that X stands in the relation L to Y; and X.LY that it does not. Let LM signify the relation compounded of L and M, so that X.. LMY signifies that X is an L of an M of Y. In the doctrine of syllogism, it is necessary to take account of combinations involving a sign of inherent quantity, as follows :— By X..LM’Y is signified that X is an L of every M of Y. By X..L,MY it is signified that X is an L of none but Ms of Y. The contrary relation of L, not -L, is signified by 7. Thus X. LY is identical with X..7Y. ‘The converse of’L is signified by L~': thus X..LY is identical with Y..L7*X. This is denominated the L-verse of X, and may be written LX by those who prefer to avoid the mathematical symbol. The attachment of the sign of inherent quantity to the symbols of relation is the removal of a difficulty which, so long as it lasted, pre- vented any satisfactory treatment of the syllogism, There is nothing more in X..LM’Y than in every M of Y is an L~’ of X, or MY))L~'X, X and Y being individuals; and nothing more in X..L,MY than in L~'X))MY, except only the attachment of the idea of quantity to the combination of the relation. When X is related to Y and Y to Z, a relation of X to Z follows: and the relation of X to Z is compounded of the relations of X to Y and Y to Z, And this is syllogism. Accordingly every syllogism has its inference really formed in the first figure, with both premises affirmative. For example, Y.LX and Y..MZ are premises stated 223 in the third figure: they amount to X..L7~'Y and Y..MZ, giving X../~'MZ for conclusion. This affirmative form of conclu. sion may be replaced by either of the negative forms X .L~'M'Z or X..4-'mZ. The arrangement of all the forms of syllogism, the discussion of points connected with the forms of conclusion, the extension from individual terms in relation to quantified propositions, the treatment of the particular cases in which relations are convertible, or transi- tive, or both—form the bulk of the paper, so far as it is not contro- versially directed against those who contend for the confinement of the syllogism to what Mr. De Morgan calls the onymatic form. _ An appendix follows the paper, on syllogism of transposed quan- tity, in which the number of instances included in one premise is equal to the whole number of existing instances of the concluding term in the other premise. Mr. J. H. Rohrs also read a paper ‘‘ On the Motion of Bows, and thin Elastic Rods.” . May 7, 1860. The Rev. Professor Sedgwick made a communication ‘“ On the Succession of Organic Forms during long geological periods; and on certain Theories which profess to account for the origin of new species.” May 21, 1860. The Public Orator read a paper ‘‘On the Pronunciation of the Ancient and Modern Greek Languages.” He gave a rapid sketch of the “‘ Reuchlin and Erasmus” contro- versy in the sixteenth century, especially the part taken in it at Cambridge by Cheke, Smith, Ascham, and Bishop Gardiner; and then proceeded to show how the proper sounds of the Greek letters may be determined from the following sources :— . Distinct statements of grammarians. . Incidental notices in other ancient authors. . Variations in writing of inscriptions and MSS. . Phonetic spelling of cries of animals. . Puns and riddles. . The value of the respective letters in other languages employ- ing the same alphabet, especially Latin. 7. The way in which Latin proper names are spelt in Greek, and vice versd. 8. The traditions of pronunciation preserved in modern Greek. He concluded that, on the whole, the method of Erasmus ap- proached more nearly to the ancient pronunciation than that of Reuchlin. oP De 224 “But,” he proceeded, ‘‘ when we consider the untrustworthiness of each of these sources of evidence taken singly, and when moreover we find them often in conflict with one another, it cannot be ex- pected that the result should be very certain or very satisfactory. There are also other considerations which enhance the difficulty of the inquiry. As there were very marked dialectic varieties in Greece, so there may have been local variations even in Attica itself. ‘«« The pronunciation, too, changed from time to time. Plato gives us proof of this in the ‘ Cratylus.’ ” After quoting several instances, and showing that great changes both in pronunciation and spelling had taken place in modern lan- guages, French, Spanish, and English, “it would,” he said, ‘be hopeless to attempt to determine the pronurfciation of any language by a reference to its orthography at a time when both were perpe- tually changing. But in the history of every nation there arrives a time when the creative energy of its literature seems to have spent itself; when, instead of developing new forms, men begin to look back and not forward, to comment and to criticise. Then it is that a language begins to assume, even in minor and merely outward points, such as pronunciation and spelling, a fixity and rigidity which it retains with scarcely any change so long as the nation holds together. Such a period in Greek history was that which began with the grammarian sophists in the fifth century B.c., and culminated in Aristarchus and Aristophanes of Byzantium. In the spelling and pronuriciation of Greek there was probably very little change from that time to the end of the third century a.p.” October 19, 1860. Dr. Paget made a communication ‘‘ On some Points in the Physio- logy of Laughter.” — November 12, 1860. The Public Orator read a paper (a sequel to that on May 21) ‘On the Accentuation of Ancient Greek.” The question of accents was not discussed in the Reuchlin and Erasmus dispute. At that time all pronounced according to the system of accents introduced by the Greeks of Constantinople, who first taught the ancient language to the Italians. It was probably in Elizabeth’s reign that we began to disuse the old pronunciation of vowels both in Greek and Latin; and concur- rently with this change we, as well as the other nations of Europe, began to pronounce Greek, not with the modern Greek, but with the Latin accent. The reasons were :— 1. Teachers speaking the modern Greek were no longer required, so the tradition was not kept up. 225 2. It saved much trouble to pronounce both languages with the ‘same accentuation. : 3. The Greek accent perpetually clashes with quantity; the Latin much more rarely; never, indeed, in that syllable of which the quantity is most marked—the penultima. Isaac Vossius (1650-60) advocated the disuse of accentual marks altogether, as the invention of a barbarous age to perpetuate a bar- barous pronunciation. After showing the meaning of the word ‘accent’ as applied to modern languages, and discussing the accentuation of the German, English, French, &c., he proceeded to say: *«There are three methods of emphasizing a syllable :— 1. By raising the note ; 2. By prolonging the sound ; 3. By increasing its volume. “‘Scaliger, De Causis Lingue Latine, lib. ii. cap. 52, recognizes this division when he says that a syllable may be considered of three dimensions in sound, having height, length, and breadth. *« Now in our own language, when we accent a syllable, which of these dimensions do we increase? Generally all three, but not necessarily ; for when the prayers, for example, are intoned, i. e. read upon one note, the accent is marked by increasing the volume of sound (the third method), which involves also a longer time in utterance, 7. e. a lengthening of quantity. In speaking, all three methods are employed, but one more prominently than the other, according to individual peculiarities of the speakers. What we blend, the Greeks kept distinct. «« We cannot understand the Greek system unless we bear this in mind, They never confounded accent with quantity. Ineradicable habit prevents us from reverting in practice to their method, just as they would have been unable to comprehend ours, «Tt is clear from Dionysius, De Comp. Verb. lib. xi. cap. 75, that the dialogue in tragedy preserved the ordinary accentuation, which was disregarded only in choral passages set to music.” The practical conclusion was this: that while it would be desirable, if possible, to return to the Erasmian system of pronunciation, it would be extremely absurd to adopt the barbarous accentuation of modern Greek, which has quite lost the old essential distinction ‘between accent and quantity. In this respect, as we cannot recover practically the ancient method, it is better to keep to our own system of the Latin accent, which does not confuse the learner’s notion of quantity in verse as the modern Greek does. An Athenian boy has the greatest difficulty in comprehending the rhythm of Homer or Sophocles. Hence it is not blind prejudice (as Professor Blackie asserts) which makes us keep to our old usage, but a well-grounded conviction that we should lose more by changing than we should gain. 226 November 26, 1860. Professor Challis made a communication ‘‘ On the Solar Eclipse of July 18, 1860.” December 10, 1860. Mr. Seeley read a “‘ Notice of Opinions on the Red Limestone at Hunstanton.” Professor Miller also described “ An Instrument for measuring the radii of arcs of Rainbows.” February 11, 1861. Mr. H. D. Macleod read a paper ‘‘ On the present State of the Science of Political Economy.” The writer took a general survey of the science as it at present exists, testing several generally received doctrines by the principles of inductive logic, and earnestly enforcing the necessity of a thorough - reform of the whole science, which must be constructed on prin- ciples analogous to those of the other inductive sciences. February 25, 1861. Dr. Humphry made a communication ‘‘ On the Growth of Bones.” March 11, 1861. The Master of Trinity made a communication ‘‘ On the 'Timzus of Plato.” PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. IPA Rennes snr October 26, 1863. A communication was made by Mr. H. D. Macleod “On the Theory of Banking.” November 9, 1863. Communications were made by Dr. Humphry, ‘‘ The Results of Experiments on the Growth of the Jaw.” By Mr. Todhunter “ On a Question in the Theory of Probabilities.” November 23, 1863. A communication was made by Professor Challis ‘‘ On the Meteor of August 10, 1863.” December 7, 1863. A communication was made by Dr. Akin ‘‘ On the Origin of Elec- tricity.” February 8, 1864. A communication was made by Professor Liveing “On the new Metal, Thallium.” No. XVI.—Procerepines or THE CAMBRIDGE Pui. Soc. 228 February 22, 1864. Communications were made by Professor Cayley “Ona Case of the Involution of Cubic Curves,” and ‘‘ On the Theory of Involution.” By Mr. Spencer “ On Vitality.” March 7, 1864. Communications were made by Mr. Harry Seeley— 1. ‘On the Significance of the Succession of Rocks and Fossils.” 2. “On Pterodactyles, and on a new Species of Pterodactylus macherorhynchus.” April 18, 1864. Communications were made by Mr. Rohrs ‘‘ On the Strains and Vibrations of Ordnance under the Action of Gunpowder.” By Professor Cayley ‘On the Classification of Cubic Curves,” and ‘‘On Cubic Cones and Curves.” May 2, 1864. Communications were made by Mr. Harry Seeley— 1. “Note on Paleobalena Sedgwicki (nob.), a Whale from the Oolite.” 2. “On the Fossil Birds of the Upper Greensand, Paleocolyntus Barretti (nob.) and Pelargonis Sedgwicki (nob.).” 3. “On the Osteology and Classification of Pterodactyles, Part II., with descriptions of the new species P. Hopkinsi and P. Oweni.” May 16, 1864. . Papers were read by Professor De Morgan— 1, “A Theorem relating to Neutral Series.” The theorem is as follows. If a,—a,+a,—... be a convergent series which has the limiting form 1—1+1—..., and if a, be of 229 “continuous law, so that a,;,: a, finally approaches towards a limit, the limit towards which the series approaches as its form approaches neutrality is a,—az4; divided by a,—a,42. And this limit is always 3. If there be in the series a cycle of laws involving an even number of terms, so that Fanz—Aanz+1s enz+1— Ganz+2) ++ + T2nz+2n—1— Ganz+2n approach in ratio to k,, k,,...ken—1, then the two series a4,—a,+4,—... and a,—a,+a,—..., which have unity for their sum, have the ratio of k,+k,+...+hon-2 to k,+k,4+...+en-1. But if the cycle have an odd number of terms, each of these series is 3, just as if the law had been continuous. The demonstration is founded upon the following theorem :—If P,+P,+...andQ,+Q,+... be di- verging series, whether of increasing or decreasing terms, their two infinite sums are in the final ratio of P, to Q,._ Applications of this theorem are given to the determination of a large number of terms of 1"4+2”+4 ... when 2 is —1 or greater, and to the determination of the usual approximation to 1.2.3... when z is great, 2. «On Infinity, and on the Sign of Equality.” The author professes himself satisfied of the subjective reality of the notions of infinitely great and infinitely small. His paper, as far as it deals with various objections by various modes of answer, is not capable of abstract; but four points, on which he especially relies, may be stated as follows :— 1. The concepts ‘of the mind are divisible into imageable and un- imageable: the first can be pictured and placed before the mind’s eye; the second cannot. The mathematician, dealing in great part with imaged concepts, is apt to repel the unimageable, as if it could not be a legitimate object of mathematical reasoning. But all that is necessary to reasoning is knowledge of the connexion of sub- . jects and predicates. Infinite quantity is unimageable in its relation to finite quantity, but not therefore inconceivable, nor destitute of known attributes. A million of cubic miles is as destitute of image as infinite space ; nevertheless it is a conception the attributes of which give known propositions. 2. Number, or enumeration as distinguished from multitude, is a concept from which no notion of infinity can be gained; but much perplexity has arisen from the attempt to make it a teacher of this subject. Abstract number has more than one affection which is de- rived from the concrete in such manner that the two abstractions, number and its affection, cannot have their function explained ex- cepting by return to the concrete. Such affections are the divisible unit, on which the doctrine of fractions is founded, and the opposi- tion of positive and negative. The representation of infinite and of infinitesimal number is a third, affection of the numerical, which cannot be explained on purely numerical notions. 3. The infinite is not a kind of terminus to the finite, but another status of magnitude, such that no finite, however great, is anything but an infinitesimal of the infinite. And the same may be said of each order of infinity with reference to the one below it. 230 4, The symbol }, the infinite of common algebra, represents an extreme of infinite which can no more be attained by passage through orders of infinity, than any infinite by passage from finite to finite. Each of these positions comes into conflict with some of the usual arguments for or against the introduction of infinites. The second part of the paper is on the meaning of the sign of equality. Mr. De Morgan contends for the ultimate attainment of a purely formal algebra, in which every transformation shall have meaning and validity in every possible case. He points out certain difficulties and inconsistencies in the ordinary use of the sign of equality, which can, he affirms, receive a consistent explanation on the extension which he proposes, and which, to some extent, he con- siders as virtually adopted. His notion is that equality, strictly so called, is but a species of the genus undistinguishable; and that the actual use of the sign (=) shows a leaning to the generic definition. Every order of infinites or infinitesimals has its own metre, and the sign (=) indicates undis- tinguishability with reference to the metre, which is often in thought, but for which no symbol is employed. Algebraical changes may or may not demand or permit changes in the metre. It would be im- possible to give any further account, with justice to the subject, in a short abstract. 3. By Mr. Harry Seeley “On Saurornia, and the Classification of Pterodactyles, Part III.” 5 af October 31, 1864. Papers were read by Professor Selwyn “On Autographs of the Sun.” By A. R. Catton, B.A., St. John’s College, ‘“‘ On the Constitution of Chemical Compounds.” November 14, 1864. A communication was made by G. F. Browne, M.A., St. Catha- rine’s College, ‘‘On certain Ice-Caverns.” November 28, 1864. Papers were read by Dr. Humphry on the question, ‘‘ Is the Ver- tebral Theory of the Skull to be abandoned ?” This communication was intended partly as a reply to the opinion expressed by Professor Huxley in his lectures on comparative ana- tomy, that the vertebral hypothesis of the skull has been abolished by the recent discoveries in development. Dr. Humphry commencd by calling attention to the Laws of Uniformity of Plan, and Variety in Detail, which prevail throughout the animal kingdom, and, indeed, throughout the material system, and which the recent discoveries 231 by the microscope have shown to rule over the ultimate structure and formation of all the tissues of the body. The discovery of the illustration of these laws in the plan of cell-formation of the tissues, and in the development of all animal and vegetable structure from the simple cell-form, he regarded as the grandest discovery in phy- sical science that has taken place in our time. Of late years, the attention of anatomists has been much directed to the exemplifica- tion of these laws in the vertebrate classes, to tracing the uniformity of plan, especially in the skeleton, through the variety in detail which the members of these classes exhibit. This constitutes the branch of anatomy called “‘ Homology.” The general features of the plan m which vertebrate animals are constructed are clear enough in all of them. Osseous segments, or vertebrae, with neural and vis- ceral processes, enclosing respectively the neural and visceral cen- tres, constitute the trunk, including neck, chest, loins, &c. Proba- bility is in favour of the view propounded by Goethe and Oken, and worked out by Oken and Owen, that the skull falls in with the Law of Uniformity, and corresponds with the rest of the frame in having a vertebral composition. It is by all anatomists admitted to be segmentally constructed. Most anatomists are agreed as to the number of segments. Ought not, therefore, these segments to be described by the same name as those of which they form a conti- nuation, especially as they bear the same relations to the neural and visceral centres, and the same or nearly similar relations to the nerves and blood«vessel? In their mode of development, too, the segments of the skull show a marked general correspondence with those of the trunk. The chorda dorsalis, around which the verte- bral centres are formed, extends at any rate halfway along the base of the skull; and the bodies and arches of the cranial segments are evolved from a continuation of the same embryonic structure (the “yertebral plates”) as the trunk segments—the chief difference being that in the trunk segmentation takes place at an earlier period than in the head. Inthe trunk, it is observed in the vertebral plates ; and these primitive segments are called ‘‘ protovertebre.” They ap- pear not to exist in the head. The segmentation, however, takes place in the cranium as soon as ossification begins, even if it does not do so before ; and the significance of the protovertebre as dis- tinctive features between the skull and the trunk is diminished, first, by their being related to the formation of the nerves as much or more than to that of the vertebre; and secondly, by their not really corresponding with the vertebra, each permanent vertebra being formed by a half of two protovertebre. Dr. Humphry expa- tiated on this and other points in the development of the skull, and expressed his decided opinion that the differences between the deve- lopment of it and of the trunk vertebrae were by no means sufficient to controvert the view—which coincides with the Law of Uniformity, and which is confirmed by the segmental construction of the skull, by the relation of its components to surrounding parts, and by so many fundamental resemblances in development—that the same name may be applied to the segments of the skull and of the trunk, 232 and that the one, as well as the other, consists of vertebrae modified to meet the requirements of the parts in which they are found. He concluded by stating that the greater number of those anatomists to whose observations we are indebted for most of our knowledge of the development of the skull and of the trunk, are agreed that the differences between the mode of formation of the segments in the two form no real argument against the vertebral character of either ; and he thought stronger reasons must be adduced than had yet been shown before the anatomists could be called upon to abandon the vertebral theory of the skull. By Professor De Morgan “ On the Early History of the Signs + and —.” An account is given of the work on arithmetic of John Widman, printed in 1489, in which the signs + and — are used to denote more and less. The use made is twofold: a+6 signifying that 6 more than a is wanted, infers a direction to add b toa. But a+b in the old rule of false position is used to signify that the assumption of a for the answer gives 6 too much in the solution. This last usage was continued by many writers through the greater part of the sixteenth century. Some account is given of the Die Coss of Chr. Rudolf, which passes for the first work in which + and — are used. ‘The first edition of his work being lost, a question is raised as to how far the second edition, edited by Stifel, is a fair reprint of the first. A Latin translation of this first edition is said to be in the Imperial Library at Paris. From the mannner in which Widman introduces his signs, Mr. De Morgan thinks there is some ground to suspect that they were originally warehouse marks, indicating the scale into which smaller weights were introduced to make the balance, when the nearest number of larger weights had been put in. ‘This point and others require the examination of older works, print and manuscript. February 13, 1865. Communications were made by Professor Cayley ‘‘ On Abstract Geometry.” By Professor Clifton,‘ Note on the Early History of the Signs + . and —.” February 27, 1865. Mr. Alfred Newton, M.A., F.L.S., communicated some ‘ Notes on Spitzbergen,” of which the following is an abstract. The author stated that last summer he accompanied Mr. Edward Birkbeck on a voyage to Spitzbergen, in that gentleman’s yacht, the ‘Sultana,’ R.S.Y.C. After giving a slight sketch of some of the principal voyages which had been made to that country, he pro- 233 ceeded to say that the ‘Sultana’ left Hammerfest on the 3rd of July, in company with a Norwegian sloop which had been specially fitted to encounter the ice, and chartered to attend upon the yacht. On the 6th they arrived at the entrance of Stor Fjord, which was found to be entirely blocked by the ice. Horn Sound and Bell Sound were subsequently discovered to be in the same condition. They then made for Ice Sound, and anchored in Safe Haven on the 9th. Ice Sound was described as a very much larger inlet than itis represented in the charts to be, extending at least fifty miles into the interior. All the valleys on the north side, and consequently having a southern aspect, are completely occupied by large glaciers, which, with one exception, are only terminated by the sea. The single exception consists of a small but remarkable glacier suspended on a hill-side, some 360 yards from the beach, resting conformably on its own moraine, and having no apparent means of discharge. The author supposes this last was effected by filtration through its bed. The south side of Ice Sound contains several bays of consi- derable size; and the valleys opening upon it, and therefore having a northern aspect, were entirely free from glaciers; the observa- tion being directly opposed to the account given of the Spitzbergen glaciers by Sir John Richardson in his ‘ Polar Regions.’ A great many reindeer frequent this part of the country, and countless numbers of sea-fowl breed on such of the high cliffs around the Sound as are inaccessible to the Arctic foxes. In Ice Sound Mr. Birkbeck’s party had the pleasure of meeting the Swedish Scientific Expedition under Professor Nordenskjold, who are engaged in measuring an arc of the meridian. On the 4th of August the party separated, some going to the eastward in the Norwegian sloop, while the yacht made an- other ineffectual attempt to ascend the Stor Fjord. The sloop sailed as far as Ryklis Islands, but was stopped by the ice. She then pro- ceeded further east in the hope of getting round the pack, and came in sight of ‘Commander Gile’s land,” the existence of which had been so long doubted, it having been ignored by Sir John Richard- son in his work, and in the Admiralty Chart of Spitzbergen. It appears to lie about sixty miles east of the entrance to Walter Thy- men’s Strait, and its flat or round-topped hills (so different from those © of Spitzbergen) were very plainly seen. The author stated that in 1859 the master of a Norwegian vessel landed upon it, and he pro- duced a pebble which had then been brought thence, in proof of the reality of its existence as land, and not either ice or fog-bank. He then proceeded to remark on the driftwood with which the shores of the ‘‘ Thousand Islands” are strewn, which he believed to be cer- tainly of Siberian origin, and not brought, as sometimes imagined, by the Gulf-stream,—stating that though often worm-eaten, he had never observed any signs of barnacles upon it. He then commented on the discovery of the passage from the top of Stor Fjord to Hin- lopen Strait, of which there had long been a traditionary knowledge, though it was not effected till 1859; and showed, from the Swedish surveys in 1861, that this passage must lie some thirty miles further south than the position assigned to it on the Admiralty charts, thus 234 affording another instance of our imperfect knowledge of the geo- graphy of Spitzbergen. In conclusion, the author stated that he left Spitzbergen on the 21st of August, the sun having seta night or two previously for the first time, and the-salt water begun to freeze; and he warmly urged his audience to support the further cireumpolar exploration which has been lately proposed by Captain Sherard Osborne; and said that, as a zoologist, he could declare there were many questions of the very highest interest which could only be solved by a new Arctic expedition. Professor Cardale Babington and Mr. Harry Seeley made commu- nications respectively on the plants and on the fossils brought by Mr. Newton from Spitzbergen. March 18, 1865.~ A communication was made by Professor Liveing “On Gun- cotton,” March 27, 1865. A communication was made by Professor Miller “On the Crystal- lographic Methods of Grassman, Hessel, Frankenheim, and Uhde, and on their employment in the’ investigation of the general geo- metrical properties of Crystals.” May 1, 1865. Communications were made by Mr. Harry Seeley— 1. “On the Cambridge Greensand.—Part I. The Rock and its Origin.” 2. “On the Gravel and Drift of the Fenland.—Part II. Theory.” May 15, 1865. A communication was made by Professor Churchill Babington “‘On the Coinage of England before the Norman Conquest.” May 29, 1865. A paper was read by Mr. Todhunter ‘“‘On the Method of Least Squares.” The object of this communication is principally to demonstrate a very remarkable result which Laplace enunciated, without demon- stration, in the first Supplement to his work on ‘ Probabilities.’ An exposition is also given of the process adopted by Laplace for inves- tigating the method of least squares. Laplace’s process is genera- lized and extended; and results which he obtained for the case of two elements are shown to hold for any number of elements. The ‘mathematical part of the investigation consists chiefly in the evolu- tion of certain definite multiple integrals. 235 October 30, 1865. The following officers were elected :— President .... Rev. W. H. Cookson, D.D. : Mr. I. Todhunter. Vice-Presidents.< Dr. Paget. Professor Challis. Treasurer .... Rev. W. M. Campion. Professor Cardale Babington. Secretaries .. < Professor Liveing. Rev, T, G. Bonney. Professor Selwyn. New Members } Rev. W. G. Clark. the Council. | Mr. R. Potter. Rev. N. M. Ferrers. 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 short- est 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 (Mdbius), be four genera- ting lines of an hyperboloid ; and if 12 denote the moment of the lines 1 and 2, and similarly 13 the moment of the lines 1 and 3, &c., the forces are as o/ 23.34.42 :4/34.41.13: 4/41 12. 24 :4/12.23. 31. Calling the four forces P,, P,, P,, P,, it follows as a corollary that we have P,P, . 12=12. 844/13 .42 .4/14. 23=P,P, .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. 236 November 13, 1865. 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 south- wards 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 Lune 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 rainfallin some por- tions of the Lake mountains is not less than 150 or 160 inches in the year. Among the neighbouring carboniferous mountains the rain- fall 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 decks, 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 @ 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) :— “ Harbourgill, 6th of the 2nd month, 1752. ** Dear Bro’ and Sister,— “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 provi- dence 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 usin Dent. I hope I shall never live to see the like again: for we had the greatest storm of wind and snow that conti- 237 hued 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 (#. e. gate) to the sycamore trees, before the stable, peat-house, and all the calf- parrack (7. 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 pe- rishing ; 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 inthe 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 lace to another to escape their fury, which was with difficulty: for it continued raining extremely, that we were wet to the naked skin, not daring to come in any house. And it drawing towards night, we resolyed to make an attempt to get to brother John’s, and accordingly set forwards, and got up at our pasture head on to the moor, and with difficulty got over Harbourgill, and so forwards to the Mun- keybeck. But we knew that the bridge was broke down, so that we must be obliged to pass it somewhere on the moor, and we waded through the water and snow till we were almost spent in extreme wet and fatigue; and at last got over a little below where our peat- fell is (tho’ with very great, hazard of our lives), at last, my poor old Father and Betty being almost quite spent, he having only one * Slack (coom or hollow in the hill-side). 238 shoe on one foot the greatest part of that time. Then when we were got over, it gave us some fresh encouragements, and we arrived at Bro’ Johns just before it was dark, where we were thankful to see the faces of one another in a place of more safety. We went three nights successively to Bro’ John’s to lodge, not daring to stay about the old place. Old Francis Swinbank [rect. Swithinbank] and Thomas Stockdale’s whole family perished in a moment about the same time that the thing happened with us, being seven in number. Likewise John Burton, Stone House, had a barn swept away and a cow killed. “I hope these few broken hints will be excused, for I am not very good at writing at this time, all being so in confusion. Sr. greatly desires you would come to see us as soon as well can. For our love is very much towards you. You perhaps may think I have out- stretched, but if you please to come your eyes will convince you to the contrary. For I have not told you one half. Soshall conclude your very loving Brother, “THomas THISTLETHWAITE.” “ Betty’s kind love is to you both, but Sr. in particular.” _November 27, 1865. Mr. J. W. Clark, Trinity College, read a paper upon the Rib of a Whale found by some fishermen near Sherringham (four miles N.W. of Cromer). It was discovered after a high tide, which caused a fall of the cliff; it was reported to have been imbedded in drift gravel. Mr. Clark stated he had compared it with the rib of Physalus, with which it did not agree ; for the tubercle and head of the rib were very much wider apart in the fossil specimen than in Physalus. It resembled Balenoptera more nearly, and still more closely Balena mysticetus. It was probably the fourth or fifth rib of the left side. -He remarked that a few months since some cervical vertebre had been found at Plymouth which probably belonged to Balena Biscayensis, a whale which was nearly extirpated in the sixteenth century. It was, however, still occasionally met with, a cow and cub being not long since seen near S. Sebastian, and the latter cap- tured. He could not positively refer the rib exhibited to any known species. ; Professor Sedgwick remarked that large cetacean vertebra had been found at Landbeach, Cambridgeshire, and in the Crag of Nor- folk and Suffolk. This was the largest rib that he had ever seen; but he had great doubt, from its general appearance, whether it could have come out of the gravel. Mr. Newton remarked that our knowledge of whales had of late been greatly enlarged; for of the bodies which had been stranded so many new species and even new forms had been observed, that hardly two had been found to be identical. 239 Professor Miller exhibited two new forms of heliotrope, explaining at the same time the difficulties in signalling, which they were in- tended to overcome, commenting upon the relative merits of those invented by Gauss, Steinheil, and others, and explaining the spe- cial advantages of the two which he exhibited. Professor Miller also communicated a Supplement to the Crystal- lographic method of Grassmann. Mr. G. F. Browne, St. Catherine’s College, communicated some Notes upon some Ice-caves explored during the summer of 1865. Two of these he had visited during the previous summer, and he found that there was a somewhat greater quantity of snow in the caverns than there had formerly been. In the first cave he again examined a pit in the ice about 70 feet deep, but, owing to the dan- gerous condition of the ice, was unable to descend into it. In the second cave he had again cut through a curtain of ice into an icy tunnel ; but this year the diameter of the tunnel was so much smaller that he was unable to descend it, although provided with ropes for the purpose. He described some flies found inside the tunnel. The third cave had not previously been explored ; it was an oval in shape, with a level floor of ice. He had descended for about 12 feet between the ice and the rock, and there found a narrow tunnel which appeared to lead to a subglacial reservoir containing water. He ascertained that the ice was at least 24 feet thick ; but it was impos- sible to descend the tunnel. Mr. Bonney, who had accompanied Mr. Browne, made some ob- servations on the general character of the country, expressing his opinion that the glaciéres were formed by the accumulation of snow in suitable fissures; and remarking that the prismatic structure of the ice noticed last year by Mr. Browne was very conspicuous. rinks ee tt SHAE Poet % wine ta edi eoite een aly ah boiine RGR fistosisceemniitin ot 33% Polo se isge Bae ae “ek ; aw sat shit wee hiey ‘et. eMeO' ‘ee pian re re Liat «. WEE -locinied oft abie aL . ous ni evoliieewdty -} hon ieeel aE tis oda ‘oh by iid Bs oe eee agi 3 BM, it “aif “de atv shank hwonths aid GAAS HE Z92 TM UEy woe te wetiatianrtie Oy fay SUT pms oii a INDEX OF NAMES. VOL. & N.B.—The larger figures indicate separate papers ; the smaller, incidental references. Adams, J. C. F.R.S. 119, 192 Airy, G. B. 27, 62, 205, 218 Akin, Dr, 227 Amsler, Prof. 1 2 Angstrom, Prof. 136 Babington, C. C. 127, 234 Babington, Prof, Churchill, 234 Barrett, L. 168, 218 Bashforth, Rev. F, 173, 182 Baxter, H. F. 200 Bezont, M. 53 Blakesley, Rev. J. W. 218 Bonney, Rev. T. G. 23 Browne, Rev. G. F. 230, 239 Calvert, Rev. F. 74 Candy, F. J. 182 Carnot, M. 66 Catton, A. R. oo 235 Cauchy, M. 1 Cayley, "Prof, 298, 298, 939, 235 Challis, Prof. 26, 27, 73, 75, 75, 120, 122, 123, 132, 133, 134, 146, 179, 203, 204, 216, '219, 226, 227 Clark, Prof. M.D. 148 Clark, J. W. 238 Clark, W. G. 204, 223, 224 Clifton, Prof. 232 Cox, Homersham, 88 Decher, Prof. 192 De Morgan, Prof. 1, 6, 13, 29, 64, 90, 106, 121, 123, 136, 145, 155, 172, 190, 194, 194, 208, 222, 228, 232 Denison, E. B. 77, 121 De Senarmont, 120, 136 Dobson, 127 Donaldson, Rev. J. W. D.D. 175, 199, 206, 217 Earnshaw, Rev. 8S. 14, 25 Ellis, R. L. 5, 63, 147 Fisher, Prof. 132, 146 Fisher, Rev. O. 147, 158, 194 Forbes, Prof. 4 Goodwin, Rev. H. 15, 19, 28 Graham, Prof. 200 Grassmann, Prof. 234, 239 Harrison, A. A. 169 Hayward, R. B. 166 Hegel, 84 Hopkins, W. F.R.S. 2, 9, 116, 148, 149, 181, 209, 217 Humphry, G. M. M.D. 177, 181, 206, 226, 227, 230 Kemp, G. M.D. 21 Kingsley, Rev. W. T. 117, 145, 182 Lescarbault, M. 219 Lionville, M. 136 Liveing, Prof. 307, 234 Mc Coy, F. 64 Macleod, H. D. 226, 227 Maxwell, J. Clerk, 134, 149, 160, 163, 173 Miller, Prof. 116, 117, 119, 136, 177, 192, 203, 206, 209, 218, 226, 234, 239 Mitscherlich, 136 Monge, M. 136 Moon, Rev. R. 75, 79 Morgan, see De Morgan Munro, Rev. H. A. J. 218 126, . Newton, Alfred, 232, 238 O’Brien, Rev. M. 33, 38, 56 Paget, G. E. M.D. 29, 148, 200, 224 Phear, J. B. 133 Porro, M. 206 Potter, Prof. 20 Power, Rev. J. 179 Pritchard, Rev. C. 122 ault, M. 66 Rohrs, J. H. 115, 223, 228 Sarrus, M. 109 Sedgwick, Prof. 158, 187, 218, 223, 236, 238 Seeley, H. 226, 228, 228, 230, 234, 234 ree a 230 Steele, W Stokes, G. . 16, 19, 35, 37, 61, 78, 82, 83, 85, 95, 104, 110, 111, 115, 117, 119, 120, 122, 132, 181, 209 Thompson, Prof. W. H. 191 Thomson, Prof. W. 66 Todhunter, I, 227, 234 Tozer, J. 2 Warburton, H. M.P. 43, 145, 173 Wedgwood, 127 242 Wetli, Prof. 192 Whewell, W. D.D. 4, 74, 81, 84, 89, 99, 100, 103, 110, 148, 149, 158, 176, 2038, 204, 217, 226 Wiedemann, 136 Willis, Prof. 148 Young, Prof. 36 SUBJECTS. Aberration of light, 19 Absolute thermometric scale, 66 Adjustments of a transit, 123 Algebra, foundation of, III, 1 i IV, 13 Algebraic Equations, Geom. repre- sentation of their roots, 28 Analysis, continuity in, 36 Angle, infinite, 14 Antithesis of Philosophy, fundamen- tal, 4, 74 Arbitrary constants, discontinuity of, 181 Arctic Regions, geological discoveries in, 219 Argument, origin and meaning of the word, 217 Aristotle’s account of induction, 89 Astronomical instruments, construc- tion of, 206 Athenian trireme, structure of, 175 Atwood’s machine, 148 Aurora Borealis, 20 Avalanche in 1752, 236 Axis of a zone, direction of, 203 Balance, knife edges of, 119 Banking, theory of, 227 Barrow, and his academical times, 203, 204 Biela’s double comet, 120 Body and Space, knowledge of, 97 Bones, growth of, 226 Bows, motion of, 223 Calorimeter, a new, 217 Camera lucida prisms, 218 a, action, theories of, tested, — period, fossil fish of, Caustic, intensity of light near, 62 Chemical compounds, 230 Clock escapements, 77, 121 Clouds, height of, 119 Coinage of England, 234 Colours, experiments on, 149 », Of thick plates, 110 Combinations, &c. of numbers, 43 Comet, Biela’s, 120 » Donati’s, 204 » Klinkerfue’s, 132 Composition of forces, principles of, 208 Conductivity of substances, 181 Consonances, imperfect, 190 Continuity, principle of, 36 Crystals of an anorthic system, 203 5 of the oblique system, 136 Crystallized minerals, 116 Crystallography, sphere in, 209 “ investigation of, 234, 239 Cubic cones and curves, theory of, 228 Curve, equation to, 81, 83, 100 Curves, on singular points of, 155 Cyclones, theory of, 127 Dent, geology of the valley of, 236 Development of vertebral system, 132, 146 Diffraction, dynamical theory of, 85 Dispersion of light, 119 Divergent series, &c., 6, 181 Dorsetshire, deep pits in, origin of, 194 s Purbeck strata of, 147 Earth, interior temperature of, 126, see 181 », superficial do. 116 », changes in do, 148 Earth and other planets, external temperature of, 149 Earthquake in Switzerland, 158 Eclipse of the Sun, Mar, 15, 1858, 203 sé July 18, 1860, 226 Elastic beams, impact on, 88 » medium, equation for motion of, 56 » rods, motion of, 223 » solids, friction of, 16 Electricity, origin of, 227 Elliptic analyzer, an, 111 Entozoa, discoveries respecting, 148 Equality, sign of, 229 Equation, algebraic, proof of root of, 194, 205, 209, 218 ii relating to bridges, 83 . to a curve, 81, 83, 100 3 differential, of the first order, 101 solution of, 172 theory of, 136 ” ” ” 9? 243 Equation for an elastic medium, 56 , partial differential, 64 numerical, 79 Equilibrium of forces, theorem on, 235 Erratic blocks, transport of, 9 Escapements, ‘clock, 77, 121 : Euclid, geometry of, 127 _ Eye, affected by malformation, 27 Fall, tendency to, 148 Fen-land, gravel and drift of, 234 Fluids, effect of friction on pendu- lums, 104 s, in motion, friction of, 16 Force, Faraday’s Lines of, 160 Forces, composition of, 209 » equilibrium of, 235 Formic acid, synthesis of, 235 Fossil birds, 228 » fish, 64 Fossils, succession of, 228 Fusion, temperature of, under pres- sure, 126 Geology of the Lake District, 187 ” Suffolk, 133 f the Valley of Dent, 236 Geological discoveries in the Arctic Regions, 219 Geometry, abstract, 232 re &c. new notation for, 33 sy symbolical system of, 38 mS of Euclid, 127 Gipping, valley of, 133 Glaciers, on the motion of, 2, 209 Gnomonic projection of sphere, 209 Goniometer, reflective, 117 Ge “s substitute for, Gravity, variation of, 82 _ Great-cirele sailing, chart for, 203 Greek, ancient and modern, 223 » accentuation of, 224 Green sand, Cambridge, 234 »» upper, fossil birds of, 228 Growth of bones, 226 e ,», the jaw, experiments on, 227 Gun cotton, 234 Gunpowder, effects of, 228 Haidinger’s brushes, 117, 179 Halos, parhelia, &c, 122 Heat, theory of, 169 Heliotrope, new portable, 218, 239 Human frame, on the proportions of, 181 Hypotheses, transformation of, 110 Ice-caverns, 230, 239 Induction, Aristotle on, 89 Infinite angle, values of sine and cosine, 14 Infinity, on, 229 Instruments, surveying and astrono: mical, 206 ee calculus, on some points of, Integrals, numerical calculation of, 95 Integration of diff. equations, 64 Internal friction of fluids, &c. 16 Involution of cubic curves, &c. 228 Isomorphous substances, 120 Jamaica, survey of, 218 Jaw, growth of, 297 Jupiter, occultation of, 179 Lake mountains, geological disloca- tions in, 187 Laminated structure, production of, 53 Laughter, Physiology of, 224 Least squares, on, 5, 234 Light, aberration of, 19 », dispersion of, 119 », double refraction of, 235 » intensity of, 62 » polarization of, 27, 115, 117 » reflected from crystals of pot- ash, 132 » refrangibility of, 122 transmission of, 73 Limestcne, red, at Hunstanton, 226 Logic in general, 194 », of relations, 222 »» symbols of, 90, 121, 123 Longitude of Cambridge ‘Observatory, 146 of that of St John’s Col- lege, 119 Luminous appearance, 132 $5 rays, theory of, 26 iy vibrations, theory of, 73 Mathematical exposition of Political Economy, 99, 103 - part of the ‘“ Meno,” 217 Matter, theory of, 63 Mean values, principle of, 123 Mechanics and geometry, connection between, 15 Mercury, planet within its orbit, 219 Meteor, of Aug. 10, 1863, 227 Metre of an inscription in Algeria, 218 Microscope, photography applied to, 117 Mollusca on the coast of Norway, 168 Moon’s motion, models illustrating, 173 », orbit, eccentricity of, 134 method for obtaining, 133 Morbid rhythmical movements, 29 ” 9? 244 Nervous system, 177 Newton’s Principia, Hegel on, 84 A rings, central spot of, 78 Notation, new, for Geometry, &e. 33 Numbers, partitions of, 43 Occultation of Jupiter, 179 a4 Saturn, 216 Optical instruments, theory of, 173 _ Ordnance, strains and vibrations of, 228 Organic forms, succession of, 123 Oscillations of suspension bridges, 115 Oscillatory waves, theory of, 37 Paleozoic rocks, 158 Partitions of numbers, 43 Pendulums, effect of friction of fluids on, 104 Periodic series, 61 Permutations, &c, 43 Philosophy, fundamental antithesis of, 4, 74 Photography, applications of, 117, 145, 182 Physiological alphabet, 182 Planimeters, various, 192 Plato, cosmical system of, 207 » mathematical part of the Meno of, 217 », Sophista of, 191 » Limeus of, 226 » notion of dialectic, 149 of the intellectual powers, 158 », | Survey of the sciences, 148 theory of ideas, 176 Polarity, organic, 200 Polarized light, streams of, 115, 117 Political economy, 99, 103, 226 Potash, reflective properties of, 132 Potato disease, analysis of, 21 Prisms, for surveying, 218 ” ” Probabilities, theory of, 29, 90, 147, 227 Pronunciation of Greek, 223 Pterodactyles, 228, 230 osteology, &ec. of, 228 Purbeck strata, 147 Quicksilver, how to make tremorless, 122 Railway bridges, 83 ee te instrument for measuring, 2 Refraction, double, 73, 120, 235 Rock masses, internal pressure on, 53 Rocks, Paleozoic, 158 », succession of, 228 Roots of equations, rational, 79 Saturn, occultation of, 216 Saurornia, 230 Sciences, hypotheses in, 110 » of magnitude and direction, 19 R 5 mechanics and geometry, 5 ; Series, divergent, &c. 6 », infinite, 95 », Maclaurin’s, 145 », neutral, 228 »» periodic, 61 self-repeating, 145, 173 Signs + and —, history of, 232 Skull, vertebral theory of, 230 Solar eclipse, 203, 226 Solitary waves, math, theory of, 25 Sound, theory of, 75 Space, knowledge of, 97 Species, nearly allied, 127 * theories on new, 223 Spitzbergen, notes on, 232 Squares, Least, on, 5 St John’s College Observatory, long. and lat, of, 119 Standard of weight, 177 Statue of Solon, 199 Storm tracks of the South Pacific, 127 Strata, Purbeck, 147 Sun, autographs of, 230 Surfaces, transformation of, 134 Surveying instruments, 206 Suspension bridges, oscillations of, 115 Syllogism, 29, 90, 121, 194, 222 Symbolical system of geometry, &¢, 38 Thallium, a new metal, 227 Theory of probabilities, 29 Thermometric scale, absolute, 66 Transit instrument, adjustments of, 123 Transport of erratic blocks, 9 Trebbia, battle of, 204 Trigonometry, plane, and its formule, 74 Trireme, Athenian, 175 Undulatory theory, 26, 27, 73 Velocities, &c. direct method of esti- mating, 166 — system, researches on, 132, 6 Vertebrate animals, limbs of, 206 i skeleton, 177 Vibratory motion of a medium, 56, 73 Vitality, on, 228 Voluntary muscles, defects in, 200 Waves, solitary, theory of, 25 Whale from the oolite, fossil, 228 Wood engraving, photography applied to, 182 February 12, 1866. The PRESIDENT (W. H. Cooxson, D.D., Master of St Peter's College) in the Chair, The following new Fellows were elected: H. Russexy, B.D., St John’s. T. H. Canpy, M.A., Sidney Sussex. On the Functions of the Air-cells, and the Mechanism | of Respiration in Birds. By W.H. Drostzr, M.D. Caius alae He Sccerkod that, although, during the last two centuries and a half, a great many distinguished anatomists and physio- logists (whose names he mentioned) had directed their special attention to the conformation of the respiratory organs of birds, and their functions, nevertheless our knowledge in regard to these matters was in a very unsatisfactory state. For instance, with regard to the number and relations of the large air-vessels in connexion with the lungs of birds, anatomists seem to be not yet agreed; and with respect to the uses of these, the views held by different physiologists were very contradictory, and at best only conjectural: all which shews that the subject in ques- tion was one of great difficulty, It is not a little singular that the views, which have been chiefly relied on as explaining the principal uses of the air-cells and their continuations into the cavities of the hollow bones of birds, will not bear for a moment a close scrutiny. The first 2 2 and principal of these was originated by Camper, who in 1771 discovered accidentally that the hollow bones of birds commu- nicate with the great air-sacs; and conceived that the air con- tained in these, being warmed, lessened the specific gravity of the bird, and so rendered it fitter for flight. Since the time of Camper this view has been almost universally adopted. Thus Owen in a work just published adopts it’. In disproof of this view, the lecturer gave a calculation to shew that a pigeon weigh- ing 10 ounces, or 4375 grains, would have its weight in air diminished by only a fraction of a grain in consequence of the rarefaction of the air in its. air-sacs and hollow bones from the warmth of the body; so that the floating power resulting from such rarefaction would be almost inappreciable. We cannot then for a moment suppose that the extensive and complicated system of air-vessels has for a principal object a saving in weight so trifling: not to mention that this system is found in all birds, even such as do not fly. Another theory that has been very generally adopted was propounded and developed by Cuvier’, who assigned’ to the air-cells an office directly supplementary to that of the lungs; ‘so that the air having passed through the lungs and reno- vated the blood in the pulmonary capillaries, is brought once again into contact with the blood in the systemic capillaries in every part of the body, and again exchanges oxygen for carbonic acid with the vital fluid. This theory was founded on a great anatomical mistake, into which, strange to say, many eminent anatomists have fallen, viz. that the air passes from the air-sacs into the cavities of the peritoneum, peri- cardium, and pleure, and even extends itself between the muscles, and beneath the skin: whereas it is in reality confined 2 Owen, On the Anatomy of Vertebrates, Vol. 11. 1866, p. 216; also Art. Aves in Todd’s Cyelop. * Anatomie Comparée, 1805, t. IV. p. 330, and Owen, 1. c. cum multis aliis auctoribus, 3 within the membranous walls of the air-sacs, and the various. definite prolongations of these. This latter was the description given two centuries ago by Harvey and Perrault, and more re- cently by Colas, Guillot and Sappey; and its correctness does not admit of a doubt. An interchange of gases may, indeed, take place between the air in the air-cells and the blood in the vessels of their walls: to a very trifling extent however; for the blood-vessels of these delicate membranes are very minute, and sparsely scattered. We cannot therefore but believe that the extensive, elaborate and (as regards the class of birds) ubiquit- ous system of air-cells must have some far more important office than that of conveying oxygen to these minute vessels. The question is, What is this office ? Bearing in mind that the air-sacs occur in all birds; that their number and position is very uniform; that they always have very free communications with the lungs; that the greater part of the air inspired passes into them, and is again expelled in expiration, we can scarcely doubt that they are a necessary part of the respiratory apparatus of birds, The respiration of birds, even when in repose, has been shewn to be much more active than that of mammals. But in order that birds may be equal to the enormous exertion required of them for sustaining themselves in the air for considerable periods of time, very ample provision must be made for respiration. If therefore the lungs were constructed after the mammalian type, they would require to be very large, and powerful muscles must be provided for the respiratory movements. But this would add unduly to the weight of the body. The lungs therefore are small, very porous, and light; and yet nevertheless they are more efficient respiratory instruments than mammalian lungs of greater weight and volume. This we know from experimental comparisons of the quantity of oxygen absorbed, and of carbonic acid evolved by birds and mammals, for each pound weight of their bodies, in equal times. The increased efficiency is due to 2—2 4 three causes; first, to the greater rapidity of the circulation, on’ account of which more blood passes through the lungs in a given time ; secondly, to the more abundant supply of air drawn in at each act of inspiration; thirdly, to the more complete exposure of the blood in the pulmonary capillaries of a bird to the action of the respired air. It must be observed, that it is necessary that these three conditions should all be satisfied at the same time, to make the lungs more perfect as respiratory organs. For if more blood pass through the lungs in a given time, it would be of no avail unless more fresh air were at the same time supplied for its aeration. And again, if these two con- ditions: were satisfied, the quicker transit of the blood would necessitate the provision of increased facility for interchange of the gases by osmose through the walls of the pulmonary capil- laries; otherwise the blood would pass on before hematosis was sufficiently effected. Now in the bird the first of these eondi- tions is satisfied by the more rapid action of the heart; the second by the greater quantity of air inspired; the third by the peculiar structure of the lungs, in which, as was first pointed out by Mr Rainey, in 1848, the whole circumference of the almost naked pulmonary capillaries is exposed to the action of the air in the lungs, which passes between and around them, in the areole that answer to the cells of the mammalian lung. The lungs of birds being very small cannot contain much’ air. When however the ample chest expands, a large quantity of air rushes in by the trachea, and passing through the larger bronchi, that open into the great air-sacs, fills them. It has long been known that the air-sacs do not all expand at the time of in- spiration, but only the two pairs that lie at the sides of the thorax, and are in contact with the ribs and the intercostal muscles: these may be called the middle sacs. The others, namely, the abdominal, the cervical, and the anterior thoracic (which may be called the extreme sacs) contract at this time. In expiration the action of both sets of cells is reversed. During the time of inspiration, then, the middle sacs become filled with air. The pulmonary diaphragm is at this time in a state of contraction, and its tense aponeurosis, which is attached to the inferior surface of the lungs by a fine cellular tissue, causes the lungs to expand to such an extent as they are able, for they are not very elastic. In so expanding they receive a part of the inspired air, which how- ever does not remain stationary in them, but passes on with a gentle motion towards the middle air-cells, in which the pres- sure has been reduced by the expansion of the walls of the thorax. A much greater amount of air however reaches the middle air-cells by the more direct, and more open course of the larger bronchi, that open into them; and a third portion passes into them, at the same time, from those extreme cells that are nearest to them, and which are contracting, as we know, at this time. Inspiration being completed, expiration commences, and endures for a longer time than inspiration. The pulmonary diaphragm relaxes, and in consequence of the partial closure thereby of certain openings, the middle air-sacs slowly empty themselves by openings in part differing from those by which they were filled; so that the air passing through the smaller bronchi and the substance of the lungs, goes principally out by the trachea, but partly into the extreme air-sacs, which at this time are dilating. Hematosis is going on therefore during ex- piration as well as during inspiration; so that birds are truly animals having a double respiration, though not in the sense in which Cuvier so defined them’. It is exceedingly instructive to observe the way in which the organization of birds has been modified to meet the require- ments of their marvellous powers of rapid and long-sustained flight. Muscles capable of very powerful and rapid action are necessary to support the bird in the light and yielding air. This again requires very ample provision for respiration. The amount of muscular force must also be proportioned to the weight of the 1 Cuvier, Régne Animal, 1817, t. I. p. 290. 6 animal to be sustamea in the air. If we can diminish the weight of the osseous system, we may lessen, in a greater pro- portion, the weight of the muscular; for there will be less muscle, as well as less bone to carry. In birds the osseous framework is made as light as is consistent with the strength necessary, by reducing as far as possible the quantity of osseous matter, and so disposing this on mechanical principles as to offer the greatest possible resistance to the action of the muscles. ‘To this end the bones are hollow, and the bony matter deposited in superficial lamella, dense, but of no great thickness. Air is admitted into their cavities, to absorb the aqueous matter se- creted by the endosteum, and being renewed gradually by the respiratory movements, carries off the moisture that would other- wise collect, fill the cavities of the bones, and add greatly to their weight. The cellular tissue of the lungs is reduced to a minimum, with the double effect of rendering them lighter, and facilitating hematosis; and, finally, voluminous air-vessels, of great tenuity, and containing large supplies of air for re- spiration, without materially increasing the weight of the body, conduce to make the respiratory apparatus of birds one of the most striking examples of the perfect adaptation of means to ends to be met with in the animal kingdom. Our great physiologist, John Hunter, believed it impossible that the ribs and sternum of a bird could move while the powerful pectoral muscles are engaged in flight. He therefore thought that the air-sacs of birds might be intended to act as reservoirs of air to be used in respiration during flight. These sacs, however, do not hold enough air to support the respiration of a bird for two minutes; for in that time, if the trachea of a bird be tied, it dies; yet many birds continue on the wing for hours together. Sappey has endeavoured to explain the difficulty of Hunter by pointing out that the great pectoral muscles of birds arise exclusively from the sternum, and not at all from the ribs, as they do in mammals. But this explanation only removes 7 a part of the difficulty ; for the ribs are so articulated with the sternum, that they cannot move unless the sternum moves also. Now the sternum in respiration moves at its articulations with the two coracoid bones, these bones being fixed in regard to the sternum and humerus in the movements of flight. It might seem, therefore, that when the pectoral muscles contract, the sternum would be drawn powerfully upwards as the wings are drawn downwards, and so the sternum and ribs fixed. But this is not so; for the fibres of these muscles converge towards and pass over the coracoid pénes on their way to be inserted into the ridge of the humerus, and they act, when the wing is extended in flight, in a line very nearly parallel to the axis of the coracoid, but a little below it. Their principal effect on the sternum, therefore, is to draw it more closely against the coracoid; so that they do not much interfere with the action of the respiratory muscles, rather assisting however the inspiratory muscles in depressing the sternum ; a circumstance favourable to deep inspi- ration during flight. The author gave a mathematical as well as an experimental proof that the external intercostal muscles raise both the ribs to which they are attached, and that the internal intercostals depress both ribs. A frame of wood, in the form of a parallelogram with hinges at the angles, represented two ribs, the spine, and the sternum. An india-rubber ring was passed over a peg in the upper rib and another in the lower rib, at different distances from the spine, to represent the intercostal muscle. Both ribs were elevated or depressed according as the upper peg was nearer to or further from the spine than the peg in the lower rib. Dr Humpury made a few remarks upon some points in the paper, expressing himself not quite satisfied with the proof of the aeration of the blood by means of the double current of air from the cells. February 26, 1866. I. Topuunter, M.A., F.R.S., Vicz-PRESsIDENT, in the Chair. The following new Fellows were elected : B. W. Beatson, M.A., Pembroke. J. R. Lumpy, M.A., Magdalene. F. J. Brarruwaite, M.A., Clare. On the Papyrus of the Lake of Gennesaret. By Cuaries C, Basineron, M.A., F.R.S., Professor —_ Botany. Tue object of this communication was to point out the ex- istence of two species of Papyrus, and to explain that the plant in cultivation as the “ Egyptian Papyrus” is not the true Cyperus Papyrus (Linn.), but was probably introduced into our gardens from Sicily, where it had been planted (in the opinion of Prof. Parlatori of Florence) shortly son the 10th century of our era. It was also shewn that the Papyrus of the plain of Genne- saret (from whence a specimen gathered by Mr Tristram was exhibited), and of the vast and deep marshes of Hfleh, by the Lake Merom, in the north of Palestine, is not the plant found at Syracuse and on the coast of Palestine, but is identical with that which grows in an extensive swamp or shallow lake connected with the White Nile at about the seventh degree of 9 north latitude. Also that Bruce found it at thé same place in Palestine, and in two lakes of Abyssinia. As the plant bears a heavy head upon a lofty stem, and does not root strongly in the ground, it can never have been common in Egypt, for the high winds and strong current of the river would be too powerful for it. It was probably brought to Egypt chiefly from Nubia, and only grew in the marshes and back-waters of the river when within the true limits of Egypt. The communication concluded with a few remarks upon the technical characters of the plants derived chiefly from Parlatori’s paper in the Memorres of the Institute of France. Proressor Livertne exhibited an echinoderm from the coral- line crag of Aldborough, which he referred to the genus Rhyn- copygus (D’Orbigny). ‘Two imperfect specimens of the same species from the red crag have been figured and described by Forbes, who referred them doubtfully to the genus Echinar- achnius, but these specimens evidently did not shew the peri- stome. | Professor Liveing’s specimen is a depressed urchin, con- vex above, concave below, the concavity shovel-shaped, the posterior lobes being more developed and descending to a greater distance below the mouth than the anterior lobes. The apex sub-central, somewhat anterior, dorsal ambulacra sub-petaloid, the poriferous zones nearly parallel, extending nearly to the margin and open. The mouth (which in the specimen is partly erushed and one side gone) sub-central but somewhat anterior, the ambulacra about the mouth sharply defined, leaf-like, shallow depressions, with crenate margins, the interambulacral spaces terminating in small tubercles. The anal opening marginal, transverse and overhung by a projection of the back. Genital pores somewhat large and three in number, the left anterior pore wanting. The whole test covered with thick-set tubercles, 10 each surrounded by a depressed areola and a circle of granules. Some of the tubercles shew a ligamental cavity in the centre. The want of the 4th oviduct may be only an individual pe- culiarity, as some recent species are known of which individuals have 2, 3, and 4 genital pores respectively; and in other respects the characters of the specimen agree with those of the genus Rhyncopygus, except as to the position of the anal opening, which in the other species of that genus is supra-marginal, With respect to this character, as soon as the perfectly radiate type in which the opening is in the apex is departed from, it is merely a question of degree, whether the opening be higher or lower in the posterior interambulacral space, and no connexion is known to exist between the position of the opening, when excentric, whether marginal or extra-marginal, with any im- portant modification of other parts. The margin is not defined by any anatomical characters, and moreover in some tolerably well defined genera the anal opening is found to vary in position in different species. Now we ought not to take as a generic character any external peculiarity which is not either known to be connected with some important modification of the ana- tomy of the creature, or which does not belong to some feature which in some well-defined genera is so constant as to lead us to suppose that the law of the creature’s development will not per- mit that feature to vary without an otherwise important change in the anatomy of the animal; the position of the anal opening with reference to the margin satisfies neither of these conditions, and for these reasons Prof. Liveing considers the marginal posi- tion of the anal opening an insufficient reason for separating the specimen from the genus Rhyncopygus. The other species of this genus named in Desor’s Synopsis are cretaceous, except one from the quaternary deposits of Guadaloupe, so that the occurrence of this genus in the crag is a link rather with the older than the more recent formations. 11 On a new Theory of the Skull and of the Skeleton; with a Catalogue of the Fossil Remains of Vertebrate Animals contained in the Woodwardian Museum. By H. Szzzey, F.G.S., Sidney Sussex College. Tue author thought that it was not possible to discuss the theory of the Skull till the theory of the Vertebre had been determined; and that it was impossible to arrive at a theory of a vertebra without considering the theory of the growth of a simple ossification; and believed that when the theory of ossification was once arrived at, then the theory of the verte- bra, the theory of the skull, and the theory of limbs would follow from the conditions which determined the multiplication and co-ordination of simple bones. _ First, then, he endeavoured to show what the possible me- chanical forces acting upon bones were; and having discussed the source of these forces, concluded that all growth must be due to vibrations of pressure and tension; and that the intensity of growth depends on the amount of pressure and tension in the direction of the increase; and that the bones owe the origin of their first particle and form to the same causes which add to their bulk: illustrations of these views being drawn from Me- chanics, Pathology, and Comparative Anatomy. Secondly. The nature of compound bones was considered, as illustrated by the comparative osteology of the humerus, and by the carapace of the turtle, and the so-called tarso-metatarsal bone of birds. And the author arrived at the conclusion that a primary ossification might, if the pressure and tension. were sufficient, develope upon itself other ossifications or epiphyses in any direction; and that these epiphyses might themselves assume the nature of separate bones and also develope epiphyses: 12 or, in other words, one ossification may develope another if suffi- cient pressure and tension can be applied to its surface: a law which appeared to be as true for the entire animal, as for a single bone. . And hence the vertebra was supposed to consist of a centrum or centre of ossification which typically developes three pairs of epiphyses: one pair, in front and behind, being the epiphyses commonly so called; another pair above to enclose the neural cord, called neural-epiphyses; and a third pair beneath to en- close the viscera, called hemal epiphyses. ‘The author then discussed the extent to which these epiphyses actually do pro- duce other epiphyses. And thirdly, the comparative osteology of the skull, and the morphological and embryonic development of the brain were considered; and the conclusion drawn, that since the skull was the entrance to three distinct regions of the body, the ner- vous system, the respiratory system, and the digestive system, it - must be considered in the relation of its three several regions to the corresponding parts of the organism before it can be com- pared with the body as a whole. The author then considered the ossification of the trachea, and showed how by the laws which had been arrived at, ossifica- tion would be greater at its termination where it is in contact with other bones; and so the circles of bones which surround the anterior termination of the respiratory region in the skull were regarded as only the modified end of the trachea, and therefore could not correspond with any part of an ordinary vertebra. The brain-case was recognized as consisting of three seg- ments; and such that the basisphenoid corresponded to an ordinary vertebral centrum, and developed the presphenoid and basi-occipital for its epiphyses in one direction, and the alisphen- oids in another; and by a law previously determined it was considered that the whole occipital segment was the posterior epiphysis of the whole parietal segment, while the whole frontal 13 segment was its anterior epiphysis. The brain-case therefore being modified from the plan of a single vertebra. The lower jaw, both by homology in embracing the respira- tory and digestive systems and by development, would be the hemal epiphysis or representative of a rib to the parietal segment of the skull. Tn conclusion, the author regarded the skull as the terminal segment of the body, each of its three regions being modified on the plan of the corresponding structures of the adjacent and continuous regions of the body. | March 12, 1866. The Presipent (W. H. Cooxson, D.D., Master of St Peter's College) in the Chair. The following new Fellows were elected : W. J. Beamont, M.A., Trinity. N. Goopman, B.A., S¢ Peter's. On the Homeric Tumuli. By F. A. Patzy, M.A. THE object of this paper was to shew that most of the facts which have been ascertained from the exploration of sepulchral barrows in this and other countries may also be verified. histo- rically, and with considerable minuteness, from the Homeric poems, in which much that appears authentic is said about the tumuli of the Grecian and Trojan heroes, and also about the ceremonies of interment. These facts were collected and compared with a view of shewing the very great antiquity 14 of this method of sepulture, which prevailed for ages, and ap- parently when no other method was in vogue, all over the north of Europe, and especially in the neighbourhood of the Euxine and about the Crimea; and it was also shewn that barrows existed which were regarded as of unknown antiquity even in the Homeric ages. The burning of the body of the deceased, the collecting of the bones in sepulchral urns, the burial of the arms or other ornaments in the same mound, the method of making the mound, with or without a circle of foundation-stones, and the surmounting of the tumulus itself by a ste/é, or stone pillar, were all described from Homer; and it was shewn that in all these respects the written accounts generally agreed in a remarkable manner with observed facts. Sir W. Gell’s identi- fication of the Homeric tumuli with existing barrows was dis- cussed; the etymology and significance of several of the Greek words relating to this form of burial were explained, and the institution of games, with the sacrifice of living victims to the spirit of the deceased, illustrated by examples. And a compari- son was made of the two detailed accounts in [iad xximl. and xxiv. of the funeral rites performed at the burning and burial of the bodies of Patroclus and Hector respectively. Lastly, the shape of the ancient barrows, whether oval.or circular, was con- sidered, in reference both to the expressions in Homer, and to modern examples in the Troad, in Sweden, and elsewhere. The occasional use of ancient tumuli as land-marks, as posts of observation, or as places for holding councils, was proved from passages in Homer; and some points were suggested as worthy of particular notice in the event of ancient barrows being ex- plored. - PROFESSOR SELWYN inquired whether any differences had been observed in the forms of existing tumuli, according to their positions, which would strengthen the arguments for their hay- ing been used as land-marks; he also made some remarks upon the connexion between trees and tumuli. 15 Mr Bonney (St John’s) mentioned some tumuli in Brittany, Scandinavia, and Constantine (Algeria) which illustrated the use of stele, and the method of building the mound. He discussed the probable antiquity of some in the first country, and shewed that they belonged in all probability either to the Neolithic or to the Bronze age, exhibited a progress in Art, and were certainly pre-Roman. He also described the various forms of their tomb chambers, and made some remarks upon the geographical dis- tribution of these remains. The Pustic Orator (Mr W. G. Ciark, Trinity) gave some interesting particulars from his personal recollections concerning the barrows in the Troad, and mentioned that all which had been explored lately had been found to have been previously opened. He also commented upon the difficulty of deciding upon the age of tumuli from the remains contained, owing to the fact that one material did not necessarily wholly replace another ‘in the marfufacture of weapons, &c.; instancing the heaps of flint chips at Marathon, which are comparatively modern, being the remains of threshing instruments still used in agriculture in that part of the country. He also doubted whether the various ‘ages’ could always be maintained. In reply to a question by Mr Candy (Sidney) as to the re- lation of Mr Paley’s paper to Stonehenge, Professor CARDALE Basineron briefly indicated the commonly received opinions concerning the date and purpose of Stonehenge; and brought forward instances from Denmark and the north of Europe to shew that the distinction of the ‘ages’ could not be entirely set aside. 16 On the Method of demonstrating some Propositions in Dynamics. By \. Topuunter, M.A., F.R.S., St John’s College. Suppose a particle moving in a straight line; let s be the space described at the end of the time ¢, v the velocity, f the acceleration; then we have the equations dv d’s f= Se And similar equations hold in more complex cases of motion. Thus we have theoretically the choice of two methods when we wish to determine f; namely we may first find the velocity and then f from the relation f= si or we may find / without | attending to the velocity from the equation f= 7 The propositions given in Newton’s second and third sec-. tions are in effect treated in the latter of the two methods. It is however quite possible to treat them by the former method ; and the following advantages seem to follow by adopting the first method. The results can be obtained without requiring so much knowledge of the properties of Conic Sections. | The theory of limits is used in a more simple and convincing manner. The illustration of mechanical principles is more varied. As an example take the most important proposition, namely that of motion in an éllipse round the focus. The figure may be easily constructed. Let S be the focus of force, H the other focus, P and @ two points on the ellipse. It may be shewn in the most elementary manner that the 17 velocity at any point can be resolved into two, perpendicular to the radius vector and axis major respectively; and both constant. Call these v, and v, respectively. Now suppose the body to move from P to Q; when it arrives at Q it has velocities equal and parallel to those which it had at P, together with the velocity generated by the central force during the motion from P to Q. Call this last velocity u, and suppose its direction to make an angle yw with SQ. Let PSQ= ¢. Thus at @ we have three component velocities in assigned directions ; and these must be equivalent to v, and v, perpen- dicular to SQ and the major axis respectively. Hence v, sind —u cosy =0, v,cosd+usiny=v»,; therefore v, (cos d ee) =,; therefore cos y = cos (6—wW); therefore v= h¢. ‘This result is exact; it shews that as the body moves from P to Q the effect of the central force is to generate a resultant velocity the direction of which bisects the angle PSQ. Let f denote the acceleration at P, then when the time, #, of motion is made small enough we have fi=u, tht = "sin $ , where r is put for SP; therefore fr’ sin d = hu v, sing cosy ” or ultimately Je: 18 This shews that the force varies inversely as the square of the distance. It will be seen on examining this demonstration that it involves very few properties of the ellipse, and those only of the most elementary kind. It introduces an important result, namely that involved in the relation y=4¢. And the theory of limits is only required in a form which may be easily under- stood and admitted. After arriving at the result 4 =4¢ we might complete the demonstration thus: Let PSQ be any finite angle as before; let p be adjacent to P and qg to Q, such that the angles PS&p and QSgq are equal, and p and q fall between P and Q. Then since the angles pSg and PSQ are bisected by the same straight line, the central force produces as much effect while the body moves from P to p as it does while the body moves from g to Y. But the times of describing these portions are ultimately as SP? to SQ; and therefore the forces at F and @ are ultimately as ie to ae ‘ ProFEssoR ADAMS made a few remarks on this communica- tion, describing a somewhat similar investigation which he had used in his own Lectures. HYTOC OF TH a WES XAT vee eit orrety. April 23, 1866. ‘The Presipenr (H. W. Cooxson, D.D., Master of S¢ Peter’s College) in the Chaivr. The following new Fellows were elected : R. Morton, B.A., St Peter’s. J. Stuart, B.A., Trinity. On Capillary Attraction. By Ricuarp Porrer, M.A. Tus being a mathematical paper, it admits of only a very imperfect abstract. For the history of the subject up to 1834, reference was made to the very complete report drawn up by Professor Challis at the request of the British Association, and published amongst their reports. After noticing some points in the history of the subject, such as M. Clairaut’s proposition of the relation of the attraction of the solid for the particles of the liquid in contact with it, com- pared with the attraction of the liquid for the same particles, in order that there may be capillary elevation or depression of the liquid; the views of Segner, Monge, and Dr Young, that the phenomena were due to the tension in a flexible sheet forming the capillary surface; and the mathematical discussions of La- place and Poisson; the author stated his agreement with those who considered these latter as only having obtained their results in accordance with Dr Young’s hypothesis of a constant tension in the sheet of inappreciable thickness forming the capillary surface. 22 MM. Frankenheim, Sondhaus and Brunner have shewn the phenomena of capillary attraction to be functions of the tempera- ture and not of the density of liquids. The author stated that he had employed no new principle in his method of solution of the problem of the forms of the capil- lary surfaces, but had employed in a different manner the recog- nized properties of fluids, and treating each case as a distinct hydrostatical problem, solutions were obtained in the first instance for the cases of vertical parallel plates near together, and tubes of small diameter, with their lower ends immersed in liquid, by the statical property of forces acting on bodies, that the vertical and horizontal forces must balance each sepa- rately amongst themselves; and that a fluid by Pascal’s prin- ciple may be considered as separated into distinct portions by imaginary rigid films without the state of the fluid being altered. The vertical force exerted along the line of contact of the solid and liquid is shewn by Laplace to be the force which supports the weight of the liquid in capillary elevations abeve the level of the outside fluid, and the same must be the case when the liquid is separated by imaginary rigid films into portions like- wise. ‘This vertical force is balanced by the vertical component of the tension in the capillary surface of the liquid at the line of contact. The tension which may exist in the surface is limited by the attraction of aggregation from which it must be distinguished, as it is a force transmitted from the impressed force at the line of contact, and as in flexible solids, such as cords and sheets, it may vary from nothing to the utmost the attrac- tion of aggregation permits, The horizontal forces are only the horizontal components of the tension, which must therefore balance amongst themselves as in the catenary curve. For the capillary depression the procedure is the converse of that for capillary elevations. With these considerations it was found that between vertical parallel plates for a first approximation the form of the perpen- 23 dicular section of the capillary surface was a parabola, from which a second approximation furnishing a small correction of the first was easily obtained. In tubes of small caliber a like method shewed the section of the capillary surface by a ver- tical plane through the axis to be a parabola, as a first ap- proximation, with a small correction of the result as a second approximation. , Success with these cases led the author of the paper to in- vestigate the form of the capillary surface for a liquid of indefinite extent in contact with a single plane vertical surface of a solid of which the lower edge was immersed in the liquid. This is the first case treated in the paper as the fundamental one of capillary attraction. By taking an elementary vertical prism in the liquid held above the original level, and finding the condi- tions for equilibrium amongst the forces acting at ifs upper sur- face, a differential equation for the tension was obtained, and from the consideration that the tension must be the minimum pos- sible at each point when the liquid is at rest, the calculus of variations gives the form of the section of the surface by a ver- tical plane perpendicular to the plane of the body as an ex- ponential curve with the equation z=h. e where / and & are constants, z the vertical and y the horizontal ordinate of any point in the surface, with the origin the point where the ver- tical solid meets the level of the liquil at a great distance. An addendum to the paper contained investigations for the forms of the capillary surfaces between vertical parallel plates at any distances and for tubes of any diameter without approxi- mation For parallel plates at any distances the equation of the section of the surface by a plane perpendicular to them was. found to be expressed in finite terms as follows; ’ y v z -* fe™ + e “I. where h’ and m are constants, and y the vertical and hori- 3—2 24 zontal ordinates of any point in the surface, with origin a point equally distant from the plates, and the axis of y perpendicular to them in the level of the outside liquid. For tubes of any diameter held vertically in a liquid a differential equation was found and a relation between the ordinates expressed in an infinite series, but an equation in finite terms was not found. The equation for parallel plates is similar to that for the cate- nary curve for a wniform chain, but differing from it in having the constants different from each other. The first terms of the expansions in series of these two cases give the results which were before found by approximate methods. . In the discussion which followed, ProrEessor CHALLIS made some remarks upon the difficulty of the subject which Mr Potter had been investigating, but reserved his approval of the method followed until he could give the argument a fuller consideration. Proressor STOKEs objected to the method pur- sued, and was unable to agree with the results obtained. May 7%, 1866. Pro rEssor StoKxes (Senior Member of the Council present) in the Chair. On the Root of any Function; and on Neutral Series, No. II. By Prof. De Moreay. Tue author divides algebraical thought into guantitive and structural. A quantitive proposition is seen in ‘A value of x can be found, so that dr=a’: a structural proposition in ‘If y =x, there exists a form, ¢”’, such that e=qy.’ These 25 propositions are co-extensive: but the first is never either af- firmed or denied; the second is always tacitly assumed. Mr de Morgan sets out as much of. the differential calculus as he wants, from quantitive considerations alone: he demands nothing but values of his functions, and does not need to know even the form of the differential coefficient of a product. Argument on difficulties apart, such difficulties as are always watched with keen eyes by those who examine a new theorem as it ought to be examined, the proof that every function has a root is so simple that it is its own abstract. If 6 (@+y/—-1) =P+Qv-1, and if dP : dx be represented by PB, &c., we have P,= Q,, P,=—Q,. It is easily shewn that the families of curves P,=h, Q,=k, are orthogonal trajec- tories; that is, individuals, one of each family, always meet at right angles. Here / or k may be nothing or infinite. Conse- quently, P, and P, cannot both vanish, or both become infinite, except at isolated points. And it is easily shewn that one of the two, without the other, can only become infinite at isolated points. Hence a curve may be drawn from any one point to any other, so as not to pass intermediately through any point at which there is impediment to simple quantitive reasoning upon the equations ; dP=Pdua+Pdy, dQ=—Pdx+Pdy, or their consequences (R= P,?+ PB?) Rda=PdP—PdQ, Rdy=PdP+ Pad. Let it be required to solve ¢ (a+ y/—1)=a+b/—1; or to find x and y so that P=a, Q=b. Let w=), y=p, give P=l, Q=m; and, P and Q being co-ordinates, draw a curve which shall avoid all impediments from the point (/, m) to the point (a, b). Divide a—1, b—m, into d,P+..+d,P, and d,Q+...+4,Q, as conceived in a common integration, each value of dQ being obtained from the (P, Q) curve by means of a value of dP. 26 From i, uw, @,P, d,Q calculate dx, dy: from 1+d,2, w+dy, d,P, d,Q, calculate dx, dy; and soon. We thus arrive at A+daet..t+de=a wtayt...+dyg=P, the values of x and y which solve the required equation, This proof involves, though in a laborious way, the approximate: de- termination of the value of «+ y4/—1 required, by use of small instead of infinitely small subdivisions. | The part of the paper which relates to neutral series is. recon- sideration of some difficulties connected with the method of the former paper on the same subject; it admits of no separate abstract. The chief point is the double approach of . , — A, + A, — A, + vee + Aon — Gans to the limiting form 1—-1+1-—-1+... ad inf. May 21, 1866. The Prestipent (H. W. Cooxson, D.D., Master of S¢ Peter’s College) in the Chair, Notes on the Cetacea which have lately been taken on the Welsh coast, and on some other additions to the Museum of Comparative Anatomy. By J. W. Cuark, M.A. 3 Arter a short sketch of the present state of knowledge of the Cetacea, and a few words on the difficulty of obtaining speci- mens in a perfect condition, Mr Clark described some examples of Delphinus Tursio which had recently been obtained for the Anatomical Museum at Cambridge. These were from a shoal of sixteen which came ashore near Holyhead, of which two 27 tolerably perfect examples had been purchased, parts of which were exhibited. In their stomachs were found whelks, crabs, and a considerable number of pebbles; and in one a conger eel. He drew attention to the fact that the teeth were not universally blunted as it had been stated. On certain alleged misrepresentations and discrepan- cies in Plato's Theatetus. (Grotr’s Plato, Vol. i. c¢, xxvi.) By E. M. Copz, M.A. THe Author commences by expressing the obligation which he, in common with all scholars, feels toward Mr Grote for the light which, by his extensive learning and independent criticism, he has thrown, in general, upon the Platonic doctrines and writings. Plato stood in especial need of an impartial statement and thorough sifting of the doctrines attributed to him; owing to the preconceived notions which had so influenced those who had un- dertaken the task of interpreting him, that students were too often led to look for what Plato might have said or ought to have said rather than what he did say. Mr Cope gives two instances of these bars to the right un- derstanding of Plato. (1) The theory held by Schleiermacher, that Plato’s dia- logues are not only all pervaded by one spirit and tone of feeling, but also can be fitted together so as to form essential parts of one complete system. (2) A tendency to unduly harmonize and interpret by inference rather than by the direct statements of the author, and so to represent the miscellaneous collection of the Platonic 28 dialogues as one body animated by one soul, as a harmonious whole, pervaded in all its parts by a central unity of purpose governing the whole. Mr Cope then proceeds to state the professed object of the Thestetus: viz. the determination of the nature and character- istic distinctions of knowledge as compared with other modes of apprehension belonging to the human intellect, namely, sensa- tion or sensible perception; opinion or belief, Sofa or admis 8é£a; or this accompanied by Aéyos. He considers it to be the first serious attempt at a psychological analysis of the faculties or modes of apprehension of the mind, of which it suggests that there are three, sensation, belief, knowledge. _ Mr Cope maintains that the question involved in this discus- sion of the Protagorean dictum ‘zravtwy pétpov avOpwrros’ is no mere dialectical encounter of wits, but one of the highest interest, namely, “Is there any such thing as truth? if so what is it? is there any standard of truth and knowledge independent of our- selves our own feelings and momentary consciousness?” Mr Cope then proceeds to consider the following questions: (1) Has Plato misrepresented Protagoras’ theory? (2) Has he refuted it, or is it really true as Mr Grote - holds it to be when properly interpreted? (1) Mr Cope argues against the probability of Plato having misrepresented Protagoras’ theory either wilfully or through ig- norance, by a close examination of the discussion in the Thezete- tus; and further brings forward evidence from Aristotle, Diogenes Laertius, Sextus Empiricus, and Simplicius, to prove that he has not done so. Mr Cope then shews, from an examination of the sense in which words expressive of mental faculties or processes are used in the Thestetus, that there is an a priori probability that Protagoras was ignorant of any distinction between sen- sation and thought or knowledge; and therefore confined his theory of the subjective standard of truth to the apprehension of objects by sensation. 29 (2) Mr Cope examines Plato’s arguments against Protagoras which are mainly these: (a) Protagoras asserts that the views, opinions, beliefs of everyone are true to himself as he conceives them; but the majority of mankind think Protagoras’ theory wrong; therefore this opinion is true to them and yet is contradictory to the theory; therefore the theory is in conflict with itself and so is false. (8) Even if there were no other than the subjective standard of present truth and present right, there must be of future ; for in that all men’s opinions and thoughts are not equally true and valid. The man of science can, in certain cases, predict what will happen in future time—the iSiétns cannot; his fancies and impressions are here of no value; and are hazait to be false even to himself. Mr Cope developes these arguments and shews why a con- siders them to be satisfactory. _ The paper concludes with a defence of Plato against a charge of self-contradiction in the statements and views of the Thezxtetus and Sophist on the subject of ‘false opinion.’ The Master or TRINITY made a few remarks; in the course of which he stated that, in his opinion, the ideas of Protagoras took a somewhat wider range than the senses alone; and that Plato was not primarily attacking Protagoras in the Theetetus, but some contemporary Sophist. NOTICE. At a Meeting of the Council of the CamBripGe PHILOSOPHICAL Soctery, Nov. 12, 1866, the following Resolution was carried : “That every Fellow of the Society, who has paid his sub- scription for the year, or the composition in lieu of annual payments, be entitled to receive a copy of every Part of the Transactions published during the year to which his subscription refers.” Any Member of the Society can have his Part on personal application, or on sending an order (with stamps for postage, if necessary) to one of the Secretaries. —** A Part will be ready at the commencement of the Lent Term, 1867. PROCEEDINGS | bition peg a a ! ; y¢ Philosophical Society. 33 October 29, 1866. (ANNUAL GENERAL MEETING.) The Prestpent (H. W. Cooxson, D.D., Master of St Peter's College) in the Chair. The following Officers were elected for the ensuing year : President. Rey. H. W. Cooxson, D.D., Master of St Peter’s College. Vice-Presidents. G. E. Pacet, M.D., Caius College. Rev. Jas. CHALuis, M.A., Plumian Professor. G. G. Stoxes, M.A., F.RS., Lucasian Professor. Treasurer. Rev. W. M. Campion, B.D., Queens’ College. Secretaries. C. C. Basineton, M.A, F.RS., F.LS., Professor of Botany. G. D. Liverne, M.A., Professor of Chemistry. Rey. T. G. Bonney, B.D., F.GS., S¢ John’s College. The following were elected new members of the Council : F. A. Parry, M.A., St John’s College. I. Topaunter, M.A., F.R.S., St John’s College. J. W. Cuark, M.A., Trinity College. The following continue Members of the Council : A. Newton, M.A., F.LS., Professor of Zoology. G. M. Humeury, M.D., F.RS., Professor of Anatomy. A. Caytery, M.A., F.RS., Sadlerian Professor. Rev. H. A. J. Munro, M.A., Trinity College. Rev. G. F. Browne, M.A., St Catharine’s College. Rey. W. Setwyn, D.D., Margaret Professor of Divinity. Rev. W. G. Criark, M.A., Public Orator. R. Potter, M.A., Queens’ College. Rey. N. M. Ferrers, M.A., Caius College. 34 Note on the Halo of 22°. -By W. H. Miuier, M.A., For. Sec. Rk. S., Professor of Mineralogy in the University of Cambridge. Ir had long been conjectured that the halos of 22° and 46° were produced by the refraction of the light of the sun through prisms of ice descending slowly through the atmosphere. The physicists who undertook to compare this hypothesis with observation assumed the value of the refractive power of ice, and in some cases they also assumed the existence of forms in ice which had never been observed. M. Bravais in an exhaustive Memoir on Halos, inserted in the 31° Cahier du Journal de VEcole Polytechnique, and in Annuaire Météoro- logique de la France, Année 1851, gives an account of a very careful determination of the refractive power of ice made by himself, and shows that the radius of the halo of 22° agrees with sufficient accuracy with the minimum deviation of light through the alternate faces of the regular six-sided prism 0 1 1, and that of the halo of 46° with the minimum devi- ation through a face of the form 1 1 1, and a face of the form 0 1, 1, faces making a right angle with one another, By measuring the minimum deviations through a prism of ice, Bravais obtained for the indices of refraction the following values :— Middle of red ....... a ee | 1:307, Middle of orange ............ 13085, Middle of -yellow ............ 13095, Middle of green ............. 13115, Middle of blue and indigo 1°315, Middle of violet .............. 1°317. As the observation presents many difficulties, and the value of the research depends entirely on the accuracy of these 35 numbers, it may be worth while to compare one of them, the yellow or the brightest part of the spectrum, with the value obtained by myself in a different manner. A prism cut from a thick plate of ice, with its refracting edge normal to the surfaces of the plate, was placed in a vessel having parallel ends of plate-glass filled with water ob- tained by melting a portion of the same plate of ice, to a depth of twice the thickness of the prism. The vessel containing the prism floating in water was placcd so that a distant slit having a salted spirit flame behind it could be viewed with the telescope of a theodolite through the compound prism of ice, water and the plates of glass, and the image of the slit bisected, the prism was then depressed to the bottom of the vessel, and the slit as seen through the. water and glass plates alone bisected. In this way the refraction out of ice into water was obtained unaffected by any small error in the de- termination of the angle of the ice prism, the impurity of the ice, and the want of perfect parallelism of the plates of glass. The resulting index for the line D out of ice into water at 0° was 1°01952. According to Jamin (Comptes Rendus, Tom. XL. p. 1191) if w,, mw, denote the indices of refraction of water at 0°O and °C respectively bt, = fly -— 0°000012573¢ — (0:000001929) #, up to 30°. With the aid of this formula, which yields results agreeing closely with those obtained by Dale and Gladstone (Phil. Trans. for 1858, p. 889), the indices of refraction of water observed at various different temperatures may be re- duced to their value at 0°. Of the observations by Fraunhofer, (Denkschriften der k. Akademie zu Miinchen, B. V., 8. 223, 225), the third and fourth at 8° R and 9}° R respectively, are the indices of a ray which he denotes by J, coincident with the double line D of the solar spectrum. The observations by Powell (Zhe Undulatory Theory 36 as applied to the Dispersion of Light), and by Dale and Gladstone, were made with the instrument described in the Proceedings of the British Association for 1839. It differs from the instruments usually employed in researches of this kind in having the circle vertical instead of being horizontal. This arrangement interferes with the possibility of measuring a double deviation, and renders the telescope liable to flexure ; the absolute results are consequently less deserving of con- fidence than those which are comparative. Van der Willigen’s observations are given in Poggendorff’s Annalen, B. 122, 8. 191. Observer. Temp. C. Index for D at t®? C. Index at 0° C. Fraunhofer ...........- 18°75 1333577 1:33449 Fraunhofer ...:\ sss. 18°75 1°333577 1:33449 Fraunhofer ............ 10 1:33358 1:33390 Fraunhofer .........+++ 1156 —-1:33359 1:33399 Powoll sis cies ss ok 15'8 1:3343 13350 — Dale and Gladstone 0 1:3330 1:3330 Van der Willigen ... 16°58 133332 133406 Van der Willigen ... 22°37 133282 133400 Mean 1:33412. But the index for D out of ice into water is 101952. Hence the index for D out of air into ice is 130858. The value of this index found by Dale and Gladstone is 13089. Fraun- hofer estimates the place of the brightest part of the solar, spectrum to be at a distance from D equal to one-third or, one-fourth of the distance between the lines D, #. Taking this distance equal to (0:29) DH = (0:062) DH, and observing that according to Dale and Gladstone the index of refraction of water at 0° for the line H is 0:0108 greater than for the line D, the probable index of refraction of ice for the brightest part of the solar spectrum will be 130925. The value 1°3089 of the index of ice for the line D gives for the index of the brightest part of the spectrum 13096. These values agree 37 so closely with the value 1:3095 obtained by Bravais as to leave no room to doubt the adequacy of the received hypo- thesis. Its truth has been still further confirmed by the occurrence of a solar halo formed on the ground. (Listing, “Ueber einen in Russland von Herrn Korsakoff beobachteten terrestrischen Sonnen halo, nebst bemerkungen iiber das krystallisirte Wasser.” Poggendorff’s Annalen, B. 122, S. 161.) M. Korsakoff found the radius of the middle of the red ring - equal to 22°15’ by measurement with a sextant, and traced the light to small prisms of ice scattered on the ground. When examined under a microscope these were found for the most part to be hexagonal prisms terminated at one end by a plane normal to the axis of the prism, and at the other end by a hexagonal pyramid. They were from 0°38 to 0°66 millimétres long, and from 0°13 to 0:19 millimétres thick. Crystals of ice exhibiting combinations of hemibedral forms with inclined faces had been approximately measured and described by N. A. E. Nordenskidld, (Poggendorff’s Annalen, B. 114, 8. 662). They were combinations of the rhombohedral forms 111,011, «231, «120,«153. The angles were Pha, 0 1.1=90°; 121,23 1=38' 57’; 11.1,120=58°14; 111,15 3=81°S!’ Another instance in which the formation of a halo was distinctly traced to the refraction of the light of the sun by an assemblage of crystals of snow is recorded in the first Jahrbuch des Schweizer Alpenclub. On the 3rd of January, 1864, Professor Ritz saw a halo in the cloud of loose snow stirred up by a violent wind at the entrance of the upper valley of Kandersteg. This view of the formation of halos is supported by the observations of Arago and Sir David Brewster, who ascertained that the light of halos was polarized in the direction of a tangent to the are at the point observed. Bravais says that 38 the evidences of tangential polarization are frequently more or less masked, especially near the ends of a horizontal diameter, by the atmospheric polarization the direction of which is radial. The polarization of a halo of 22° observed by myself at Rosen- ‘laui, in August last, was very distinct, and did not appear to be sensibly enfeebled at any part of the circle by the polar- ization of the atmosphere. This was probably due to the dimi- nished amount of atmosphere above the place of observation, the height of which was about 1340 métres above the level of the sea. The polariscope employed was Savart’s. On account of the small breadth of the ring of light it was difficult to deter- mine the plane of polarization by noting whether a dark or a light band occupied the middle place in the system of bands. But by holding a piece of varnished paper at a proper in- clination, so that. when viewed through the polariscope it ap- pearéd to be in contact with the arc of the halo, the dark bands parallel to the radius of the halo produced by the light reflected from the paper, coincided with the light bands seen in the halo. Consequently, the light from the paper being polarized by reflexion in the direction of a radius of the halo, the light of the latter was polarized in the direction of a tangent to the arc, i.e. was polarized by refraction. PROFESSOR CHALLIS stated that the explanations of the die: nomena of halos were simple, and ought to be included in the University course of Mathematical study ; he also adverted to the importance of the instance of the halo formed on the ground. Mr Potrer (Queens’) gave a brief sketch of the history of the observation of prisms of ice. He mentioned that in the course of a voyage from New York a halo 22° was observed in a cloudless sky, which was followed by stormy weather. Further Experiments on the Synthesis of Organic Acids. By A. BR. Cartoy, M.A., F.R.S.E., St John’s College. November 12, 1866. The Prestpent (H. W. Cookson, D.D., Master of S¢ Peter’s College) in the Chair. The following new Fellows were elected : J. V. Durewt, M.A., St John’s College. G. W. WELDoN, M.A., Trinity College. S. W. MitcHeLt, B.A., Caius College. Abstract. The Laws which have determined the distribution of Life and of Rocks. By Mr H. G. Srszey, F.G.S. In the first part of this paper the author, having assumed that the Earth had cooled from Igneous fusion, discussed the nature of faults, earthquakes, changes in the level of land, &e. The second part was an investigation into the influence of water in distributing and modifying the relations of the material of stratified rocks, so discovering principles of their necessary sequence in space and in time. The third part discussed the possibility of modification and development of the groups of life called species, &c. by the in- fluence direct and indirect of the Sun’s energy, accumulated in the organism as work; and the influence of this force on the distribution of life. The fourth part consisted of an investigation of the effect produced by the forces discussed in Part I. on the distribution of rocks and of life arrived at in Parts II. and III., whereby the author explained the present distribution of life on the Earth, 40 Abstract. On the Potton Sands. By Mr. H. G. Szeuey, F.G.S. THE author described the English deposits between the Kim- meridge clay and the Upper Greensand and Gault ; especially as seen in the sections from Hunstanton to Oxford and Farring- don. The author proposed to restrict the name “ Carstone ” to the district where Sands are the only deposit from Kimmeridge clay to the Red Rock of the Wolds. He proposed that by Shanklin (or Lower Green) sand, as it is commonly called, should be understood the deposit between the Wealden and the Gault. But by “Potton sands” he proposed to distinguish the strata which occur as sands between the Gault and the Kim- meridge clay. The author thought that they represented the whole series of strata from Portland sand upwards, and com- pared them with the German series described by Dr Oppel. The deposits are separated into Upper, Middle and Lower, the latter two being very fossiliferous. The fossils resemble in many ways those of the Farringdon sponge-gravels. The origin of the rock-material and of the fossils were dis- cussed in detail by the methods of the previous paper. November 26, 1866. The PresipENT (H. W. Cooxson, D.D., Master of St Peter’s College) in the Chair. The following new Fellow was elected: T. GwaTKIN, M.A., St John’s College. The comparatively late date of our Homeric Texts. THIS paper was an attempt to show, that although the cha- racters and events of the Trojan war are undoubtedly of great 41 antiquity, the composition of the Iliad and the Odyssey, as we now have them, may probably be referred to a period even sub- sequent to the age of Herodotus. The author argued, from an examination of the numerous passages in Pindar that refer to the Trojan war, that that poet could not have known our Iliad, but that other ancient Epics took their place in his time, the same, in fact, as the Greek Tragic writers used for the themes of their plays. Out of about fifty passages in Pindar, and even more than that number of Greek Plays, the titles of which, and to some extent the subjects, are known to us, the writer contended that only two, the Rhesus and the Cyclops, could be referred to our texts of Homer; while out of at least fifty passages in Plato, where Homer is mentioned by name, and citations made from him, there are none which do not refer to our present texts. Hence it was argued, that the compilation of our Homer from those older epics was made between B.c. 450 and 400; that its preservation was due to the fact that it was from the first a written work, and that it finally superseded the more ancient epics, which were recited by the rhapsodists, from the superior merit of the poetry as well as from its being better adapted to the advanced literature of the period when it was written. The more comprehensive poems on the T’roica, which are commonly held to be later in date, secondary in importance, and merely supplementary to the Iliad and the Odyssey, were shown to be in all probability the original poems, reduced to writing probably after the composition of the Iliad and Odyssey, but wholly unable to compete with them in merit, and therefore regarded in much later ages, in which we first find any separate’ account of them, as the inferior productions of post-Homeric poets. The author further argued from the remarkable resem- blances between the style and diction of Homer and Herodotus, and from the generally Asiatic character of the Homeric similes and scenery, that the compiler of our poems was an inhahitant of Asia Minor, and that possibly the poet Antimachus of Colo- 42 phon, who is recoided to have been an “editor” of Homer in the age of Pericles, may have had more to do with the actual authorship than is commonly thought. This theory was shown to be supported by the very modern style of a great deal of our Homeric Greek, combined, as might be expected, with archaic forms retained from the earlier epics. Evidence to the same effect was adduced from the early Greek Vases, The Pusiic Orator (Rev. W. G. Clark, Trinity) stated that Hipparchus was said to have introduced the Homeric poems into Greece, and to have had them committed to writing; that it was very improbable that a work of such magnitude should have been introduced without detection at an age such as that assigned by Mr Paley as its probable date. The impression produced ‘upon his mind by reading the Iliad and Odyssey (which he had done on the spot) was that they were poems belonging either to different ages or to different states of society. He accounted for the occurrence of modern forms in the poems by supposing that they might have crept in from time to time until the text became fixed. Writing had been in use in Egypt at a very early date; for instance, there was a Hieratic papyrus in the British Museum to which the date 1100 B.c. was assigned. Good subjects for Tragedies were not to be found in the Homeric poems, and this was sufficient to account for the rarity of Tra- gedies founded on them. Mr Patry replied that he did not suppose that the Homeric poems mentioned in connexion with Hipparchus were those which we possess ; and that there were no words for ‘ reading’ and ‘ writing’ in early Greek, so that although they might be known in Egypt, they were not in Greece. NOTICE. AT a Meeting of the Council of the CAMBRIDGE PHILOSOPHICAL Society, Nov. 12, 1866, the following Resolution was carried : “That every Fellow of the Society, who has paid his sub- scription for the year, or the composition in lieu of annual payments, be entitled to receive a copy of every Part of the Transactions published during the year to which his subscription refers.” Any Member of the Society can have his Part on personal application, or on sending an order (with stamps for postage, if necessary) to one of the Secretaries. *.* Part I. Vol. XI. is now ready. (Postage Gd.) The Index to Vol. X. will be published with the neat Part. re ; Ke Se, Beta a ; ‘ sa wae : ae " Hespanie GAN FA Ts os aaa TK ie Heth aha ¢ ees. ebin aie Pp tape’ GH hase "Barn ican by a % ant: — Laiped BETS PS bead 5 BON edit ee Aen oa UN tise § net a Mise one - i February 18, 1867. The Prestpent (H. W. Cookson, D.D., Master of St Peter’s | College) in the Chair. The following new Fellow was elected: Rev. Josepu Cast.ey, M.A., Pembroke College. On the continued change in an Eye affected with a peculiar malformation. By Grorce Bippew. Ary, M.A., LL.D., D.C.L.; formerly Fellow of Trinity College; and late Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge ; Astronomer Royal. ON two occasions, of which the first occurred forty-one years ago, I have communicated to the Cambridge Philosophical Society the description and numerical elements of a malforma- tion affecting my left eye. In the first communication, dated 1825, February 5, I gave as the characteristic of the mal- formation this result of experiment; that, on viewing a very small luminous point, formed by the light passing through an extremely minute hole in a blackened card, the luminous point, at one distance, presented to the malformed eye the appearance of a sharply defined line nearly horizontal, and, at a greater distance, presented the appearance of a sharply defined line at right angles to the former line, and therefore nearly ver- tical. And, as numerical elements, I gave the measures of the 5—2 48 two distances of the luminous point from the front of the cornea (with sufficient accuracy) when these two appearances presented themselves; the card being made-to slide upon a graduated scale whose end rested against the orbital bone of the eye, and the eye view being directed steadily forward. I may here men- tion that the method of experimenting in the subsequent obser- vations has been precisely the same. The second communication was dated 1846, January 14, or more than twenty-one years after the first. In this, besides giving the elements of astigmatism of the left eye, in the same manner in which I had given them in 1825, I gave the measure of the focal length of the right eye, which is perfectly stigmatic. After another period of more than twenty years, I am now enabled to give similar measures for both eyes. The comparison of the similar elements at the three different epochs may perhaps be interesting. I. Distance from the cornea of the left eye at which the luminous point presents the appearance of a nearly hori- zontal line. In 1825, 3°5 inches; Reciprocal = ‘286. In 1866, 47:-... 25 ee ‘213. In 1866, 54 ... ; = 185. ts 028. Difference — ‘078. II. Distance from the cornea of the left eye at which the luminous point presents the appearance of a nearly vertical line. In 1825, 60 inches; Reciprocal -166. in 184, 89 op ‘112, In 1866, 106... 3 a Difference — 054, III. Measure of the astigmatic power of the left eye at dif- ferent epochs; estimated in each case by the differences of the reciprocals in the last table. et, 49 In 1825, “120. In 1846, “101. In 1866, 091. * — 010. Difference — °019. IV. Focal length of the right or stigmatic eye, measured from the front of the cornea. In 1846, 4°7 inches; Reciprocal ‘213. In 1866, 55 ... ; uA 189. Difference — ‘031. The following conclusions appear to be supported by the examination of these numbers. 1. The change in both focal elements of the left eye is far greater between early manhood and full manhood than between full manhood and approaching age; and the change in the astigmatic power is decidedly greater between early manhood and full manhood than between full manhood and approaching age. 2. The most constant of the elements is the astigmatism. 3. Through the period 1846—1866, the distance for the left eye corresponding to the appearance of a nearly horizontal line is sensibly the same as the focal length of the right eye. G. B. AIRY. Roya, OsservatTory, GREENWICH, 1866, November 19. On some points in the Anatomy of the Chimpanzee, and the consideration of the term ‘ Quadrumanous,’ as applied to that animal. By Prorsssor Humpury, F.R.S. His remarks were the results of the recent dissection of two Chimpanzees, and referred chiefly to the differences between their lower limbs and those of Man. He pointed out that the 50 outer condyle of the thigh bone is round instead of being pro- longed from before backwards, and flattened beneath, as in Man. Hence there is comparatively little security afforded by the ligaments in the straight position, and little provision for the maintenance of the erect posture. The bones and joints of the ankle were shown to be constructed so as to permit free move- ment rather than to bear weight. With regard to the term “hand,” and the objections which have been urged to its appli- cation to the lower limb of the monkey, Professor Humphry remarked that if we use the term to designate a certain modifi- cation of the fore imb—a certain deviation, that is, from the ordinary fore foot—we may with equal propriety apply it to a corresponding modification of the hind limb—a corresponding deviation, that is, from the ordinary hind foot. We must not expect it closely to resemble the human hand, but merely to present such a similarity to it as the special features of a hand, viz. the shortness, mobility and opposeableness of the thumb and the relative length of the other digits, would give it. Judging by this rule, it is as correct to speak of the “hind hand” of a monkey as of the “fore hand,” though, forasmuch as both are employed in progression, it may, perhaps, be better to use some other term, such as “cheiropod” for the ‘designation of the class, leaving the term “bimanous” to indicate the characteristic feature of man. The psychical qualities should not be omitted in con- sidering the distinctive features of man; and the importance of the long, strong, firm great toe in this respect was pointed out. Some other peculiarities in the limbs and in the prostate gland of the Chimpanzee were described. ‘March 4, 1867. The Presipent (H. W. Cooxson, D.D., Master of St Peter’s College) in the Chair. On Roslyn or Roswell Hill Clay-pit, near Ely. By the Rey. O. Fisoer, M.A., F.G.S. Rostyn or Roswell Hill Clay-pit has long been a standing puzzle to Cambridge geologists. I have visited it several times, and have notes upon it made in 1853 and 1856. I was there in November of the present year, having by Professor Sedg- wick’s permission the assistance of H. Keeping. The pit is probably well known to you. It covers several acres of ground, and extends in a direction N.W. and S.E. The material has been used for the purpose of making up the banks in the fens, and the section is comparable to that of many natural cliffs. The northern side of the pit is occupied by horizontal Kim- meridge clay, which is or used to be capped here and there by a thin covering of lower green sand. At the western end of the pit boulder clay of a typical character abuts against the Kimmeridge clay, the plane of junction running nearly east and west, and dipping at a high angle under the boulder clay. The southern side of the pit, as at present exhibited, shows a blueish grey cretaceous clay, flanked at either end by nearly vertical chalk marl, which becomes somewhat argillaceous towards the eastern end of the pit. The chalk marl and clay are evidently in true sequence. The question which I propose to discuss relates to the sin- gular collocation of these several beds. I will first of all consider the presence of the chalk marl and the clay on the northern side of the pit. I have said they are clearly in sequence. Their junction is perfectly natural. The chalk marl becomes sandy, and contains 52. a few scattered nodules of phosphate of lime, with some of the fossils of the upper green sand usual in the neighbourhood of Cambridge, and then the clay succeeds. The character of the beds here has been so well described by Mr Seeley’, that I need not say more about them, except that I think it open to question whether the clay is really gault. There is about Cam- bridge a band of clay in the lower part of the chalk, which was well shown in the construction of the waterworks at Cherry- hinton, and I am rather inclined to think that the Ely clay belongs to the same bed. I recollect phosphatic nodules oceur- ring in connection with it at Cherryhinton. I think the abun- dance of shells, and especially of Perna, in the Ely clay, rather militates against its being gault. But I merely throw out this as a suggestion’, Boring by Mr Docwra at Cherryhinton waterworks noted in 1854. “Soil ' ; «10 Le Plastic . i ‘ te 8S 0 Upper green sand with fossils . Or ee Clunch j : 6: ge Sand. : ; ; -> Ore Gault not pierced Lt eel ae 82 I visited the spot and found belemnites in the “Plastic,” and coprolites from the supposed upper green sand. From a subsequent cutting made to convey water from near Mr Okes’s house, I recollect observing that the above-mentioned “plastic” was a stratum in the clunch. This section makes a coprolite layer beneath the clay. There may also be one above it. 1 Geol. Mag. Vol. 11. p. 529. * At a subsequent visit, 26 Apr. 1867, I saw the lower green sand in sequence to this clay, which would make it the true gault. In another part of the pit the gault reposed on boulder clay with chalk pebbles. At any rate we may look at the chalk and clay near Ely as a single mass, and whatever accounts for the presence of one will equally account for the other. In short they are a large mass of cretaceous beds in a nearly vertical position, with boulder clay abutting upon them. The curved lines of junction as seen in the section are nothing more than the curves formed by the in- tersection of the surface of the workings with a nearly plane surface of junction between the chalk and the gault, dipping at . a very high angle towards the north. Now there are two ways of accounting for the presence of this cretaceous mass. It is either brought up by a fault with reference to the boulder clay, but down with reference to the Kimmeridge, or else it is a huge boulder, forming as much an integral part of the boulder clay as any block of oolite or flint which it contains. Mr Seeley appears to consider its presence best accounted for by a fault, but I think I shall be able to show that the other is the more probable explanation. And to clear away any possible a priori objection drawn from the magnitude of the mass, I would beg to remind you that chalk boulders occur in the Norfolk drift so large that quarries and limekilns are worked in them. Extract from a letter by the Rev. John Gunn :— IRsTEAbD, Dec. to, 1866. “Of the masses of chalk you enquire about, that near Castle Rising is now exhausted and used for top-dressing land. Only the large flints remain to prove that the mass belonged to the upper chalk which does not remain anywhere in that part of West Norfolk. “The largest detached mass I know of is between Cromer and Overstrand. I do not know the precise boundary of the 54 parishes. It has been for years used for lime and a kiln is on the premises. Mr Prestwich as well as myself noticed a layer of sand beneath it. (Obs. They evidently looked for proof that the mass was not in situ, showing how nearly it simulated a natural bed of chalk.) “In several places north of Cromer from that place to Sher- ringham are large masses of bouldered chalk proved to be so by the underlying beds. “On the south side also at Barton and Happisburgh there were some but they have all been washed away. In North Walsham, Worstead, and Witton large bouldered masses have from time to time been worked either for making lime or for top-dressing. A tooth of Hlephas prim. was obtained by me at Witton in connection with one. “The large masses at Trimmingham figured in Lyell’s ele- ments are part of the fundamental chalk, remnants of an upper bed from which the gravel of East Norfolk is derived.” These instances show that the mere size of the mass of cretaceous strata at Ely is no argument against its having been carried thither by ice, and the fact of its consisting of portions. of two distinct beds is a mere accident. There is nothing singular in so large a block of chalk be- coming detached from its parent bed. For some miles along the coast west of Lyme Regis landslips on a large scale have occurred where masses of chalk and green sand fully equalling in bulk the mass at Ely have fallen from the cliff. The last of these falls occurred not many years ago. The lower portion of the disengaged strata consisted of a sandy loam, the upper of chalk. If we could conceive such circumstances under a glacial climate that this mass could have been floated away, as for in- stance by snow blowing over the top of the cliff and being frozen on to its face, we should have all the conditions necessary for the deposition of an immense boulder like that at Ely; dd and on this supposition we might expect it to have been dropped in a similar position of verticality, for the float would have been attached to its edge. But without making such a supposition, knowing how frequently icebergs roll over in the process of thawing, we may expect them to drop their loads indifferently in all positions. I was originally disposed to think this mass a boulder, when 1 saw it ten years ago. It was then much less exposed than it is at present. I was quite confirmed in that opinion by what I saw the other day’. The mass seems to be much of the shape of a great punt, the prow of which is directed towards the west. The mass is so thick towards the stern, i.e. at the eastern end, that they have not dug through it, but towards the centre the workmen told me they found a “ snuff-coloured hard clay” at the bottom of the pit beneath the chalk, “hard clay” being the term by which they designate the boulder clay. Towards the western end of the exposure of the chalk this clay may be seen beneath it in the section, enclosing angular lumps of chalk. Towards the cottage on the bank the chalk thins out to nothing, the boulder clay passing beneath it. I believe the lower green sand blocks which occur on the north side hereabouts to be no more én situ than the chalk. The boulder clay between the chalk and Kimmeridge clay shows tortuous streaks of bedding, some of them chalky, in a highly. inclined position. They seem to have been originally horizontal as well as the chalk. But no one who has seen the Cromer cliffs will think any mode of bedding too strange to occur in the boulder clay, or call in the aid of a fault to account for it. The second part of the enquiry relates to the occurrence of this mass of boulder clay in juxtaposition to the Kimmeridge. The question lies between a fault and a great channel of erosion, made for itself by the glacial drift. | The only evidence upon this point to be obtained in the pit 1 See also note 2, p. 52. 56 is by examining the junction. If the country were mapped, and a fault affecting other strata traced through the pit, this would settle the question in favour of -a fault. There seem to be disturbances in the neighbourhood. We have Oxford clay, for instance’, at the bottom of the hill near the railway station, where Kimmeridge clay would have been more natural, And other places might be named (Aldreth and Alderforth), But that faults affect the oolites affords only a slight presumption that they will also affect the boulder clay. With regard to the evidence to be obtained at the spot itself, I first of all attempted to examine the junction by digging in the side of the pit; but I found that, owing to a line of springs thrown out by it, the boulder clay has slipped, so that I could not reach the undis- turbed ground. This circumstance misled me when I examined the place in 1856, and made me suppose the junction showed slickenslide, which was really due only to a recent slip. I then searched for and found the junction in one of the banks left by the workmen to exclude the water as they dig. Here I found it well defined, but I could not discover any of those symptoms of pressure, or the polished surfaces, which are always observed to accompany a fault. As far then as the evidence goes it is against the occurrence of a fault, and points to the boulder clay occupying a trough, which it has ploughed out for itself in the old sea bottom of Kimmeridge clay. Such troughs I believe to be not uncommon in districts bordering upon extensive spreads of the boulder clay. I have made diagrams of sections seen in two boulder clay pits at Gillingham in Norfolk, and at Bulchamp in Suffolk, which illustrate the manner in which the sea bottom has been eroded by icebergs, and the cavities filled with boulder clay. In the instance at Bulchamp, which I saw with Professor Liveing last summer, the sea bottom has consisted of a sand, 1 I have since learned, however, that a well at Ely commenced in the Kim- meridge soon reached the Oxford clay with a thin stony band containing Nerinea intervening. t a4 of an age one degree anterior to the boulder clay. This case has been, like that at Ely, adduced as an instance of faulting; but we noticed sand of the same character as that at the side of the section, clearly continued beneath the clay. In the other case the boulder clay has been originally depo- sited upon the same sand, but has been subsequently itself eroded down to its very base, and the channel filled again bya fresh deposit of slightly different materials. Ground Plan of the Ely Clay-Pit. The width from N. to S. is exaggerated. Railway. --" aoe om aed - =e oe nn € é 4 ag (a) Lower green sand. (b) Kimmeridge clay. (ce) Erratic clay, with boulders of granite, oolite, large flints, &c. (d) Chalk. (e) Gault (?) (f) Lower green sand. (gh) Line of junction. Note on a Case of Prismatic Structure in Ice. By T. G. Bonney, B.D., F.GS. On Jan. 26, 1867, the attention of the writer was attracted, while walking with a friend in the Fellows’ Garden of Christ's College, by the appearance of the ice on a pond. On pro- ceeding to examine a fragment from near the edge, he says, “it was about } inch thick, and was a mass of prisms with their axes perpendicular to the surface. The ends of these in one part were very irregular polygons; and the lines join- 58 ing opposite angles, speaking roughly, were on an average | about one inch in length; but in another part the forms ap- peared rather more regular—hexagons being common—the diameters of which were about 4 inch. Another fragment was then obtained rather further from the edge; this was a little more than one inch thick, and consisted of prisms whose ends were about the same size as those just described. Though the number of sides in their polygonal ends was not constant, six was certainly a common number, and this appeared to be rarely exceeded; the angular points of the polygons were a little blunted, so that the sides were slightly curved. The angles were thus difficult to estimate, and I had no instrument, but I do not think that they were constant. The ice broke very easily along the sides of these prisms, and never through or athwart one of them; so that each fragment had a beauti- fully dentate edge (see Fig.), and its side resembled a miniature group of basaltiform columns. Single columns could be easily detached quite perfect, with the point of a penknife, or even with the finger-nail. The ice contained a few air-bubbles and a chance bit of weed or bark here and there, but was in no other way remarkable. We carried a large piece of this last described to my rooms; and after hastily improvising a freezing bath of snow and salt, subjected some fragments of it to a temperature of about — 5° Cent. (for a short time the thermo- meter sank 3° or 4° lower, but this was the general tempera- ture). After leaving it here for some time we examined it carefully ; the prismatic structure was entirely obliterated, the only traces of it being the slight surface depressions here and there which marked the edges of the prisms, and certain vertical chains of small air-bubbles which had formed in the interstices at their angles. The ice was very hard, and, when broken, ex- hibited the usual conchoidal fracture. I was anxious to see whether the prisinatic structure would return as the ice thawed again, but unfortunately the fragment laid aside for this pur- pose got in contact with some of the melted salt and snow of the bath, and was dissolved too rapidly.” The author then expresses his opinion, derived from the examination of this case and of the ice in some of the glaciéres of Savoy, that the axes of the prisms are always perpendicular to the ice-surface, whatever its form may be; that the structure is only developed under peculiar conditions, viz. when thawing © takes place slowly and without disturbance in a place sheltered from wind and sun. The following table, from the register kept by Mr Pain, Sidney Street, shews the temperature in the neigh- bourhood during the days of thaw preceding Jan. 26. Jan. 22. | Jan. 23. | Jan. 24. | Jan. 25. | Jan. 26. At 9 A.M. 28°2 44°5 515 46 40 During past 24th. maximum ...| 30 44 53 55 50 minimum ... 26 22 43°5 42 34 March 18, 1867. G. E. Pacet, M.D., Vick-PRESIDENT, in the Chair. The following new Fellows were elected : G. H. Evans, M.A., King’s College. E. Carver, M.A., St John’s College. J. B. Brappury, B.A. and M.B., Downing College. R. K. Miter, B.A., St Peter’s College. 60 J. E. Frvcu, B.A., Trinity Hall. J. F. WALKER, B.A., Sidney Sussex College. B. W. Earte, B.A., Jesus College. W. K. Cuirrorp, B.A., Trinity College. On the difference of Longitude between the Society's Clock and the transit Clock of the Cambridge Observatory. By Proressor CHat.is, F.R.S. THE author described the instrument used and the means adopted in determining the difference, and stated that it was 5°88”, and that the longitude of the Society’s clock was 28°52” east from Greenwich. On the Meteoric shower of November, 1866. By Proressor Apams, F.R.S. THE author described the instrument used in the observation of the Meteors, and mentioned the various hypotheses which have been advanced concerning the orbit of these bodies; he explained the calculations which he had made to determine this, and shewed that the attractions of the Earth, Jupiter, Saturn and Uranus were nearly sufficient to account for a hitherto unexplained change of about 29 minutes in the position of the nodes of the orbit in each period of 33 years. He called at- tention to the fact that the orbit calculated appeared to coin- cide very nearly with those of certain comets; and held that the latter were elongated ellipses with a periodic time of 33 years, PROFESSOR CHALLIS and ProrEssoR MILLER made some remarks on this communication, in which the former expressed himself not quite convinced by the arguments of PROFESSOR ADAMS. Erratum. To title of Second Paper on page 40, add ‘by F. Patny, M.A., St John’s College.’ ¢ Philosophical Society. April 29, 1867. The PresipENT (H. W. Cookson, D.D., Master of St Peter’s — _ College) in the Chair. The following new Fellow was elected: T. W. Dunn, B.A., St Peter’s College. It was announced to the meeting that the Hopkins Prize was adjudged to Professor G. G. Stokes, Sec. R.S., for numerous memoirs on various questions in Pure Mathematics and Mathe- matico-physics ; for his discovery of the change of the refrangi- bility of light and the application of spectrum analysis in Optico- chemical investigations; and particularly for his Paper on the long spectrum of electric light published in the Philosophical Transactions for 1862, Part 11, printed and circulated in 1863. THE FOLLOWING ARE THE REGULATIONS FOR THE “HOPKINS PRIZE.” I. Txir the Prize he called “Tur Hopxiys Prizz.” II. That this prize be adjudged once in three years. III. That it be adjudged for the best original memoir, invention or disco- very, in connexion with Mathematico-physical or Mathematico- experimental science that may have been published during the three years immediately preceding, but that the adjudicators be at liberty, if it seem to them advisable in any particular case, to award the Prize for a discovery in Mathematics alone, or in Experi- mental Physics alone, or for one which has not been published within the fore-mentioned period. IV. That it be confined to those who are or have been Members of the University of Cambridge. F 6—2 ed 64 V. That the fund be vested in the Cambridge Philosophical Society, and the prize adjudged by three Fellows of that Society, nominated by the Council of the Society for each occasion. VI. That, in the event of any difficulty arising in carrying out the above provisions in any particular instance, either from lack of a prize- subject of sufficient merit, or from any other cause, the Council of the Cambridge Philosophical Society be at liberty to carry over the amount of the Prize for that term towards augmenting the fund for future prizes, or to award it to some one not a Member of the University. On Capillary Attraction. By Mr Porter. May 13, 1867. ' The Prestpent (H. W. Cookson, D.D., Master of St Peter's College) in the Chair. The Treasurer made his financial statement, his accounts were passed, and the thanks of the Society were returned to him. On Modern Musical Scales. By Harvuy Goopwiy, D.D., Dean of Ely. THERE are two points connected with the system of musical scales universally adopted in Europe in modern times, upon which I have long desired to have clearer notions than I have been able to gain from books. I. The first is the principle upon which we pass from one key to another by the introduction of a new sharp or flat into the signature, 65 _ Il. The second is the reason why the division of the notes’ In the ordinary diatonic scale, artificial as it manifestly is, has become the universal division of European music, and appears so simple and natural. Upon these two points I propose to offer some suggestions in this paper, premising to professional musicians (if the paper should fall under the eye of such) that musically I write only as an amateur. This, however, is perhaps of no great import- ance, as the question is one rather of numbers and mathematics than of technical musical knowledge. I. It is well known to every one acquainted with music that the ordinary musical keys, having for their signatures respectively no sharps, one sharp, two sharps, &c. are formed each from the other by sharpening the subdominant of the scale, and so bringing it within half a tone of the dominant to which in the next scale it becomes the leading note. Thus we pass simply from any key to the key of the dominant, and the transition is so easy that even in the least complicated com- positions, as for example in hymn-tunes, the modulation con- stantly takes place. The tonics or key-notes in the scales thus formed are C, G, D, A, E, B. Now it is not very easy to see the manner in which these successive scales are related to each other, nor why the system is so complete as it appears to be. But the relation may be exhibited to the eye, and the symmetry of the system con- sequently made plain, in the following manner. Let us regard the semitone intervals of the chromatic scale as being all equal, which, though not true upon any theoretical principle, is true according to that system of temperament, upon which pianofortes have long been tuned and upon which (as I understand upon good authority) it is now becoming the practice to tune organs. Then the musical interval between each note and its octave will be divided into twelve equal intervals, and these intervals may be conveniently represented 66 by angular spaces of 30° each, the whole twelve thus amounting to 360°, and so representing what may be regarded as the actual coincidence to the ear of the tonic and its octave. The meaning of this will be seen from the annexed figure. With centre O describe a circle, and divide the cireum- ference into twelve equal parts. Join the dividing points with Fig. I. the centre, and put letters as in the figure. It will be observed that each diameter has the same letter at each extremity, the letters at the two extremities being distinguished by an ac- cent affixed to one of them; it will also be observed that one diameter has an ambiguity, being either the diameter BOS’, or the diameter FOF"; this arises from the fact that in the natural diatonic scale F is only a semitone removed from E, and B only a semitone from C. In the figure the angular spaces BOC, EOF have been shaded, to indicate at once to the eye that these are the two semitone intervals in the diatonic scale. For the sake of distinctness, call COC’ the tonic line, and GOG@ the dominant line. Then it will be seen that the tonic line and the dominant line are inclined to each other at an angle of 30°: they are in fact next to each other in the group of note-lines which have been drawn through O. It will be seen also that the shaded spaces corresponding to the two semitones are situated symmetrically with respect to DOD’, the line corresponding to the second note of the scale. Now let us see what effect will be produced upon our figure by sharpening the subdominant F. The arrangement will then 67 be as in Fig. mu. Comparing Fig. IL. Fig. m1. with Fig. 1. it will be seen that the shaded spaces of Fig. Il. are symmetrical with respect to AOA’, as those of Fig. I. were with respect to DOD’; hence A is the second note of the new scale, in other words the key-note is G. For it will be observed that Fig. 11. represents a scale of precisely the same kind as Fig. 1.: in each case the shaded portions occupy one side of a diameter, leaving one semicircular space wholly unshaded, and containing between them an unshaded section of 120°. The eye will in fact at once perceive that if we start from G@ in Fig. 11. and pass round the circumference of the circle in the sense of the motion of the hands of a clock, we shall come to exactly the same succession of tones and semitones as we should in Fig. 1. if we started from C. GOG’, then, has now become the tonic line; but this line, regarded merely as to its direction, (I mean, GOG’ being re- garded as the same line as G’OG,) is removed only one division, or 30° degrees, from the original tonic line. Hence it may be said, that the effect of sharpening the subdominant is to turn the tonic line through one semitonal angular space, or through 30°; and as the sharpening of the subdominant of the key of C has brought us to the key of G, and turned the tonic line into the position GOG", so the sharpening of the subdominant of the key of G will turn the tonic line through another angle of 30° into the next position DOD’, or will bring us to the key of D, and so on. Hence if we drop the accents, the figures above drawn will give us, by looking at the extremities of the successive radii, the key-notes of the consecutive major scales, namely, C, G, D, A, BB. 68 After this there is an apparent discontinuity, but not a real one, as the next key would be that of F%, and this is in fact the key indicated by the line OC’, being intermediate to OF and OG. Thus the passage from one natural key to another is repre- sented by the orderly revolution of a radius ny our musical circle through angular spaces of 30°. It is hardly necessary to say that the same method is appli- cable to the representation of the succession of flat keys: but it may be interesting to exhibit the method to the eye. The successive flat keys are produced by flattening the leading note, or seventh of the scale. If we Fig. I. perform this process upon Fig. we py Cc ¢! have the annexed figure. Here the shaded semitonal spaces are sym- metrical with respect to GOG' ex- actly as in Fig. 1. they were sym- metrical with respect to DOD’, and in Fig. 11. with respect to AOA’. Con- sequently G is the second note of the scale, or F is the tonic. Hence for the flat keys our tonic line revolves slirdlngtil one angular space of 30° in the sense contrary to the motion of the hands of a clock, or contrary to that in which it revolves for the sharp keys; and therefore Fig. 11. will give us for the key-notes corresponding to the signatures one, two, three, four, five flats, the following F, EZ’, A’, D, G, or, as will be seen by inspecting the figures, F, Bp, Eb, Ab, Db. II. I now pass on to the second subject which I proposed to discuss in this paper; and I shall endeavour to exhibit by the machinery already introduced the great convenience of the 69 arrangement of the notes in the diatonic scale, and to suggest grounds for believing that no other division would be equally convenient. Let us resume Fig. 1. It will be seen that the musical circle representing the diatonic scale is divided into twelve equal portions, and that these twelve equal portions are divided into two groups, one consisting of four portions, or two whole tones, the other consisting of six portions, or three whole tones, by the two shaded portions corresponding to the semitones. Hence it is obvious that the shifting of the semitones so as to take one from the larger division of three tones and. to add it to the smaller division of two tones, will leave a musical circle divided exactly as before; that is, there will still be two great divisions of two tones and three tones respectively, separated by semitones. The arrangement reproduces itself. Not only is this the case, but it is easy to see that no other arrangement of the semitones would produce the same result. Suppose for instance we Fig. IV. have the circle divided as in Fig. e Iv., that is, into two groups of one ‘ tone and four tones respectively, separated by the two semitones; in other words, regarding C as the tonic, suppose that we have a flat third. Then it is manifest that by no shifting of the semitones can this arrangement be made to reproduce itself. In fact the problem of making such a self-reproducing scale is merely that of dividing 5 into two parts, such that if unity be taken from one and added ‘to the other the two parts shall be the same as before. It is manifest that the division into 2 and 3 is the only solution. The division of the circle represented in Fig. Iv. is some- what interesting from the fact that it is the actual division 70 in an ascending minor scale. The arrangements of the semi- tones in the ascending and descending scales of a minor key are, as every one acquainted with the elements of music knows, different, and the signature is that which corresponds to the descending scale, the semitones being put into their proper places in the ascending scale by means of two accidentals. The arrangement of the semitones in the descending scale, if represented according to the method of this paper, is the ordi- nary diatonic arrangement: for example, Fig. I. would represent the descending scale of A minor, if we pass round the circle ; from A in the sense opposite to the motion of the hands of a clock; in other words the key of A minor has the same signature as that of C major; and as the signature is thus taken from the descending scale the modulations from one key to another in the minor scales follow the same rule as those in the major; but this would not be the case if the arrange- ment of semitones were that which, because it is more pleasing to the ear, we adopt in the ascending scale. These considerations seem valuable with reference to the question, What is the reason why in modern Europe the com- mon diatonic scale has gained such universal acceptance? It is a mistake to suppose that it is a scale founded upon any natural necessity ; if so, it would be universal, which is not the case. But this seems to be the fact, namely, that there is such a perfection in the arrangement as ensures its adoption as soon as known, and guarantees its permanence to the exclu- sion of all others, except so far as a different arrangement may sometimes produce a feeling of pleasure by its novelty or its eccentricity. Nor is it perhaps difficult to deduce the diatonic division from simple principles, and to shew that it is not so arbitrary as at first sight it may seem. The first principle must be the identity of a note with its octave. I speak of the octave, and by so doing appear to anticipate the existence of eight notes in the scale; but I do not 71 intend to make this siittebpatian: I only use the word octave to describe that note which is produced by vibrations of air twice as rapid as those which produce the fundamental note. Experience teaches us that the coincidence of two notes so related is acoustically perfect, so that they may be regarded as the same, and we may with propriety speak of the upper C or the lower C, applying the same letter C to express two notes, which, mechanically speaking, differ from each other, but | which musically may be regarded as identical. The question of the musical scale therefore resolves itself into that of interpolating a convenient number of conveniently related sounds between a note and its octave. Of all possible sounds which may be interpolated there are two which seem to have a chief claim to admission. These are the third and fifth, according to the common nomenclature; but mathematically speaking they are sounds produced by vibrations bearing a very simple numerical relation to those which produce the fundamental note, and musically speaking they are sounds Which produce a very perfect harmony with the tonic and octave ; and when the four notes are sounded successively, there is a simple and majestic progress from one to the other which every ear at once recognizes with pleasure. So far all is tolerably simple, but the problem still remains to interpolate notes amongst those four, which we admit without question as the chief in the scale ; and the problem branches out still further into the more complicated one of temperament; upon this many treatises have been written, and a compendious ac- count of the question may be found in Sir John Herschel’s Treatise on Sound in the Encyclopedia Metropolitana. It is not the purpose of this paper to enter into this difficult subject ; but without doing so, I think it may be shewn by reference to the mode of illustration which bas been adopted, that we are (as it were) forced into the arrangement of tones and semitones which constitutes the ordinary diatonic scale. For if we admit 72 the principle that a semitone is the smallest interval by which it is agreeable to the human ear that musical sounds should follow each other, and if we further admit that the third and fifth must find their place in the scale, then we find our musical circle divided into three portions which contain about (not accurately) 4, 3, and 5 semitones respectively. No other division of these intervals seems possible, except that of dividing the first two in- Fig. V. - tervals into two parts each, and the a last into three. The positions of the - two semitones, the existence of which is manifestly unavoidable, will still be undecided, but there will not be much difficulty in determining their position; for the satisfaction which the ear experiences in the sound of a leading note, or a note approaching within half a note of the tonic, is so great as to leave no doubt as to the position of the semitone between G and OC; it must clearly be immediately contiguous to C; and this being so, it will be indifferent where we place the other semitones. For suppose we put it next to £, then we have the ordinary arrangement of the diatonic scale, the tonic being C: but suppose we take the other course and put it next to G, then the result is that the arrangement of semitones in the musical circle is exactly the same as before, only the tonic will be G instead of C. Hence, granting to the third and fifth their places in the scale honoris causd, and allowing the necessity of a leading note, it appears that the arrangement of the semitones in the scale must be that with which we are familiar. And thus we seem to get at a rationale of the ordinary system of notes, which is in some respects more instructive than that which is usually given, as for example in the treatise of Sir John Herschel above cited; for the reasons there adduced depend upon considerations of the 73 number of harmonies which can be made amongst the various notes of the octave or of successive octaves; and these con- siderations are valuable; but there would seem to be a propriety in the arrangement of the notes independent of them; there is a stately march of sound in the ordinary gamut which is highly satisfactory to the ear, and for which considerations of harmony do not seem to me to account. One more point occurs to me as worthy of notice. I have spoken of the third and fifth of the scale as claiming. their places before all other notes. There can be no doubt of this as regards the fifth ; it fully deserves the title of the dominant ; when we listen to a piece of music, its sound is left upon the ear almost, if not quite, as clearly as that of the tonic itself; and mathematically speaking the numbers which denote the ratio of its vibrations to those of the tonic are simpler than in the case of any other notes. But there may be a demur to the same precedence being granted to the third, at least to the major third, because it may be argued that the succession of notes is as pleasing and satisfying to the ear if for the major we sub- stitute a minor third; that is, if we take as the basis of our system of notes the succession C, Eb, G, C, instead of C, E, G, C. Let us, then, just examine the conclusions to which we shall be led, if we start with the minor third as one of the primary intervals. Our musical circle will now be as in Fig. vi; and the question will be, where shall we put the semitones? In the ascending scale the demand for a leading note will lead us to put one of them below the upper C; and with regard to the other we easily perceive that the effect of the flat third is lost unless the note preceding it be distant by a small interval or by a semitone; not to mention that. the juxtaposition of 74 two semitones, which is our only alternative, would be intolerable. Hence we arrive at the arrangement Fig. VI. shewn in Fig. vi., which is that of the ordinary ascending minor scale; but it has been already pointed out that a scale of this kind does not admit of being changed into another, as does the diatonic arrangement; and it is fortunate that in the descending scale the ear does not by any means de- mand the small interval of the semi- tone between the eighth and seventh of the scale; on the other hand the peculiar and indescribable effect of the minor key, of which the flat third is the mainspring, is increased by dropping a whole tone instead of half from the eighth to the seventh. Taking, therefore, the descending scale and beginning with a fall of a whole note, the position of the other semitone settles itself; for, whether we put it third in the scale or fourth, it will be seen from inspection of the figure that we get no different arrangement of the notes: in each case we shall have two groups of two and three tones respectively separated by two semitones; in other words, in each case we shall have the ordinary diatonic arrangement. In concluding this little essay, I will express the hope that the views which have been propounded, and the method em- ployed for their illustration, may tend to give simplicity and clearness to a branch of science, the fundamental principles of which, though in some respects easy and familiar, are in others not free from obscurity. 3 H. GOODWIN. 75 May 27, 1867. Professor CHALLIS (VICE-PRESIDENT) in the Chair. The following were elected Honorary Members of the Society : MM. EHRENBERG, » PONCELET, PLUCKER, » AGASSIZ, » QUETELET, Dr DAUBENY, Dr Topp, Mr STEPHEN SMITH, Mr Max MULLER. (1) On a New Method of maintaining the Oscillations of a Pendulum. By W. H. Mitisr, M.A., For. Sec. R.S., Professor of Mineralogy in the Uni- versity of Cambridge. Professor Challis expressed his admiration of Professor _ Miller’s contrivance, and described the practical difficulties which he had experienced with ordinary pendulums at the Obser- vatory. (2) On the Crystallographic Method of Grassmann, and on its employment in the investigation of the general geometric properties of Crystals. By PRoressor MILER. INTRODUCTION. 1. The law to which the mutual inclinations of the faces and cleavage planes of a crystal are subject, as enunciated by a large majority of the writers on Crystallography, is essentially embodied in the following statement :— 76 If through any point within a crystal planes be drawn parallel to each of its faces and cleavage planes, and any three of the straight lines in which these planes intersect one another, not being in one plane, be taken for axes, the equation to any face or cleavage plane of the crystal will be ho +kt+1-=d, a b c where a, b,c are any three straight lines the ratios of which depend upon the species of the crystal, and the selection of axes, d is any positive quantity, and h, k, 1 are any positive or negative integers one or two of which may be zero. A very different method was invented by Grassmann, who tells us that the difficulty of following the combinations of planes in the imagination, led him to the idea of substituting for the plane surfaces of crystals, normals to those surfaces or rays as he terms them. In other words, mstead of the crystal he employs its reciprocal figure, adopting the definition of reci- procal figures given by Professor James Clerk Maxwell in the Philosophical Magazine for April, 1864, Grassmann was followed in the use of this method by Hessell in the Article Krystall in Gehler’s Physikalisches Worterbuch, reprinted separately under the title Krystallometrie, Leipzig, 1831; by Frankenheim in 1832, in a very elegant investigation of certain geometrical theorems, Einige Satze aus der Geometrie der geraden Linie, Crelle, B. 8, 8. 178; and lastly by Uhde, Versuch einer genetischen Entwickelung der mechanischen Krystalliza- tions-Gesetze, Bremen, 1833. Later, however, this method appears to have been treated with a neglect it little deserves, for it possesses all the advantages of simplicity claimed for it by Grassmann, it leads directly to Neumann’s representation of a crystal by the poles of its faces, and admits readily of the application of analytical geometry of three dimensions, ordinary geometry, or spherical trigonometry, in the investigation of the geometrical properties of crystalline forms. And though it may 77 - not have led to any result that has not been obtained by the more usual method of treating the subject, an acquaintance with it can hardly fail to impart a clearer insight into the compli- cated relations of crystalline forms, and afford a fresh instance of the truth of the remark made by Sir John Herschel (Astronomy, 7th Edition, p. 6) that it is always of advantage to present any given body of knowledge to the mind in as great a variety of lights as possible. . 2. According to Grassmann, if from any point within a crystal lines be drawn normal to the several faces of the crystal, and any three of these normals, not all in one plane, be taken for axes, the equations to any other normal will be A Ag ha kB ly’ where a, 8, y are three straight lines the ratios of which depend upon the species of the crystal and the selection of axes, and h, k, l are any integers either positive or negative or zero, one at least remaining finite. That these two enunciations lead to identical results, though not at first sight obvious, admits of an easy proof. 3. In fig. 1 let O be any point within a crystal. Let the surface of a sphere described round O as a centre meet the axes in X, Y, Z. Let ABC be the polar triangle of XYZ, and therefore OA, OB, OC radii normal to the faces 100, 010, 001; P the pole of the face h & 1; and a, 4, c the parameters of the crystal. Let ZL be the intersection of the great circles BC, AP. Through any point # in the straight line OP draw RQ parallel to OA, meeting the straight line OL in Q. Through Q draw QN parallel to OB meeting OC in N. Let QR=x2, NQ=y, ON=z. It is proved in my Tract on Crystallography (4) that * sin BAP= an GAP “sin CBP = : dn AHP [Reprinted, 1880.] 7 78 : sind OP =" in BOP. b But sin AB sin BAP =sin BL sin BLP, and sin CA sin CAP =sin CL sin CLP. Hence y __sin VOQ _ sin CL _ke sin C'A 2 smnOQN sinBL blsinAB’™ In like manner = =! ; ae Also sinBC sinCA smAB sin YZ snZX sin XY snA4- snB sinO’ smX sinY sinZ’ and, since ABC is the polar triangle of XYZ, sn YZ=sin A, sin ZX=sin B, sin XY=sinC@. Therefore the equations to OR, a normal to the face hk 1, will be ha lee | where a, 8, y are three straight lines subject to one of the four indentical conditions i TR /, | a 68. sin BC sinCA sin AB’ sinA sinB sin@’ aa bB cy an bg cry sn YZ snZX snXY’ sinX snY sinZ’ It is evident that OR is the diagonal of a parallelopiped having its edges in the lines OA, OB, OC, and _ respectively proportional to ha, kB, ly. Let G be the pole 1 1 1. Then * , SEA : ale a j sin BAG =~ sin CAG, -sin CBG==-sin ABG, Cc c a 25 rt. 7 sin ACG = b sin BCG. Therefore sin BAP ne sin CAP 7sin CBP _ h sin ABP sin BAG ‘sin CAG’ ‘sinCBG sin ABG’ 79 sin ACP — 7, Sin BC. sin BOP sin ACG ™ sin BCG’ _ This is equivalent to the form in which the law was enun- ciated by Gauss (C. F. Gauss, Werke, Band 11. S, 308). 4. It remains to be seen whether the symbol of a zone has any geometrical signification when the normals to the faces are referred to OA, OB, OC as axes. Tt appears from what precedes, interchanging a, }, ¢ and a, B, y, that hh is=i=p mh zthatl==0, being the equations to a line and plane, one referred to the axes OX, OY, OZ, and the other to the axes OA, OB, OC, the line will be normal to the plane. But (Tract 200), the equations to the zone-axis u v w referred to the axes OX, OY, OZ, are Hence, a plane through 0, normal to the axis of the zone uv w, when referred to the axes OA, OB, OC, will have for its equation u~+¢v44w==0. a B Y Let a plane parallel to the zone plane uvw meet OA, OB, OC in U, V, W. Then it is evident that 5 Oe 2g OW u— =v-5—-=w—_.. , B Y Let the axis of the zone uv w meet the surface of the sphere in K. It is ead seen that = cos AK = : cos BK = + ~ cos OR. 5. It appears then thi the notation for faces and zones suggested by the equations to the faces and zone-axes, when the crystal is referred to three zone-axes as axes of coordinates, is equally applicable when the crystal is represented by rays normal to its faces, and these are referred to three such rays as 7—2 80 axes. Since the notation is the same in either case, it follows that the expressions for the geometric relations between faces and zones, in terms of their respective indices, will be absolutely the same whether we refer the faces to three zone-axes, or their rays to the three corresponding rays as axes of coordinates. 6. Instead of deducing the properties of a system of rays referred to three rays as axes, from those of faces referred to three zone-axes, it will be better to investigate the properties of Grass- mann’s system of rays without any reference to the crystal it is intended to represent. Having once established the properties of a purely geometric system of rays we may proceed to the consideration of crystals, avoiding the mistake sometimes made of anticipating the result of a geometrical investigation in the enunciation of a physical law, and assert that measurements of the angles between normals to the faces of crystals show that these normals are subject to the law according to which the system of rays was constructed, and that, consequently, all the geometrical properties of such a system of rays are properties of the rays drawn from any point within the crystal normal to its faces. : The analytical investigation is followed, first, by one in which ordinary geometry is used ; and, afterwards, by an investigation of the properties of a system of points on the surface of a sphere, the points being the intersections of the rays with the surface of a sphere described round the origin of the system as a centre. Those propositions are omitted the investigation of which is the same whether we employ the polyhedral solid or its reciprocal figure. ANALYTICAL INVESTIGATION OF THE PROPERTIES OF A SysTeM oF Rays. Rays. 7. Let OA, OB, OC be any three straight lines not all in one plane; a, 8, y any three straight lines in a given proportion ; 81 h, k,l any three integers positive or negative or zero, one of them at least remaining finite; let OA, OB, OC be taken as co-ordinate axes; and let a system of lines be constructed by giving different values to h, &, I in the equations pcs gle Aga ha kB ly’ That portion of any line defined by the preceding equations, which lies on one side of the origin, will be called a ray. The ray containing the point «=ha, y=k8, z=ly will be denoted by the symbol hkl. The ray containing the point #=-— ha, y=—k8, z=—ly will be denoted by the symbol hkl. The integers h k 1 will be called the indices of the ray; and the lines a, 8, y will be called the parameters of the system of rays thus constructed. 'The lines OA, OB, OC are manifestly the rays 100, 010,001. Zone-planes. 8. The equations to the rays h kl, p q r are 1. OR bea! 4 ee hz kB ly’ pz QB ry Hence the equation to the plane containing the raysh kl, pqvr will be x y Zz - “~+t+w—=0, to "sy where u=kr—lq, v=lp—hr, w=hq—kp. This plane will be called a zone-plane, and will be denoted by the symbol uv w. The quantities u, v, w will be called its indices. They are evidently positive or negative integers one or two of which may be zero. The symbols of the zone-planes BOC, COA, AOB are 100, 010, 001 respectively. 82 The intersection of any two zone-planes is a ray. 9. The zone-planes hk1, p qr have for their equations B The equations to the intersections of these planes will be hook 1 pend pea eee at Y Pat ig tty where wu=kr—lq, v=lp—hr, w=hq—kp. The quantities wu, v, w are obviously integers, and therefore the intersection of any two zone-planes is a ray. Condition that a ray may lie in a zone-plane. 10. Let the zone-plane pqr contain the ray www. The equations to the zone-plane and ray are Therefore, since the plane contains the ray, pu+qu+rwo=0. Portions of two rays cut off by parallels to two zone-planes. 11. In fig. 2 let OQ, OS be the rays hkl, wvw, and let the zone-plane QOS intersect the zone-planes efg, pqr, in OP, OR. Let the planes having for their equations’ oak Seto Sas t+q24r-= ea ue Pe m, Cg bee n, and therefore parallel to the zone-planes efg, pqr, meet the ray hkl in D, Q, and the ray uyw in F, 8. Let planes passing through D, Q, F, S, parallel to the zone-planes 1 0 0, meet the ray 100 in d,q,f,s. Then Od, Og, Of, Os will be the values of # at the points D, Q, F, S. Therefore, since the equations to the rays hkl, wv w are eds, eave | ce y_% la hB hy erat 83 we shall have (eh +fk+gl)Od=mhi, (ph+qk+rl) Og =nha, (eu+fv+gw)Of=mua, (pu+quv+rw) Os =nua. But Od: Oq=OD : OQ, and Of: Os=OF: OS. There- fore eu+fv+gwOF _pu+quv+rw OS eh +fk+gl OD ph+qk+rl 0Q° When only one of the rays OP, OR lies between OQ and OS, three of the points d, g, fs will be on one side of O, and the fourth on the other side. Therefore Of. Os: Og. Od will be negative. When OP, OR are both without the angle OOS, or both within it, the points d, q, f, s will either be all on one side of O, or two on one side and two on the other side, and Of . Os : Oq . Od will be positive. Hence the expression eh+fk+gl put+qu+rw 3 eut+fut+tgw ph+qk+rl will be positive except when one only of the rays OP, OR lies between OQ and OS. Anharmonic ratio of four rays in one zone-plane. 12. Since DF, QS are parallel to OP, OR respectively, sin POQ : sin POS=sin D: sn F= OF : OD, and sin ROQ : sin ROS=sin Q : snS=OS: OQ. sin POQsin ROS _eh+fk+gl put+qut+rw sin POS sn ROQ” eu+fv+gw ph+qk+ rl’ where OP, 0Q, OR, OS are four rays in one zone-plane; efg, pqr the symbols of zone-planes containing the rays OP, OR; and hkl, uv w the symbols of the rays OQ, OS. Hence Anharmonic ratio of four zone-planes intersecting one another in one ray. 13. Retaining the notation of (12), let the zone-planes e fg, pqr intersect in the ray OX; and let a plane through Q, in fig. 3, 84 normal to OK, meet the zone-planes efg, pqr in Op, Or; KOQ, KOS in Og, Os; and planes through DF, QS parallel to OK, in df, qs. Then df, gs will be parallel to Op, Or; Od: Ogq=OD : OQ; and Of: Os=OF: OS. Therefore sin pOq: sin pOs=sind: sin f= Of: Od, and sin rOq : sin rOs = sin qg : sins = Os : Og. ee sin pOq sin rOs_eh+fk+gl pu+ qv +rw sinpOssinrOg eut+fu+gw prat+qk+rl’ where KOP, KOQ, KOR, KOS are four zone-planes inter- secting one another in one ray; efg, pqr the symbols of KOP, KOR; hkl, uvw the symbols of rays contained in the zone-planes KOQ, KOS; pOgq, pOs the angles which KOP makes with KOQ, KOS; and rOq, rOs the angles which KOR makes with KOQ, KOS. Since the order of the zone-planes KOP, KOQ, KOR, KOS is the same as that of the rays OP, OQ, OR, OS, it follows from (11) that the expression which forms the right- hand side of the preceding equation is positive except when one only of the zone-planes KOP, KOR lies between the other two. Indices of a ray when the axes are changed. 14. Let planes parallel to the zone-planes efg, hkl, pqr meet the ray mno in D, L, Q, and the ray uvw in FN, S. Then (11) eutfu+gw OF _hu+kv+lw ON _ pu+qu+rw OS em+in+go OD hm+kn+lo OL pm+qn+ro OQ” Let the zone-planes hk 1, pqr intersect in the ray OA’; the zone-planes pqr, efg in the ray OB’; and the zone-planes efg, hk1 in the ray OC’. And let m’n’'o’, u'v' w’ be the symbols of the rays OQ, OS when referred to the rays OA’, OB’, OC’ as axes. The symbols of the zone-planes efg, hk], 85 p qr when referred to the new axes will be 100,010,001 respectively. Therefore (1 » Hence, comparing corresponding terms, two equations are obtained which are satisfied by making m =em+fn+go, w=eu+fv+ gu, n=hm+kn+lo, v =hu+kv+lw, o=pm+qn+ro, w =pu+qut+rw. The coefficients of wu, v, w are integers, therefore w’, v’, w’, the indices of the ray OS when referred to the rays OA’, OB’, OC’ as axes, are also integers. Hence, the rays of the system are subject to the same law when referred to any three rays as axes, as when referred to the original axes. Indices of a zone-plane when the axes are changed. 15, Let the rays e fg, hkl, p qr meet a plane parallel to the zone-plane mn o in D, #, F, and a plane parallel to the zone-plane uv w in #, 8,7. Then (11) ue+vf+wg OR_uh+vk+wl OS_up+vqtwr OT me+tnf+ogO0D mh+nk+o0l0H mh+ng+or OF" Let m‘n’o’, uv w be the symbols of the zone-planes parallel to DEF, RST, when referred to the rays e fg, hk l, pqr as axes. The new symbols of the rays OR, OS, OT will be 100,010,001. Therefore (11) a OR v OS Ma hie Hence, comparing corresponding terms, two equations are obtained which are satisfied by making m’=em+fn+go, u=eu+fv+gw, n’=im+in+lo, vw=hu+kv+ly, o=pm+gn+ro, wW=put+qvt+rw. 86 GEOMETRICAL INVESTIGATION OF THE PROPERTIES OF A System oF Rays. Rays. 16. In fig. 4 let O be the origin of a system of rays; OA, OB, OC the rays 100,010,001; OR the ray hkl; HKL a parallelopiped having its edges in OA, OB, OC, and having OR for a diagonal. Then (7), since OH, OK, OL are the values of x, y, 2 for the point R, we shall have Zone-planes. 17. In OA, fig. 5, take OU =—a, and therefore measured from O in the direction opposite to. A. Through U draw UM, US parallel to the rays hkl, p qr respectively, meeting the plane BOC in M,S. Let MS meet OB in V, and OC in W. Draw MD, SG parallel to OC meeting OV in D, G. The lines UM, US are parallel to the rays hkl, pqr, therefore, observing that since OU =—a, UO=a, UO OD DM PO OG, GS _k Aa | ag RX OD = B, DM =>, ca mae The lines DM, GS are parallel to OW, therefore OW:0V=DM:DV=DM-—GS: O0G—OD; consequently (hq — kp) 8.0W = (lp —hr)y.OV, h(lp — kr). DV =U(hq — kp), (lp —hr). OV =(lq—kr)B, (hq — kp). OW = (lq — kr). Hence, if a plane parallel to the rays hkl, pqr meet OA, OB, OC in UVW, where u=kr—lg, v=lp—hr, w=hq—kp. A plane through the rays hkl, pqr, and therefore parallel to the plane UV W, will be called a zone-plane, and will be denoted by the symbol.u v w, or by any three integers respect- ively proportional to u, v, w; and the integers u, v, w, or any three proportional integers, will be called the indices of the zone-plane. The intersection of any two zone-planes is a ray. 18. In OB, fig. 6, take OB=£8. Let planes through B parallel to the zone-planes h k 1, p q r intersect one another in the line BM which meets the plane COA in M; and let them meet OC in L, R, and OAin H, P. Then k h op=!oL, and? op=10B=" OR. Y a B Y Therefore 1.0L=ky, h. OH=ka, r. OR=qy, p.OP =qz. Hence Ir. LR=(kr—1q)y, hp.HP =(hq —kp)z«. But (Tract 187) HM.OP.LR=HP.OR.IM. Therefore, putting u=kr—lq, v=lp—hr, w=hq—kp, we have | wl. [M=uh. HM, wl. LH =—vk. HM, uh. LH=- 2k. LM. Draw MD parallel to OC meeting OA in D. By similar triangles OD : LM=OH : LH, and DM: HM=OL : Ld. Hence —v.O0D=ua, and —v.DM=wy. Draw MF equal and parallel to OB on the opposite side of the plane LOH. Then —v.MF=v.OB=vf. The line OF being parallel to BY, is 88 the intersection of the zone-planes h k 1, p qr, and is evidently the diagonal of a parallelopiped the edges of which are respect- ively coincident with the axes of the system of rays, and equal to OD, MF, DM, and therefore proportional to —v. OD, —v. MF, -—v.DM, or to ua, vB, wy. Since w, v, w are integers, the line OF, in which the zone- planes h k 1, pqr intersect, is a ray of the system, having uv w for its symbol, where w=kr—lq, v=lp—hr, w=hq —kp. Portions of two rays cut off by parallels to two zone-planes. 19. Leta plane parallel to the zone-plane p q r’ meet the axes in I, J, K, fig.7, and the ray uwwwinS. Draw KS meeting IJ in N, IS meeting JK in L, and ST parallel to OJ, meeting the plane JOK in 7. The symbols of the rays OK, OJ, OS are 001, 100, uwvw respectively. Therefore the symbol of the zone-plane KOS will be v u 0, and that of the zone-plane IOS will be Owv. The plane IJK is parallel to the zone-plane pqr. Hence the line KN will be parallel to the ray —TU, —TR) pe 4 gy, and the line JZ will be parallel to the ray qut+rw, —pv, — pw. The lines KN, JZ are in the plane IJK, therefore (18) pu.IN=qv.JN, and qvu.JK=(qu+rw). KL. But (Tract 187) IS. KL.JIN=SL.JK.IN. Hence pu. IS=(qu+rw).SL, and therefore pu. IL=(pu+quv+rw) SL. But ST : OI[=SL : IL. Therefore pu. OL=(pu+qu+ rw). ST. In like manner, if a plane parallel to the zone-plane ef g meet 89 OI in E, and OS in F, and FG be drawn parallel to OJ meeting the plane JOK in G, we shall have ew. OF = (eu + fu + gw). FG. But OS : OF=ST: FG. Therefore (eu+fv+gw).OF : (pu+qvt+rw).0OS=e.0£ : p. OL. Hence if the ray hk 1 meet planes parallel to the zone-planes efg, pqr in the points D, Q, we shall have (eh+fk+gl).OD : (ph+qk+rl).0Q=e.0E : p. OL Therefore eu+fu+gw OF _pu+qvt+rw OS eh+fk+gl OD ph+qk+rl 0Q° From this equation the anharmonic ratio of four rays in one zone-plane, of four zone-planes intersecting one another in one ray, and the indices of rays and zone-planes when the axes are changed, may be found as in (12), (13), (14), (15). PROPERTIES OF A SySTEM OF POINTS ON THE SURFACE OF A SPHERE. Poles. 20. Let the surface of a sphere having its centre in the origin of the system of rays meet the rays 100,010,001,111 in A, B, C, G, and the ray hkl in P, fig. 1. Let the great circle AP meet the great circle BC in LZ. From any point # in OP draw RQ parallel to OA meeting OL in Q. Draw QN parallel to OB meeting OC in NY. Then (7), since QR, NQ, ON are the values of a, y, z at R, QR_NQ_ON ha kB ly’ But ON : NQ=sin BL: sinCL=sin ABsin BAP: sinCA sin CAP. 90 Therefore kB sin AB sin BAP = ly sin CA sin CAP. In like manner ly sin BCsin CBP = hasin ABsin ABP, and hasin CA sin ACP =k8 sin BC sin BCP. Hence, Asin ABsin BAG =ysin CA sin CAG, ysin BC sin CBG =asin AB sin ABG, asin CAsin ACG =Bsin BCsin BCG. Therefore sin BAP _,sin CAP sn BAG sin CAG’ sin CBP _ h sin ABP sin CBG “sin ABG’ sin ACP _ k sin BOP sin ACG ~ “sin BCG ° The point P will be called a pole, and will be denoted by the symbol of the ray intersected in that point by the surface of the sphere. The points A, B, C, G are the poles 100,010, 001, 111 respectively. l Zone-circles. 21. In fig. 8 let A, B,C, GP be the poles 100, 010, 001,111, wuw respectively. Let a great circle passing through P meet the great circles BC, CA, AB in D, EL, F respectively. Then (Townsend’s Modern Geometry, 82, Cor. 3), having regard to the signs of the six arcs, sin DP sin FE + sin PF.sin DE + sin FD. sin PE=0. Therefore sin FE sin DP sin DE sin PF sin FD sn PE ‘sin DsnPE +1=0, whence sin ACF sin BOP sin DAC sin BAP sin FOB sin ACP’ sin BAD sin CAP +1=0. 91 But (20) | sin ACP oe sin BCP as sin BAP fy sin CAP “sin ACG sin BOG’ "sn BAG “sinOAG’ Therefore, putting u__ sin ACF sin BOG d w _ sin DAC sin BAG vy sin FCB sin ACG’ * Vv sin BAD sin CAG’ uw+vv + ww = 0. Let the great circle HF pass through the poles hkl, p qr, not being opposite extremities of a diameter of the sphere. Then uh+vk+wl=0, and up+v¢g+wr=0. These two equations are satisfied by making u=kr-— lq, v=lp—hr, w=hq—kp. The great circle passing through the poles hkl, pqr will be called a zone-circle, and will be denoted by the symbol u v w, _ or by any three integers in the same proportion. Condition that a pole may be in a zone-circle. 22. It appears from (21) that when the zone-circle uv w | passes through the pole uvw, we have uu + vo+ ww= 0. Any integral values of u, v, w that satisfy this equation are the indices of a pole in the zone-circle u v w, and any integral values of u, v, w that satisfy it are the indices of a zone-circle _ passing through the pole wv w. The intersections of any two zone-circles are poles. 23. Let the zone-circles hkl, pqr intersect in the points P, P’. If it be possible, let P be the poleuvw. Then (22) hu +kv+lw=0, and pu+qu+rw=0. 92 These equations are satisfied by making w=kr—lq, v=lp—hr, w=hq—kp. It is evident that wu, v, w are integers, therefore P is the pole uv w, and P’ is the wow. Relation between the arcs AK, BK, CK, K being a pole of the zone-circle EF. 24, Let uvw be the symbol of the zone-circle HF, and let K be the pole of the zone-circle EF nearest to C. Then cos AK =—sin AF# sin H, cos BK=—sin BD sin D, cos CK =sin CD sin D=sin CE sin E. The symbol of D is 0 w v, and the symbol of Z is w0u. There- fore (20) csAK snAK sinABsin ABE _ uy cosCK sinCE sin BC sin CBE wa’ cs BK snBD_ smABsinBAD vy cosCK sinCD sinOA sinOAD wf’ Whence and * cos AK = : cos BK =~ cos OK: u Vv Ww Anharmonic ratio of four zone-circles passing through one pole. 25. In fig. 9 let A, B,C be the poles 100,010, 001 respectively ; AP the zone-circle e fg intersecting the zone- circles CA in M; KR the zone-circle p qr intersecting the zone- circle BC in N; Q the pole hkl; S the pole wow. Let the zone-circles K Q, KS intersect the zone-circle MN in 7, V; also let £7 0, 6x be the symbols of 7, V respectively. Then (21), (23) the symbol of M will be g0e, the symbol of N will be 0 rq, the symbol of MN will be er gq gr, the symbol of K will be fr — gq gp — er eq — fp, and the symbol of AQ will be keq—k fp —lgp + ler, lfr —lgq—heq+hfp, hgep—-her—kfr +k gq. - 93 _ Hence (23) f= g (gq — fr) (ph + qk + rl), n=r (er —gp)(eh+ fk +l). In like manner $= g (gq — fr) (pu + qu+rw), x=r (er —gp) (ew+ fu + gw). sin PKQ sin RKS_ sin M7 sin NV sin RKQ sin PKS sin NT sin MV _ sin ACT sin BCV _ n ~ sin BCT sm ACV fy" But Therefore snPKQsnRKS _eh+ fl +gl put+qu+rw sin PKS sin RKQ~ eu+fv + gw ph+qk+rl © Anharmonic ratio of four poles in one zone-circle. 26. Let the zone-circle QS meet the zone-circle KP in P, and the zone-circle KRin R. Then, since the anharmonic ratio of the points P, Q, &, S is the same as that of the ares KP, KQ, KR, KS (Tract 16), sin PQ sn RS _eh+fk+gl put+qu+rw sn PS sn hQ eu+fv+gw ph+qk+ri ° one of the expression forming the right-hand side of the final equations in (25) and (26). 27. In (25) the left-hand side of the final equation may be replaced by its equivalent cot PKS — cot PKR cot PKQ—cot PKR’ and in (26) it may be replaced by cot PS — cot PR cot PQ — cot PR* From the form of these expressions it is manifest that they are positive, and therefore also the expression forming the right- hand side of the equations in both cases, except when one only of the zone-circles KP, KR lies between Q and S. [Reprinted, 1880. ] 8 94 POSITION OF ANY POLE IN EACH OF THE SIX SYSTEMS OF CRYSTALLIZATION. Position of any pole in the cubic system. 28. In this system the arcs joining every two of the poles 100, 010, 001 are quadrants, and the arcs joining the pole 111 and each of the poles 100, 001, 001 are all equal. Let A, B, C, O, P be the poles 100, 010, 001, 111, hkl respectively. Then BC, CA, AB are quadrants, 0A =OB=O0C, and the right angles A, B, C are bisected by OA, OB, OC. Hence the equations in (20) become ; tan BAP =", tan CBP=%, siiovchem But tan AP =tan ACP sec BAP, tan BP =tan BCP sec ABP, tan CP = tan CBP sec ACP. Whence 2 h* (cos AP) i +P+E , 2 ké kegs BE) gaa lil (008 OP)" = ETE Position of any pole in the pyramidal system. 29. In this system the arcs joining the poles 100, 010, 001 are quadrants, and the ares joining the pole 111 and each of the poles 100, 010 are equal. Let A, B, C, G, P be the poles 100, 010,001,111, hkl respectively. Then BC, CA AB are quadrants, and AG = BG. Consequently BAG=ABG, and ACG =BCG. The arc BG intersects CA in the pole 101. Putting the arc 001, 101=4Z, and observing that the angles A, B, C are right angles, the equations in (20) become tan BAP = 4% E, tan ABP ae cot H, tan AC. m Kone h 95 AP ot ie’ BAP = tan Ecos CAP, cot BP = * tan E cos CBP = : cos ABP, cot CP = j cot E cos ACP = ; cot £ cos BCP. V+ (tan CP)? = z (tan £)?. The are £ may be taken for the element of the crystal. Position of any pole in the rhombohedral system. 30. In this system the arcs joining every two of the poles 100, 010, 001 are all equal, and the arcs joining the pole 111 and each of the poles 100, 010, 001 are all equal. In fig. 10, let A, B, C, O be the poles 100, 010,001,111 respectively; P the pole hkl. Let the zone-circles OA, OB, OC meet the zone-circles BC, CA, AB in D, E, F. Then, since BC=CA=AB and OA = OB= (OC, it is evident that BC, CA, AB are bisected in the points D, F, F; that OD=OE=OF; that the angles at D, E, F are right angles ; that the symbols of D, E, F are 011, 101, 110 respectively ; and that the six angles having their apices in O are each of 60°. The symbol of OA is 011, and that of OB is 101. Therefore (27), cot AOP—cot AOB_h-l cot AOF —cot AOB k-l' But AOB=120°, AOF=60°. Therefore tan AOB=—4/3, tan AOF=,/3. Hence _(k-) V3 tan A0P=F 71° 96 Tn like manner PR 5) te ars ah 2k—-l—h’ (h—1 k) /3 and tan COP = Seay a Hence, 2h—k—-l on aS aa (0+ 2 (0h +2 — BN’ 2k—l—h 00s BOP = FEN TF4 20 hPa DB’ cos COP = cold. V{2(k-)?+2(l—h)? +2 (h—k)}" Let the zone-circle OP meet the zone-circle CA in H and the zone-circle ABin I. The symbol of J will be h—1, k—1,0. The symbol of OB is 101, and that of CA willbe 010. There- fore (27), cotOP—cot OH _ k cot Ol —cotOH k-l Let OA =D. Then tan OF = cos 60° tan OA = $ tan D; cot OH = cot OF cos HOP l+h—2k = 2cotD To Exh a2 my aw’ cot OI = cot OF cos FOP PAP are. h+k—2l a) V{2 (kh — 1? +2 (1—h)? +2 (h—k)*} Hence tan OP = V2 (k—1)? +2 (1—h)? +2 (h—k)*} ‘tan DD 2h + 2k + 21 The arc D may be taken for the element of the crystal. Position of any pole in the prismatic system. 31. In this system the ares joining any two of the poles 100, 010, 001 are quadrants. Let A, B, C, G, P be the poles 100,010, 001, 111, hkl neenenely- Then BOC, CA, AB are quadrants. 97 The ares AG, BG, CG meet the arcs BC, CA, AB in the poles 011,101,110. Putting 010,011=D,001,101=2£Z, 100, 110=F, and observing that the angles A, B, C are right angles, the equations in (20) become tan BAP= tan aren CBP="tan E, tan AOP=" tan F, cot AP= : cot F cos BAP= : tan H cos CAP, cot BP= F cot D cos CBP= 4 tan F'cos ABP. oot CP= F-cot Ecos ACP = tan D cos BCP. Any two of the arcs D, EH, F may be taken for the elements of the crystal. They are connected by the equation tan D.tan #.tan F=1. Position of any pole in the oblique system. | 32. In this system the arc joining the poles 100,010, and the are joining the poles 010, 001 are quadrants. Let A, B, C, G, P be the poles 100,010,001,111,A2k 7 re- spectively. Let BG, BP meet CA in L, 8. Then LZ will be the pole 101, and § the pole hOl. The arcs AB, BC are quadrants, and consequently the angles ACB, CAB are right angles. Hence the equations in (20) become tanCAP k sin AL snCS_h tan ACP &£ tan CAG 1’ sin CZLsinAS 1’ tanACG A’ Putting hsin CL Usin AL’ the are AS will be given by the equation tan (AS—4$AC)=tan }AC tan (1m — 6). tan 0@= 98 But sin CL =cot ACG cot BG, sin AL =cot CAG cot BG, sin CS=cot ACP cot BP, sin AS=cot CAP cot BP. Whence tan BP _hsin CL _tsin AL tan BG ksinCS ksin AS” The arcs AL, BG, CL may be taken for the elements of the crystal. Position of any pole in the anorthic system. 33. Let A, B, C, G, P be the poles 100,010,001,111, hkl respectively; and let AG, BG, CG meet BC, CA, AB in D, £, F, the poles 011,101,110. It is easily seen that sn CA sinCAG _sinCD sin ABsin ABG _ sn AE snABsinBAG sinBD’ sin BC sinCBG sin CH’ sin BC sin BOG _ sin BF sin CA sin ACG sin AF” But (20), t sin BAP _ sin CAP sn CBP _,snABP sin BAG sinCAG@’ sinCOBG sin ABG’ sin ACP sin BOP sin ACG sin BCG ° Whence sin CAP _ k sin AB sin CD sin BAP IsinCA sin BD’ sn ABP. IlsinBC sn AE sinCBP hsin ABsin CE’ sin BCP _hsin CA sin BF sin ACP ksin BO sin AF’ 99 Putting tan 0, tan ¢, tan respectively for the right-hand sides of the preceding equations, we obtain tan (BAP —}3BAC) =tan}BAC tan (in — 6), - tan (CBP — 3CBA) =tan }CBA tan (17 — 9), tan (ACP — ACB) = tan }.ACB tan (tn — fp). Whence, knowing the segments BD, CD, CE, AE, AF, BF, the position of P can be found. _ Any five of the segments of the sides of ABC may be taken for the elements of the crystal. For five of the segments being known, the remaining segment is given by the equation sin BD sin CK sin AF =sin CD sin AE sin BF. On the Association of Potton Sand Fossils with those of the Farringdon Gravels in a phosphatic deposit at Upware on the Cam; with an account of the Super- position of the Beds, and the significance of the Affinities of the Fossils. By Mr Harry Sze ey, F.G.S. ABSTRACT. In 1860 the author had traced the Galt by Swaffham fen, west of Wicken into Soham Mere; fossils were then collected and placed in the Woodwardian Museum. But though the beds over the Kimeridge clay and under the Galt are represented in Dr Fitton’s section through Upware, they do not appear to have been seen again until the pits were opened for digging nodules of phosphate of lime last year. These have yielded about 120 species of fossils, chiefly mollusca and spunges, with a number of vertebrates. As a whole they recall beds in the same posi- tion in the north of Germany ; in part they resemble with un- expected closeness the fauna of the Farringdon gravels; while 100 the resemblance to the fossils of Potton is such that nearly all Potton types of life have already oecurred at the Wicken dig- gings. Potton however is rich in vertebrate remains and in the phosphatic casts of shells; at Wicken the silico-phosphatie con- cretions are smaller, and the mollusca, &c. for the most part preserve their carbonate of lime shells; moreover at Potton have occurred Cycadoidea microphylla, Cycad cones, cones of Pan- danus and three species of Pinites, besides much wood mineral- ized, sometimes with phosphate of lime, sometimes with silex. At Wicken but little wood has occurred. All these facts appear to demonstrate that, assuming the phosphate beds at these places to be one and the same, then Potton was nearer to the old land, streams from which brought down the animals and plants, than was Wicken. This was assumed. Then it would follow that on these deposits being traced to the south-west they would become fluvio-marine and freshwater, and finally have no repre- sentative in that direction because of the interposition of dry | land. Also if these remains were brought down by rivers (and they are in the same state of preservation as bones from the Wealden), the river banks on ceasing would become continuous with the sea shore of the land, along which would be distributed fragments of the rocks which formed the cliffs in those days, as well as rolled bones. | Such a pebble bed is found, and with it are mixed the nodules of phosphate of lime, the casts of shells, the sand nodules concreted with phosphate of lime, and most of the reptilian remains. The pebbles are chiefly old rocks, black slate, Lydian stone, brown hornstone, white and rose quartz, with an occa- sional fossil from the mountain limestone, notably joints of the column of Poteriocrinus. And it would also follow that on these deposits being traced out at sea parallel to the shore the pebbles would cease, the sands would become fine and thin, and ultimately be replaced by clay ; moreover, assuming a small river to have brought down 101 the remains of Megalosaurus, Hyleosaurus, Iguanodon, &c. into a sea tenanted with Pliosaurs, and Ichthyosaurs, and Plesiosaurs, &c., then after depositing these heavy bones and their sand, the fine mud would still be carried out to sea, and with it some of the spoils of the land, as in the case of all river deposits. The sections at Wicken were simple, being ferrous sands _ with varying courses of nodules of phosphate of lime and pebbles sometimes united into a bed six feet thick, oftener subdivided by intervening sand into two or three beds of from a few inches to a foot or two thick. Under these and not well separated is a thin band rich in caleareous matter, often making it a hard con- tinuous agglomerate ; but it is extremely variable, and some- times disappears in small isolated concretions. These beds, which were spoken of as the Potton sands and Wicken beds, rest at Wicken partly on the white Upware limestone (usually called Coral Rag, but regarded by the author as not older than the lower part of the Kimeridge clay), and partly on a blue clay with Ammonites serratus, and near the top casts of Nucula and other shells, &c. in phosphate of lime. This blue clay the author regarded as Kimeridge clay, though the Ammonites serratus usually occurs lower in the series. Above the sands is the Galt, the actual junction not seen, though there are cracks in the sands a foot or two wide into which the Galt with its characteristic fossils has been squeezed. The author followed the Wicken beds to Harrimere, near to which pluce they form the bed of the river, as a hard dark grey fine sand agglomerate of shells and phosphatic nodules. The species were numerous ; those collected are in the Woodwardian Museum. This probably represents the lower phosphate bed at Wicken. Up to the old West Water the phosphatic casts of Lucina, Myacites, Cyprina, Ammonites, &c. occurred plentifully in the bed of the river for two miles before its junction with the Cam. At Stuntney the hill is capped with a thin bed of nodules of phosphate of lime, like those at Wicken with similar 102 fossils. S.E. of High Hill the brown ferrous sands are well seen.coming from under the Galt of Soham, which has been bored for 450 feet without being pierced; near the base of the Galt nodules of phosphate of lime and fossils abounded. At Ely these beds are variable. At the Gallows Pits the rock is sa calcareous as to split with a crystalline fracture. The old walls of the city are of a fine grit conglomerate with occasional nodules of phosphate of lime, casts of fossils, and bones still to be seen in the blocks; while the Cemetery is on sand, said by the gravedigger to be about 12 feet thick with indurated beds in the middle. In Roswell Hole the conglomerate rock was at the bottom and sand above; here from the rock were obtained about six species of mollusca. At Wilburton a phosphatic band with jaws of Edaphodon occurs near the top of the brown sands. At Haddenham the sands 30 feet thick rest on the upturned and eroded clay [Kimeridge]. In the middle were found three small phosphatic concretions. At Aldreth no concretions were seen. Mr Westrupp states that in dredging the Cam he finds the bottom to be a mixture of gravel and galt between Bottisham sluice and Swaffham sluice; and that at a place near Bottisham sluice called Calves Flat the bed of the river is a hard fer- ruginous bed with a hard bed below, mixed with gravel and galt. It is highly probable that these are the phosphate beds. Mr Westrupp also states that gravel exists under the whole of Isleham Fen*. At Downham Market the top of the Kimeridge clay contains small sand concretions and phosphatic casts of shells with green grains in them, resembling in species and preservation those from the sands at Wicken. At Hunstanton the phosphatic concretions are numerous but chiefly fragments of casts of Ammonites Deshaysii, and about * Since this paper was read the author has seen this phosphate bed under ferrous sands and graduating into the clay on which it rested at Impington. It appeared to be in situ. ' 103 a dozen other mollusca. They occur very near the bottom of the sands, heretofore called the Carstone. These facts do not demonstrate the position in the geological sequence of the Potton and Wicken beds. In the south of England the beds between the Kimeridge clay and the galt are (Lower Green or) North-Down sands*, Wealden series, Purbeck series, Portland series, and to represent the whole of these there are in this district only the Potton sands. These sands in the middle of England have generally been referred to some portion of the series. Thus Smith called them the sand of the Portland rock. Conybeare, who instituted the Ironsand group, supposed the Portland series as well as the Greensand to thin away northward, and so put these deposits into his Ironsands; and Fitton, who instituted the Lower Greensand, supposed the Portland, the Purbeck, and the Weald, to thin off to the north, and threw these beds into the Lower Greensand. The author then detailed at length the physical characters of the beds between the Kimeridge clay and the Galt in all the English sections, and arrived at the conclusion that the period of elevation indicated by the pebble beds of Potton and Wicken was identical with the period of elevation indicated in the south of England by the Purbeck and Wealden group, the marine equivalents of which would be thin. And in this district the author supposed the upper part of the clay called Kimeridge to be only the necessary clay repre- sentative of beds which to the south are sands. Under these circumstances it was thought that the old * The North Downs give the types of the divisions of the so-called Lower Greensand adopted by the Geological Survey. 104 nomenclature of Cretaceous and Oolite, as divided by Professor : Forbes, could not now be sustained. And the author proposed the following physical groups as more true and convenient for English geology. Mr Seeley’s divisions, Old names. Chalk , Cretaceous series, Upper Greensand the angle made by the tangent at P with BC, $+6¢ that at Q; then, assuming the usual law of resistance due to the current, the element PQ when the power of growth is just balanced, will be in equilibrium under the following mechanical forces: — Tension rt at P along tangent at P in a direction remote from Q, tension r+6r at Q along tangent at Q in a direction remote from P, ; resistance Sout sin’ d. 6s normally inwards, pressure p.5s normally outwards. By resolving these forces first tangentially with respect to P and then normally, the following equations are obtained : —Tt+(7t+6r) 008 86+ (p— 5 ov"sin'g ) be. sin = 0, (7 + 57) sin 66 — (p-5 ov' sin’ ¢ ) 8e.cos 920; and passing to the limit when 617, 5¢, 6s are indefinitely di- minished, it is readily seen that dr de dp _ a = p—5oav'sin' $; 229 therefore + is constant for all points, and ds _ yah ‘ as p- : ov’ sin* d therefore | Pis= T oe [ Boek 1-5-5 a OF pie gate In the integration indicated by the last expression, two cases arise according as p is greater or less than Sou that is, according as the power of growth is more or less than sufficient to balance the direct resistance of the current. In the first case take a a subsidiary angle such that 2 2p’ se d.tand 1 at tT. J 1+cosa. tan’ cos sin’a = . tan™ (cosa. tan ¢) ; no constant is required if s and ¢ vanish together. Also if x, y be the rectangular coordinates of P, the axis of x being parallel to BC, _ dp ds "dp °°? 50 —am'a.sin® g)’ ae d. sin p 1 R 1+sina.sing | 7 =] Tosinta.sin’d Qsma °1—sina.sing’ ae Pp sin p dp =-{ d.cos¢ T 1—sin?a. sin’? cos* a+ sin*a . cos’ it = —_—_—_——__ . tan™ (tan acos@); sin a. cos @ : $) 5 no constants are required if 2 vanishes when ¢=0, and y when g= rol 9 230 Therefore on eliminating ¢, P si a Pte 7; Sina.© =+2seca.cos sinacosa.l.y ; T In the second case, that is, when p is less than sot assume . 2 sin’ B= —4, Diiseat d.tand _ tang tan B+ tang then Pe | Sea. tan’'d = 2 A ‘tan B~ tanh _tan8 |, sin (8+), ~~g 18+ sin (B~ 4)? p o=| cos h.dd _ sin 1 sin B+sin d ~Ji-coseeB.sin®o 2 ° ©’ sinB~sind B+o _ sin 8 log sea 2 2 tan P=? Pp -{ sin d.dd -{ d.cos@ 74 Ji —cosec® B. sin 76 ‘J cosec® 8 . cos’ ¢ — cot® B vi d.cos _ sin’ cos @ + cos 8 tear e/ = sec’ B. cod Dena 8’°. coumuemedl _ sin’ ‘yeaa AR Sg eo); = Seca . log. (cot 9 . cot after eliminating ¢, the rectangular equation is found to be cosB p,_ 1 p cosp p 1 p t + @sin’p’ 7" snp’ 7°” 4 9 sinta’ ret sin” tae an B= Ser a +e sine ry In the case when p= : ov’, s P 5 = [sce $ dg =tan g, er te = 42k. y, 231 The curves whose equations have thus been determined can be traced either from the intrinsic or rectangular equations; in the case when p> oot a series of equal detached ovals is obtained whose longest diameters all lie on one straight line, perpendicular to BC; and as the curvature continually di- minishes from the point where ¢=0, to that where $= $3; the diameters parallel to BC are the least, and those perpen- dicular to BC the greatest. In the second case when p< : ov a he i) : (2) Pas ‘a0 i‘ a ‘. i \ 3) 232 pair of equal catenary-like curves is obtained with their con- vexities opposed to each other, which in the last case, when p =5 ov’, become actual catenaries. ; Those parts of the curves obtained above which correspond to points from ¢=0 to ¢=7 are alone applicable to the original hypothesis that the resistance due to the current opposes the power of growth; parts corresponding to points from ¢=7 to d=27 are either removed from the influence of the current (when circular arcs are obtained for such portions of the margin), or if subject to the influence of the current (by a slight obliquity in the plane of the leaf, the anterior margin being depressed), would have it so as to assist rather than oppose the power of growth, and the equation for such portion of the margin would be ds _ T PUP as ’ - pt f sv’ sin’ iL ov assume tan’? y =s—, t] ee =| d. tan } sai rT 1 aaa . sin® > 1+sec’y. tan’ = cos y. tan™. (sec y. tan ¢), P .sin $ 1 a ‘ Wee ae sin® >. tany.”. ‘reg a ea p va d.cos _ cosy 1—siny.cosg sec’ y — tan’. o08' ~ 2siny’ °S*T+siny.cosd? siny p . siny p y + therefore sty’ t"" 4g costy’r'” ==" og, (tanya). T 233 Suppose that the leaf-margin after acquiring the shape as determined above ceases to be flexible, then the action between two contiguous portions of the margin will not be entirely tangential, but in addition to the tension 7+ there will exist a normal strain NV and a mechanical couple P tending to wrench the margin. Let 1,, p,, v, be the tension, the power of growth, and the velocity respectively at the time when the margin acquires its shape, then for portions of the margin subject to the opposing resistance of the current, The new equations of equilibrium will be —7+ (7+) cos 86—(N+ 8) sin 84=0, WN — (N+ 8X) cos 86 — (r + 87) sin 56 bb 3 =o +(p—5ov" sin’) Be. 00s and by taking moments about the point P P—(P+6P) —(N+6N) ds=0, 234 and passing to the limit x: 8 GN fae ) dP uy 5 aad atta (P 5 7v sin p dg’ 7p tN =9- 2 p— sav". sin? Therefore OT +T=T : dp 3 jE Pr 57% sin’ VT, 0, D— pV =T, «et —3 i Po Therefore (see Boole, Differential Equations, edit. 1. p. 383) e. 4 Te sing ie (v'p—v*p db _ tT, cosd/sin d(u,,p—v°p,) dp T=—;.T “eae deat. uv, E po—5orssin®g ‘ } po— sors sin® eS rytsing [2. & (vp out} a 14-cos9[%! 2y (pes =P) a dd. Vy ® Therefore if p a be considered constant for different points — 2: r= + MoP— Ps) LA dt EB 4 a) — (Pre — Po) 09s. (y—2), 0 Vy % where a and 6 are constants ; N= = = POP sin g. cos d. it + CPP» cos $ (e— —a) — (Pee PD 008 g. sing 55+ 4 P= 2PY in g (y — —b) = (= PPD) feos p.(e—a) +sin$.(y — I}; 0 p=-|N.ds=- Copa Pd HEE (e— a) + 2 (y a) ds U = "P= 7Pate_ (w—a)*—(y -B)) 0 where c is a constant. 235 But since from the nature of the case both r and P are unaffected when w — ¢ is written for ¢, therefore 5=0. Thus t, N, and P are determined; and it is easily seen that the variable portion in the expression for the tendency to break varies as the square of the distance from a point situated in the axis of x. The theory advanced in this paper applies to leaves whose structure is such that the marginal vigour of growth is the same at all points; and in the case of aquatic plants with floating leaves which may be otherwise organized, some modification of the same theory will be necessary; it applies during the period between the first unrolling of the leaf at the surface of the water and the completion of its growth. Each lobe of a divided leaf must be treated to a separate calculation, and new lobes may be formed at those points where the tendency to break is the greatest. Growth is supposed to proceed steadily, while the leaf is submitted to a suitable external pressure ; when violent pressures exist, as, for example, when the current is very rapid after heavy rains or otherwise, growth is probably checked for a time and fissures may be started, or the leaf- margin maintains its shape by the support of its interior which then resists the external pressure, until at a proper time growth pushes forward the margin again, and its form is matured in obedience to the above-investigated laws. It is further to be noted that the form of the leaf-margin remains the same, even if at different times the power of growth, the direct resistance due to the velocity of the current, and the marginal tension all vary, provided only that their proportional values remain unaltered. It is a tenable hypothesis and by no means improbable that, during much or most of the time when actual growth is taking place and when the velocity of the current is subject to many and various vicissitudes, the plant has the power of adapting its growing efforts to the circum- stances just necessary for its development, that is, in the no- 236 tation of the previous analytical investigation, the quantities p, ov, 7, or at all events the first two of them, maintain a constant proportion. The shape of the curve depends only upon the ratio of p to ov’, and the size depends further upon the proportional value of r. It is evident, on the other hand, that neither any one curve nor the system of curves belonging to any one of the above equa- . tions nor any portion of it or them can, except in very simple and entire leaves, delineate the whole margin of the floating leaf; for otherwise there would be no means of explaining the di- visions, fissures and incisions which are frequent even in floating leaves and which give characters for the definition of species. It does not therefore follow as a consequence of this investi- gation that all floating leaves grown in a perfectly still water (if such a phenomenon were possible to contrive) are simply circular in outline, though a circular form might be favoured; but it does follow that the several portions of the margin would be circular arcs. It has before been hinted at, that new fissures (in addition. to any previously existing ones) may be made and accounted for by the mechanical action of the current. It is a matter of common observation that many floating leaves, as for example in Ranwneulus, vary considerably in con- sequence of, or in association with, the nature of the stream in which they grow. At all events the theory discusses the forms which floating leaves would find mechanically suitable for their growth and maintenance, in order that they might dwell free from unne- cessary strains and wrenches, and under an equal distribution of their power of growth, which as we know is capable of exerting considerable force under compulsion, but is in general slow and steady. 237 ANNUAL GENERAL MEETING, OcTOBER 30, 1871. The PRESIDENT (PROFESSOR CAYLEY) in the Chair. The following Officers were elected: President. Professor HUMPHRY. Vice-Presidents. Professor C. C. BABINGTON. Professor CAYLEY. Professor ADAMS. Treasurer. Dr CAMPION. Secretaries. Mr BonNeEY. Mr J. W. CLARK. Mr Coutts TROTTER. New Members of the Council. Professor MILLER. Professor MAXWELL. Mr GopFRAY. Mr J. STUART. 238 Communications made to the Society : (1) On the Equation which determines the form of the Strata in Legendre’s and Laplaces Theory of the Figure of the Earth. By I. Topnunrsr, M.A., ERS. The object of this Memoir is to examine various investiga- tions which have been given respecting the equation which occurs in the theory of the figure of the Earth considered as a heterogeneous fluid, and from which it is inferred that the figure must be that of an ellipsoid of revolution. Especially the assumptions on which these investigations rest are discussed. The most general treatment which the equation has hitherto received is shewn to be unsound. Finally a new method is proposed, by which the required result is demonstrated with fewer limitations than have hitherto been employed. (2) On a Cirque in the Syenite Hills in the Isle of Skye. By T. G. Bonney, B.D. The Syenite hills occupy a portion of the eastern coast of Skye between the Liassic plain of the Strath (through which they have been extruded) and the great Trap district on the north, Though the date of this extrusion is uncertain, it is generally believed to have happened—as did that of the Trap—in Meio- cene times. The author stated that he had already described a number of cirques in districts of sedimentary rock (Quarterly Journal of the Geological Society, Vol. xxvit. p. 312); he was now able to. bring forward an instance from the crystalline rocks, in which good examples of such configurations, so far as his experience went, were rare. He considered that the cirques described near the heads of Alpine valleys could not be accounted for by upheaval, or marine erosion, and, by reason of the steepness of their cliffs and the limited space above 239 them, could not have been excavated by glaciers. In the above- named paper he had brought forward reasons for maintaining that cirques were excavated by numerous rather small streams, acting on rocks, suitably stratified, whose composition and ar- rangement admitted of considerable meteoric erosion. This cirque in Skye, a double one, viz. €-shaped, had its cliffs seamed by the tracks of numerous streamlets, each with its little talus of débris resting on sloping glacier-worn rocks below. He held, therefore, that this cirque had been brought to its present state by the action of streamlets, fed by rains; and had to a large extent been pre-glacial, seeing that the floor was ice-worn. Its configuration forbade him to attribute it to a glacier, unless this agent could be invested with a power of eroding vertically. Mr O. FisHeEr said the author had shewn the glacier erosion theory would not hold, but he thought that vertical cliffs must necessarily be formed by being attacked from the bottom, and that streams pouring down from above would have a tendency to produce a talus and so to mask rather than to form a cliff. He called attention to the action of the sea as evidenced in the Alps. Professor MILLER mentioned an instance shewing how slight the excavating power of water often was: at Bamberg, veins of quartz in a rock scarped some 800 years ago, and since then weathered, now protrude only from } to $ an inch. Professor LIVEING thought that streams could only cut away the bottom of a talus, when they were shot out by an overlying sheet of ice. Mr Bonney, in reply, thought it possible that in many cases there had been a pre-existing favourable configuration of the ground, but how that was produced there was now nothing left to tell. He said that streams in flood could move débris from slopes or taluses where they had deposited it when at their usual volume; that it was impossible to lay down a general rule as to either the rate of erosion at any place—(he had no faith 240 in the application of the Rule of Three to geological chro- nology),—or whether the transporting force of a stream was greater than, less than, or equal to its denuding force: each case had to be judged by itself. In many cases he thought that at the present day the taluses were increasing on the cirques ; that, however, was only a question of rainfall, strata, and the like: further it must not be forgotten that there is chemical as well as mechanical denudation. The Master of EMMANUEL asked whether there was an ob- served difference of constitution in the rocks to explain either the floor or a difference in the slope in the walls of the cirque. - Mr Bonney replied that it was in the case of most cirques difficult to say positively, seeing that their floors were often masked by talus, vegetation, &c., and that in the case of the one described in the paper, in the Syenite, there was no distinction that was conspicuous, although there very probably was some in the chemical constitution of the rock. Differences in the slope of the walls of a cirque in sedimentary rocks were doubt- less due to difference in the strata. Some further conversation took place in which Mr Potter, Mr O. Fisher, and others took part. November 13, 1871. The PRESIDENT (PROFESSOR HUMPHRY) in the Chair. Communications were made to the Society : (1) On a double action Pentagraph. By Prof. Car zy, F.R.S. The machine was exhibited and described. (2) On the experimental verification of the laws of the resistances which bodies are subject to, in moving through the air; and especially on the experiments made by Mr Robins and Dr Hutton with the Whirling Machine. By R. Porrsr, M.A. 241 The experiments made with the Whirling Machine give the resistances which bodies whirled round in the air experience to considerable nicety, and have been considered as correctly giving the resistance which the same bodies respectively would have experienced if they had moved in straight lines. In this paper it was shewn that the centrifugal foree communicated to the air, as is continually seen in various winnowing machines and blowing machines, required consideration, and that the ex- periments must be separated from those in which bodies move nearly in a straight line through the air. (3) On the comparison of the results given by the for- mula for the resistances which bodies experrence whilst moving through fluids, investigated in the paper read before the Society on 7th March, 1870, with the experimental results found by Mr Robins and Dr Hutton by means of the Ballistic Pendu- lum. By R. Porrzr, M.A, In this paper computations from the mathematical formule obtained in a paper read before the Society on 7th March, 1870, were compared with the results found experimentally by Mr Robins and Dr Hutton with the Ballistic Pendulum, to deter- mine the resistance which musket and cannon shot experienced on passing through the air. It was shewn that the computed and experimental results agreed as closely as could be expected in such a subject, and that we may now maintain that the refractory problem of resistances to motion in the air has at length been solved mathematically. | Mr GLAISHER mentioned some observations in which he had been engaged, which shewed that the formula for resistance p-=kv* did not hold in several cases, especially where p was 19 242 small, and said that the Aeronautical Society was engaged in repeating the experiments with the Whirling Machine. Prof. CLERK MAXWELL méntioned that when the Ballistic Pendulum was used, there was a difficulty in accurately ob- taining the initial velocity of the shot fired, but in other expe- riments, viz. those of Mr Bashforth, the shot was fired through frames with cotton threads; these were far better, as then variation in the initial velocity of the shot was avoided. Some further conversation took place in which Mr Stuart, Prof. Clerk Maxwell, and Mr Potter took part. Nov. 27, 1871. The PRESIDENT (PROFESSOR HuMmpPHRY) in the Chair. New Fellows elected: B. E. Hammonpn, M.A. Trinity College. W. A. Braitey, M.A. Downing College. Communications made to the Society : (1) On the Solution of Electrical Problems by the Transformation of Conjugate Functions. By Prof. Crerk Maxwett, F.R.S. The general problem in electricity is to determine a function which shall have given values at the various surfaces which bound a region of space,.and which shall satisfy Laplace's par- tial differential equation at every point within this region. 243 The solution of this problem, when the conditions are arbi- trarily given, is beyond the power of any known method, but it is easy to find any number of functions which satisfy Laplace’s equation, and from any one of these we may find the form of a system of conductors for which the function is a solution of the problem. The only known method for transforming one electrical problem into another is that of Electric Inversion, invented by Sir William Thompson; but in problems involving only two dimensions, any problem of which we know the solution may be made to furnish an inexhaustible supply of problems which we can solve. The condition that two functions « and 8 of x and y may be conjugate is a+,/-18=F(x#+/—1y). This condition may be expressed in the form of the two equations da _48 _4 a6 88 4 dx dy Laeee dy Bais If a denotes the “ potential function,” 8 is the “function of induction.” As examples of the method, the theory of Thomson’s Guard Ring and that of a wire grating, used as an electric screen, were illustrated by drawings of the lines of force and equipotential surfaces. Professor CAYLEY pointed out a theory of the translation of figures, the small parts of which are the same, which Prof. Maxwell in his paper appeared to be leading up to. Prof. MAXWELL replied that he had prepared a diagram with the purpose of illustrating this case of the transformation of Conjugate Functions. 19—2 244 (2) On a machine for illustrating the “ Parallelogram of forces.” By J.C. W. Enis, M.A. It consists of three graduated circular lamine. To the rim of each at the point 0° is fixed a weight w. Round the axle of each is wound a thread. The lamin are supported vertically on moveable stands. These stands slide along the three spokes of a horizontal wheel and may be fixed at any distance from its centre. These spokes again are capable of moving indepen- dently around their common vertical axis so as to take up any angular position with regard to each other. P is an index fixed vertically above the centre of each lamina. It can be easily proved experimentally that the tension of the thread will be proportional to the sine of the angle between W and P. The three threads are all joined to a little ring £& so as to rest horizontally on a horizontal circular graduated lamina attached 245 to the end of the vertical axis of the wheel whose moveable spokes support the lamine. Let 7, A,, P,, W, be respectively the tension of thread, the centre, the pointer and the weight fixed to the rim of one lamina, and so of the others. To use the machine—Place a pin through the ring B so as to fix the ring to the centre of the graduated horizontal lamina. Revolve the spokes so that the angles 4,RA, A,RA, A,RA, as read off on the horizontal lamina may be any pres Slide out the vertical lamine until by the tension of the threads the weights rise so that 2P.A,W,=2A,RA, 2P,A,W,=24A,RA, 2 P,4,W,= 2 A,RA, But ARC ag 8 is AN a sinP.A,W, sinP,A,W, sin P,A,W,’ ‘if T, i $3 3 2 * sin A RA, ~ sin A,RA, ~ sin A ‘Ra, , or the tensions of the threads are as the sines of the angles between the other two. Now remove the pin, and the system will be found to be in equilibrium. (3) On a mode of propelling vessels. By J. C. W. Exuis, M.A. In a fish the tail is the great propelling machine, the office of the fins is merely to guide and balance. It is rather difficult to conceive this action of the tail which is able to drive the body through the resisting medium at such great speed. The single oar at the stern of a boat as used by seamen, or one at the side as used at Venice or in the Upper Rhine, produces motion in a way not altogether unlike that of the tail of a fish. 246 The screw of a steamer acts also in a similar way, and perhaps this is the most perfect machine for the purpose of e CUE Ma «ry a Se a propulsion that can be invented. But as rapid rotatory motion is requisite to drive the screw with effect, the machinery is not of the simplest kind. A method of propelling has been tried, where motion is obtained by driving water through a tube astern. This, how- ever, is but another form of the screw. Another method has been proposed where the action of the foot of a water-fowl is imitated. The present method is one founded on an attempt at a closer imitation of the action of a fish’s tail. The following is the result of some experiments made about two years ago on a lake in Wales. Two wooden pipes about 26 feet long were covered with canvas and tarred: they were fixed parallel about 1 yard apart, and a seat raised above them. 247 This formed the boat. A crank-handle was attached to a ver- tical shaft opposite the seat. The lower end of the shaft was about 7 inches below the surface of the water. This end was so contrived that boards of various shapes and sizes to imitate a fish’s tail might be attached at pleasure so as to lie hori- zontally just under the surface of the water. A number of deal boards of various lengths, taper and shape were tried suc- eessively, but with not much success. Finally a large steel saw about 8 feet long was introduced. This was so far successful -that had the saw been broader, and stiffer at the fixed end, it was clear that fair results might be anticipated. As it was, with very slight exertion, although the boat was very heavy, being nearly water-logged by water having found its way into the pipes, the boat moved forward at the rate of about two miles per hour. The action of the saw seemed to be precisely that of the tail of a fish, lashing from side to side and driving the water astern. When the crank-handle was turned to one side the saw was bent, and in trying to recover itself produced a pressure upon the water partly to one side and partly astern: it was the reaction of this latter portion which drove the boat forward. Any other form of boat would have done equally well, the upright shaft taking the place of the rudder. A pliant steel rod covered with India rubber or gutta-percha in the form of a tail might be even more successful. Within the last few months it has been reported in the newspapers that boats pro- pelled in this way have been tried successfully on some of the American canals. If such a mode of propulsion could be introduced with success, the machinery for driving it might be very simple 248 indeed. We should not require rotatory motion. The tail might be attached directly to the middle of a cylinder sliding backwards and forwards on a fixed piston-rod, the cylinder and piston-rod forming small ares of circles, the piston- ‘rod being also the steam-pipe. (4) On a method of transforming rotutory motion into rectilinear, so that the rotatory motion remaining constant the rectilinear may be completely controlled and made to vary as to speed—may be stopped or reversed at pleasure. By J.C. W. Exuis, M.A. Two equal cones with their vertices fixed together forming a double cone have a common axis. On this same axis there are two other cones with their bases fixed together so that they form another double cone. All these cones are formed of bars, so that the vertex of the first double cone can lie in the interior of the second double cone. The axis is not attached to the first double cone, but is to the second, so that sliding the axis in direction of its length the position of the cones may be altered with regard to each other. The intersections of the cones form two wheels, the sum of whose radii is constant, but the radii may have any ratio to each other. An endless rope is passed over one of these wheels, round a distant moveable pulley A, wnder the other wheel, half round it, then round another moveable pulley B, and then under the first wheel. The two moveable pulleys may be connected by a rope passing over fixed pulleys. If the cones be now made to revolve and the double cones be placed symmetrically, the wheels they form will have equal radii and the pulleys A and B will remain in their positions as the endless rope runs round. But if the common axis of the cones be made to slide, the radii of the wheels will alter and the pulley A can be made to approach or recede from the cones with any rapidity according to pleasure, without disturbing the 250 uniform rotation of the cones. A cam might be made to act upon the common axis so as to slide it in or out according to any given law, and so produce any required motion in the pulley A, There are endless varieties of uses to which this device might be applied. It was originally designed in order to reduce the rapid rotatory motion of a circular saw into a slow recti- linear motion in order to bring the timber up to the saw, in such a way that the advance might be instantly stopped or reversed, or might be made slower to suit the nature of the timber employed. Some such contrivance might possibly be useful for ploughing purposes, so as to enable lighter tackle to be used than at present, and to regulate the speed of the plough according to the nature of the ground. Prof. CAYLEY asked whether the third machine could be made accurate enough to allow it to be applied to the work of a pentagraph, for theoretically it was capable of it. Monpay, Feb. 12, 1872. THE PRESIDENT (PROFESSOR HuMPHRY) in the Chair. It was announced by the President that the Adjudicators of the Hopkins Prize (Professors Stokes, Tait, and Clifton) had awarded it to Professor J. Clerk Maxwell. It was also announced that Vol. XI. Part 3 of the Society’s Transactions was now ready, and would be delivered to the members on application. Communications were made to the Society : (1) Further observations on the state of an eye affected with a peculiar malformation. By THE ASsTRo- NOMER Royat. ee ee ee ee ees a ee ee ae eee a ene 251 _ In this paper the author gave numerical results derived from measurement of the astigmatism of an eye, extending over a considerable period of years, which shewed that during this time there had been a change in the astigmatism. Mr Pane observed that he had found from his experience as an optician that about 1 in 100 suffered perceptibly from astigmatism, and described the mode of correcting it by glasses. He said that when astigmatism existed in the crystalline lens it was difficult to remedy it; but not so when it was in the cornea. It was generally supposed that the astigmatism did not alter with age, so that the Astronomer Royal’s observations were of much interest, as they shewed a change. Professors Miller and Maxwell, Mr Trotter and Dr Latham also made some brief remarks on the subject of the paper. (2) The comparison of measures & traits with measures & bouts. By Prof. Miturr, F.R.S. A standard of length in which the measure is defined by the distance between certain points in the surfaces by which the two ends of a material bar are respectively bounded is called a measure @& bouts. A standard in which the measure is defined by the distance between two fine lines traced at right angles to the axis of the bar is called a measure @ traits. The methods of comparing with one another two measures of the same kind are well known and need not be alluded to here. But the comparison of a measure @ traits with a measure @ bouts cannot be effected so readily. The Astronomer Royal, on being con- sulted respecting the best method of making such a comparison, recommended the following process. Two copies, AB, DE, of the original standard @ bouts are constructed with cylindrical cavities at their middle points reaching down to the points C, F in the axes of the bars, where lines are traced at right 252 angles to their axes, like the cavities adopted by Mr Baily, at the writer’s suggestion, near the ends of the standard yards and their copies. Lets denote the length of the original standard, c that of the copy @ traits. Then, the end D of the bar DEH being brought into contact with the end B of AB, the distance C/' may be compared by microscopes with c. Again, A being brought into contact with E, FC can be compared in the same manner with ec. And AB, DE may be compared with s by touch. From the data thus obtained the difference between s and c may be readily found. The cavities formed in the middle points of the bars must weaken them so as to materially injure their value except for the single operation described. The same end may however be attained without the neces- sity of sinking cavities down to the axes of the bars. The object of the Observer is to obtain a visible mark invariably © fixed for some hours or some days in the axis of each bar near its middle point. At the middle of each bar let a right-angled prism be at- tached with its two smaller faces parallel to the vertical and horizontal faces of the bar supposed to be square. Attach also a small plate of glass, having a fine line traced upon it, in such a position that the line seen by total reflexion in the prism may appear to cross the middle point of the axis of the bar at right angles to it. The marks thus obtained in the axes of the bars may now be used instead of the lines CO, F traced in cavities sunk in the bars, and the operation of comparing ¢ with s will be exactly the same as that which has been already described. In a Memoir by Steinheil on the construction of a compa- rateur for measures d bouts in the 27th Volume of the Denk- schriften der Akademie der Wissenschaften of Vienna, page 166, it is stated that the probable error of a comparison by touch is 000005 millimétres, and that of a single comparison 253 by microscopes is not less than 0°0005 millimétres. This ac- _ curacy is however illusory at temperature 0° C, unless the mer- curial thermometer used in finding the expansion of the bar, or an accurate copy of it, is preserved for subsequent use. For Regnault’s observations shew that thermometers constructed of different kinds of glass, though in perfect accordance with one another near 0° C and 100°C, may differ 0°°5 C at 50°C, as pointed out by J. Bosscha, jun. (Archives Néerlandaises, T. Iv.; Poggendorff’s Annalen, Ergdnzungsband, v., 1871, page 465). At 163°C, the standard temperature adopted in this country, the difference may amount to 0°28 C, which implies an uncertainty of 0°0045 millimétres in the length of a bronze yard, or nine times the probable error of a single microscopic comparison of two such bars, and ninety times the probable error: of a single comparison of two end yard bars by touch. The necessity for appealing to the thermometer used in the original comparison of a bar, in order to find the length of the bar at any given temperature, may be obviated by observing the expansion of the bar from 0° C to 100° C, and from 0° C to about 50°C, by the original thermometer. The same obser- vations being afterwards made with a second thermometer, even if of different glass, and differing in its readings from the ori- ginal thermometer at points intermediate between 0°C and 100° C, we have data from which we can deduce the temperature, as indicated by the original thermometer, from the reading of the new one, and thus obtain the true length of the bar at a temperature considerably distant from 0° C. Professor MAXWELL asked what the value of Whitworth’s method was. Professor MILLER replied that the instrument was so delicate that it was only of use when most carefully handled. Some further conversation occurred in which Mr Ellis, Prof, Maxwell, and Prof. Miller took part. 254 Monpay, Feb. 26, 1872. The PRESIDENT (PROFESSOR HUMPHRY) in the Chair. Fellows elected : J. W. Hicks, B.A, Sidney Sussex College. EK. H. Morean, M.A., Jesus College. s Communications were made to the Society: (1) On Teichopsia, a form of transient ‘ halfblindness ;’ its relation to nervous or sick headaches, with an explanation of the phenomena. By P. W. Laruam, M.D. The author said that the disturbance of vision referred to in this paper was a subject which had engaged the attention of Sir John Herschel, the Astronomer Royal, Dr Hubert Airy, and many members of the medical profession. He should proceed to shew that it was one stage of a complaint known under the name of nervous headache, bilious headache or sick headache ; the complaint not always accompanied by disturbed vision, but other disordered sensations being substituted for it, and on the other hand the disturbed vision not being always followed by headache ; and he should then endeavour to explain the phe- nomena. He divided the complaint’ into two stages, (i) the stage of disordered sensation, and (ii) the stage of headache. After quoting the descriptions given by those whose names are mentioned above as well as by persons who had come under his. own observation, he referred to the causes and condi- tions under which the attacks were induced. It is to be ob- served, he said, that all these causes and causes like to them are of a depressing nature, exhausting the power, and therefore lowering the tone of the system, putting it out of tune, disturb- ing the harmony of the functions, and at the same time exalting pn, eee 255 - the susceptibility of the nervous system. The result was that the power of the ganglia of the sympathetic nervous system to conduct, transfer and radiate the effects of impressions, was no longer controlled by the superior force in the cerebro-spinal centres, and instead of tranquil even harmonious action in the various organs as in perfect health, we had convulsive and pain- ful movements. After referring to the effects of irritation and section of branches of the sympathetic, the next step in his argument was that in the disorder under consideration there was first of all contraction of the vessels of the brain (probably the middle cerebral artery), and so a diminished supply of blood produced by excited action of the sympathetic, and that the exhaustion of the sympathetic following on this excitement causes the dilatation of the vessels and the headache. This he supported by various cases and comparisons. He next discussed the question, why the disorder might be sometimes unilateral and sometimes bilateral, and lastly, why in some cases there is (i) disturbance of vision without headache following, (ii) dis- _ turbance of vision followed by headache, and (iii) headache preceded by disordered sensation, but not by disturbed vision ; all of which he maintained were explicable by the theory which he had advanced. Prof. Humpury said that he had experienced a sudden attack of hemiopia, which was probably of somewhat similar origin; that on another occasion he had been conscious of considerable mental disturbance, accompanied by dilatation of a pupil. This he believed due to the eye being accidentally touched by a little atrophine, and that the mental disturbance was merely nervous sympathy. Mr Trotter asked whether Dr Latham could suggest any cause for the peculiar complicated figure appearing in this disorder. Dr LATHAM said he could not explain the special form. Prof. MILLER stated that he himself had sometimes seen the 256 zizgag outline described, very faint and shadowy, without ‘any other disturbance of the system ; it lasted about ten minutes. Dr MicHArL Foster asked whether Dr Latham had had the opportunity of seeing any patient during the attack itself, so as to see how far the blood-vessels and the pupils were affected. Dr LaTHAM said that he had recently seen two cases during the headache, the pupil was then contracted; but he had not lately seen a case during the time of the peculiar affection of the vision. Dr M. Foster said that though Dr Latham had given a general explanation, he had not brought the explanation suf- ficiently near to the particular case; that the special descrip- tion of teichopsia could only be explained when we knew some- thing more about a very complex matter, the vaso-motor centres of the brain; he also thought that the phenomena described might possibly be produced by some affection of the circulation in the retina. Dr LaTHam said he was quite aware that some difficulties - yet remained in his explanation—he had indeed thought that the minor cases of the disorder might be produced by some affection of the ophthalmic artery. The PRESIDENT said that he was not inclined to attribute it to the ophthalmic artery, but rather to the brain, whether vascular or not could not yet be said. (2) A machine for tracing Curves described by points of a vibrating String; namely, curves of the Sorms 1 tp on . (2a \ n= a0cos( t+a), y=bsin(t+8), 257 = and curves of the form w= acos (—"t+a) ~Beos (Ft+8), 1 y=asin (Fe+a)—bsin (=t+8), T, T, when a and B are constants, or «~8 a fraction of t+ a constant. By J. C. W. ELLIS, M.A. A driving wheel (4) drives by means of a band a disk B,. B., C,, C, ave three other disks equal to B,. A, B,, B,, C,, C, are all in a vertical plane. The centres of B,, B, are fixed in the same horizontal line at a distance of about four times the diameter of the disks. The centres of C,, C, are similarly placed in a vertical line. B,, B, are so connected by rods or otherwise so as to revolve simultaneously. So are C,, 0, The axis of B, is also the axis of a cone with the vertex pointing from B, and revolving with B,. This cone drives by means of a band (b) the disk C,, which is attached to a similar cone with its vertex pointing towards C,. In the disks are bored a number of equidistant holes along the diameters, into which 20 258 can be inserted pegs P,, P, and Q, Q, PB, P, are fixed at equal distances from the centres in B,, B,: Q, Q, in OG, C,. Now as B,, B, revolve through equal angles in equal times, the line P,P, will always move parallel to itself, and similarly 2,9. Hence, if O be the intersection of the lines joining the centres, and P of P,P, and Q,Q,; and straight lines in the plane of the disks through O be taken as axes of co-ordinates, the equation to the locus of P will be x =a cos (mé + a,), y =bsin (n0 +8), where 6, which can be varied at pleasure, is the distance of the peg P, from the centre of B,, and a of Q, from the centre of C,. m:n are the velocity ratios of rotation of the disks C, and B,. This ratio may be altered at pleasure by shifting the band along the cones. Either m or m is negative if the bands are crossed. The above equation to the locus of P may be written (as in Donkin’s Acoustics) : w=asin(t+a), y=bsin (e+), Ty T where ¢ is the time, 7, the time of revolution of the disk C. 7, of B,, a=5+ a,, B=B,. If 7, and 7, are commensurable then the locus of P is a re-entering curve; if not, not. If 7,, 7, are nearly in the ratio of two small numbers m and ‘n, the curve though not re-entering after the time mn may be expressed by the equation . (29 :. (Sar w=asin(“"t+a+it), y=bsin (“"t+8), where & is a small quantity, so that a+kt represents a slow 259. change of a during the course of each revolution. The curve described, instead of re-entering exactly at the time mn, does so nearly but not accurately, owing to the small change in 4, so that it starts on a slightly different course after each interval mn. So that the appearance is that of a curve, slowly changing its character and position, until, if +, and r, are commensurable, it finally returns to its primitive state. If 7, and 7, are not commensurable it never does so. All these points can be clearly indicated by fixing a screen so as to cover the disks: having in it two slits at right angles to each other, one in which P, and P, slide, and the other in which Q,, Q, slide. It is manifest that any point in this screen will trace out the curve a=asin (t+), y=bsin(—t+8), Ty Ts when a and 6 can be arranged at pleasure by shifting the pegs, T, T, by shifting the band, and a, 8 by arranging the position of the disks at starting. A pencil pressed against the screen will trace out the curves. The form of the curves may also be represented to the eye by piercing a hole in the screen, and placing a strong light behind it; or by means of the electric spark passing between the ends of fine wires, the ends being fixed close together at any point on the screen. Another way of representing the curve to the eye is this: two screens are provided, one is attached to the disks B,, B, by the pegs P,, P,; P,, P, do not slide in a slit, but there is a slit extending nearly from P, to P,. Similarly in the screen attached to C,, C, there is a slit extending nearly between Q, and Q,. Light if placed behind can be seen through both screens at once only at the point where the slits cross, namely, at the point P, whose locus is the point we have been treating of. 20—2 260 If to one of these screens a pencil be attached it will trace out on the other screen the curve x= acos (t+) — bcos (—"t+8), Ty T, y=asin(—"t+a)—dsin(t+8). T; T. The constants in this can be arranged at pleasure as in the former curve. It is manifest that if a=6 and T,=T, the curve is résuiced to a point, and the corresponding vibration to rest, ze. the composition of the two motions of the paper and pencil pro- duce rest. This may be taken as an example of interference. The pencil and every. point ‘in the paper are describing the same circle, so that there is no relative motion, and the pencil does not travel over the paper. The above curve is the epitrochoid, which ineludes the epicycle as a particular case. By crossing the band we make T, negative, and obtain the equation to the hypotrochoid i in- cluding the hypocycloid as a particular case. In this case the curve cannot be reduced to a point, but may be equal to zero during the motion, or y may be, 7.e. we may reduce the vibration to either one of two straight lines at right angles to each other. Prof. CAYLEY mentioned a machine by M. Perigal for de- scribing curves in a somewhat similar manner. Dr Hubert Airy had drawn similar curves with a pendulum. Mr GLAISHER said that the above machine, exhibited in 1848 at the Royal Society, drew curves of more complexity than that of Mr Ellis. He described the machine and gave a brief sketch of its origin. ee i ew _ ee ee a eS ee 261 March 11, 1872. The PRESIDENT (PROFESSOR HumpuRy) in the Chair. Fellow elected :— J. W. CaRTMELL, M.A. Christ's College. Communications were made to the Society: (1) A Monograph of the Ebenacee. By W.P. Hiern, M.A. The family Ebenacew was first established by Ventinat in 1799; it was revised by Jussieu in 1804; and in 1810 it was reduced to its present limits by the great botanist Brown. In 1837, Geo. Don, in his “General System of Gardening and Botany ” vol. iv., gave an account of the whole family as under- stood by him; he enumerated about 80 species which he dis- tributed among 8 genera. In 1844, Alphonse De Candolle monographed the family in the eighth volume of the “Prodromus systematis Naturalis regni yvegetabilis,” and produced 160 species and 8 genera. Three of these genera were new and several of Don’s genera were not maintained. | In the present monograph 5 genera only are recognized, namely, Royenu and Luclea from Africa, Maba and Diospyros from various countries, and Tetraclis from Madagascar, the last: of which is new; and among these are distributed about 250 species. An account is also given of the fossils that have been ‘published as members of the family, but little confidence is placed in the determination of the genera or family in the case of the great majority of the fossil species, and they are not included in the above-mentioned estimate. 262 For the purpose of preparing the present paper, the great collections both in this country and on the Continent have been examined. : The economic properties of the various members of the order are fully described. ; The head-quarters of the family is India, where the species are numerous, but of the 5 genera which compose the family. only 2 (though these are by far the largest genera) occur in the whole of the East Indian regions. Two genera are peculiar to the continent of Africa, and one, a new genus, is peculiar to the island of Madagascar. Not a single species is indigenous to Europe; one however is naturalized in the countries bordering on the Mediterranean Sea; this one species is indigenous to the Steppes-region of Asia and to China and Japan. Tropical Africa, including Natal, has above 40 species; the Kalahari region of South-west Africa south of the tropic and north of the Orange river has 6 species; and the Cape of Good Hope has above 20 species. Australia has about 16 species, none of which occur on the western coast. The Forest region of the Western Continent of Griesbach has only Diospyros virginiana, L.; the Prairie region has 2 species: the Californian coast- region none; the Mexican region 8; and the West Indies 6 species. The South American region north of the equator has about a dozen species; the region of equatorial Brazil 9 ; and the remaining portion of Brazil 14 species. Madagascar has 283 species; the Mascarene Islands 6; the Seychelles 2; Sandwich Islands 2; Fiji Islands 2; and New Caledonia 11 species. Lists are given, arranged in numerical order, of collections of Ebenacess made by the principal botanical travellers. A chronological list is also given of the published specific names, with references and localities. The natural orders bearing the closest affinities to Hbenacew are Olacinew, Styracee, Anonaceee, Ternstraemiacee, Sapotacee and 263 Ilicinew ; a plan is given exhibiting the affinities including these families and others which at a greater distance also bear some affinity to Zbenacee. A detailed description of the natural order, the genera, and the species forms the chief bulk of the paper. An alphabetical list of local names of the species, and diagrams for each genus exhibiting the numbers of stamens in ‘each species, conclude the monograph, which is illustrated by several plates. (2) The influence of human degenerations on the pro- duction of insanity. By Dr Bacon. The object of this paper was to shew that insanity was a result of degeneration in the race, produced by overcrowded dwellings, vitiated air, insufficient nourishment, interbreeding and the like. The author called attention to the circumstances under which Cretinism existed in the Alps and other places; and shewed that insanity in England was most prevalent in those counties where the agricultural labourers were the worst paid. Thus in Wilt- shire 1 in every 12 was a pauper, 1 in every 327 insane: but in Westmorland and Cumberland, where the paupers were 1 in 28 and in 24 respectively, the insane were one in 517 and 543. Hence he held that for the diminution of insanity more must be hoped from measures tending to raise the condition of the people, than from any increase of medical skill. The PRESIDENT remarked on the importance of the commu- nication, and said that as the town population was increasing at the expense of the rural, it was important to ascertain whether there were any signs of mental degeneration accompanying the asserted physical degeneration among them. Prof. Pacer enquired whether Dr Bacon had detailed facts in the case of one village which he had mentioned, and made some remarks on the physical degeneration in towns, a larger 264 percentage of recruits being rejected from towns than from country places. ae Dr LATHAM asked whether increasing luxury might not also tend to increased insanity, and this be an equal danger with poverty. Dr Bacon said that no doubt there were evils attendant on civilisation, but that he had founded his remarks on statistics ; and that these pointed to the poorest counties being the most liable to insanity. Dr CAMPION said it must be remembered that the more actively disposed persons left the country, so that the feebler, and those most degenerated, remained behind, and that statistics which did not take account of this could not be trusted. Some further conversation took place upon the same subject. (3) Supplement to a table of Bernoulli's numbers. By J. W. L. Guatsner, B.A. April 29, 1872. The PRESIDENT (PROFESSOR HUMPHRY) in the Chair. The Treasurer (Dr CAMPION) gave a statement of the So- ciety’s accounts for the past year, which had been audited by Mr PreTers and Mr Main. The thanks of the Society were proposed by Prof. LivEine, seconded by Prof. BABINGTON, ma unanimously accorded. Communications made to the Society : (1) On certain effects on light on Portland Stone. By F. A. Parzy, M.A. Mr Patey described the tendency of Portland stone and the’ odlites, as Bath, Barnack, Ketton, &e., to contract, in different a ee. ee a eS ee ee 265 degrees, blackness by exposure to the air. He shewed some reasons for doubting if this was due simply to the effects of smoke, and shewed that in all cases, but more markedly in the Portland than in other stones, the blackness was either pre- vented or removed by the incidence of the sun’s rays. Many ceases of this were adduced from the buildings in the University, and the black and white portions of St Paul’s Cathedral were shewn to be referable to the same causes, or apparently to fol- low the same law. That the black was not due solely to smut or smoke-marks, was inferred from portions scraped from the blackened surfaces being found to be quite unaffected by soap or solution of soda, and presenting a changed appearance under the microscope. It was suggested, as a question of scientific in- terest, that the potash or phosphate of lime in some of these stones might, in the course of years, undergo some chemical change analogous to oxidisation ; at all events, difficulties were pointed out in the common and obvious conclusion, that the blackening of buildings was in all cases due to the effects of smoke alone. Mr TROTTER made some enquiries as to the positions of the blackened surfaces with reference to the channels down which rain might run. _ Mr O. FisHer had only seen these blackened surfaces in London and Cambridge; and that the black was removed by slight exfoliation of the stone when frozen. The stone which lay about near the quarry was not blackened. Possibly the “quarry water” might have something to do with it. Professor MILLER thought that some kind of vegetable. growth was the chief cause, instancing the black stains common on limestone and dolomite cliffs. The red sandstone of Stras- burg Cathedral—though apparently not a favourable stone— was covered with vegetation. Professor LivEING mentioned the blackness on parts of the white marble in the floor of King’s College Chapel. This he 266 thought due to vegetation. The action of heat in drawing out- wards various crystallizing substances in the stone might keep parts clean that were exposed to the sun. Mr Bonney thought that the dampness or dryness of the stone, where sheltered from or exposed to the sun, was the chief cause; as favourable or unfavourable to both the growth of vegetation and the lodgment and chemical action of soot. Professor BABINGTON said that it was true there was much vegetation on the stone on the north side of King’s Chapel, but he attributed this blackness to smoke. (2) On Faye’s method of comparing métres a traits ; and an improvement of it suggested by Professor Mier, F.R.S. Diagrams of the instruments were exhibited and described, and a few remarks were afterwards made on them by Professor Maxwell. (3) On certain lithodomous burrows in the Carbonifer- ous limestone of Derbyshire. By T. G. Bonyery, B.D. The author referred to two previous communications on the same subject, and stated that some doubt having been expressed as to the accuracy of his observations in the most important case described in one of them, he had again visited the same neighbourhood. Not only had he confirmed his previous ob- servations, but he had found a large number of other burrows, which he described, exhibiting a very fine specimen; and he maintained these could not be (as had been said) the work of marine mollusca, as Pholades. It was very improbable that they would have lasted so long in limestone rocks; they were 267 unlike Pholas burrows in shape; they were in positions where it was wholly impossible that Pholades could burrow, as, for exam- ple, driven vertically upwards into overhanging slabs of rock ; they were at the bottom of valleys of river erosion, such as Miller's Dale and Tideswell Dale, and in one case on a scarp of rock which he was now convinced was artificial He had some additional evidence for their being the work of snails, and thought that Helix nemoralis and lapicida as well as H. ad- spersa made them. . Mr NEVILLE GoopMAN described the Monte Pellegrino (Sicily) where the stone all over the mountain is perforated, in situations where the Pholas could not bore, and in rocks which had probably not been submerged since secondary times. He quite agreed, from what he had seen, that these burrows were the work of snails. Mr O. FisHerR asked whether possibly the lime was needed by the snails. Professor HUMPHRY thought that the mode of making the hole was mechanical, by the odontophore, rather than by che- mical action. Mr O. FIsHER exhibited a flint flake from Crayford, which was taken from the old brick earth; it was associated with re- mains of E. Antiquus and R. Megarhinus, below beds with | Cyrena fluminalis and Unio littoralis. May 13, 1872. The PRESIDENT (PROFESSOR HUMPHRY) in the Chair. New Fellows elected : G. F. Sams, M.A. Dav, B.A. \ St Peter's College. 268 Communications made to the Society : (1) On a method proposed by M. Fizeau for comparing a métre & bouts with a metre a traits. By Profes- sor MILER. (2 ) On the section exposed at Roslyn Hill Pit, Ely. By T. G. Bonney, B.D. The author stated that hitherto two hypotheses had been proposed to account for the extraordinary collocation of Boulder clay, Cretaceous beds and Kimeridge clay in this pit; (1) which had been advocated by Mr H. G. Seeley and others, that this was the result of faulting ; (2) that, as had been suggested by Mr O. Fisher, the cretaceous beds were a boulder-like mass, that had been dropped in boulder clay times from an iceberg into a depression which it had excavated in the Kimeridge clay. He stated that during the last three years he had fre- quently visited the pit with a view of testing these theories, He pointed out that if the collocation were the result of a fault we should have in the space of about a hundred yards two cor- responding down-throw faults bringing down the boulder clay, _ and an inner pair of (relatively) up-throw faults for the creta- ceous beds, which latter were reversed faults. He also shewed that the lower greensand at the E. end of the pit was not, as had been supposed, in situ, and that the boulder clay at the S.E. corner formed a wedge-like mass that ultimately disappeared, allowing the gault to come in contact with the Kimeridge clay. He exhibited plans and sections, and argued that the collocation was in the highest degree improbable on a theory of faulting. There was a third hypothesis possible, that the creta- ceous beds had slipped from above the Kimeridge clay into - their present position, but, though some appearances favoured that, he thought it, on the whole, less probable than the ae el ite eh ll 269 boulder hypothesis ; he only differed from Mr Fisher in think- ing that the valley existed before the iceberg came. He quoted some instances of large included boulder-like masses, especially one recorded by Professor Morris, in Lincolnshire. Mr O. FisHER expressed his pleasure at the corroboration which his hypothesis had received. He thought the valley could hardly have existed before, because the clay would have formed sides sloping more than the limits of the Kimeridge clay appeared to do. He had, since writing his paper, sometimes thought that the boulder might have been dropped on the top of the Kimeridge clay and crushed its way down into its present position. | Mr Bonney, in reply, gave reasons for the supposed pre- existence of the valley, and thought it doubtful whether the boulder would be heavy enough to crush out the beds below. May 27, 1872. The VicE-PRESIDENT (PROFESSOR BABINGTON) in the Chair. Communications made to the Society : (1) On some properties of Bernoulli's numbers, and, in particular, on Clausen’s Theorem respecting the fractional parts of those numbers. By Professor J. C. Apams, F.R.S. The author stated that the theorem enunciated by Clausen for the determination of Bernoulli’s numbers had not been proved by him or by any other mathematician—the memoir proposed by Clausen not having ever been published. The author gave a comparatively simple proof of the theorem. Thirty-one of Bernoulli's numbers are already known; the 270 author has calculated 22 additional numbers. He also had proved that if m were a prime number other than 2 or 3, the numerator of the » in Bernoulli’s number was divisible by 2. Professor CayLey called attention to one or two points con- nected with the paper. Mr GLAISHER said he had observed that in dividing B, B,, = and a" the period of all the circulation was the same; he had verified this for about 28; he had not yet proved it, but conceived it would follow from hinaan's theorem. (2) On some of the symptoms produced by Uremie poi- soning in chronic disease of the kidney. By P. W. Latuam, M.D. | The object of this paper was to shew that many of the symptoms, as to the mode of production of which in chronic Bright’s disease much discussion has hitherto arisen, might rea- sonably be explained. That the factors involved were :— (1) The impeded passage of the blood through the minute arteries of the system, caused by excessive contrac- tion and hypertrophy of the muscular walls of these vessels, as has been demonstrated by Dr George Johnson. (2) The hypertrophy of the heart, developed by the re- sistance offered to the circulation from the contrac- tion of these small arteries; and (3) The impoverished state of the blood, which is the necessary accompaniment of the disease. 271 _ The author first dwelt upon the occurrence of paroxysmal dyspnoea or asthma, and after discussing the effects which would be produced if the minute branches of the pulmonary artery were suddenly contracted, and the general symptoms and phy- sical signs which would accompany such an event, he shewed by reference to cases recorded by other observers, and from in- stances which had come under his own observation, that the theory was supported by facts. He next referred to epilepti- form convulsions and uremic coma, and pointed out why, in some cases, convulsions might occur, and not in others; owing to the predominance of one or other of the above-mentioned factors. He then went on to say, that, although cerebral apo- plexy not unfrequently occurred in chronic Bright’s disease, where there was atheromatous degeneration of the arteries; yet that, independently of this, the apoplexy might be caused by the velocity of the blood through the minute tubes being re- tarded, (the velocity through a tube varying as the square of the radius of the section,) and so leading to the formation of a small coagulum ‘of fibrin or a thrombosis. There would then be com- plete obstruction, and consequently the greatest possible pres- sure would be brought to bear on the arterial wall and result very probably in rupture. This also, he contended, explained _the production of pulmonary apoplexy, and minute apoplexies in the kidneys and spleen, or hemorrhagic infractions occurring in chronic Bright’s disease, where no valvular mischief of the heart or endocardiac disease existed. Dr Brapsury thought the symptoms mentioned by Dr Latham were explicable on the supposition that after Bright's disease had set in, thrombosis of the heart had taken place. He described a case of pulmonary apoplexy which he had recently examined, where a large blocking had been caused in the pul- monary artery, and commented upon one or two points in the paper. Dr LatHAm thought the condition found post mortem in the 272 case quoted by Dr Bradbury supported the theory ‘he had ad- vanced, for as there was no valvular disease of the heart, the obstruction had most probably been caused by some of the minute branches of the pulmonary artery contracting, so as to retard the velocity of the blood through them to such an extent as to allow it to coagulate. . ‘ a ee ee UAL GENERAL MrevING, OcToBER 28, 1872. . RESIDENT (PROFESSOR Humpury) in the Chair. ollowing officers were elected: Let Bea de 33 ‘ ~ ae ye a 4 . = eee _ President. 2 re Professor HUMPHRY.| - Vice-P; sid. ts. : " ? a ot Treasurer. ‘Dr Capron. Mr Bonney. Mr J. W. CLARK. Mr TROTTER. New Members of the Council. Professor BABINGTON. Professor STOKES. Mr Hort. Mr M. FOSTER. 7 Bea 276 Communications made to the Society : On the form suggested by M. Tresca, and adopted by the Commission Internationale du systeéme métrique, for the Metres Internationauz. By Prof. Miner. The instrument was described. (1) On Methods of drawing in Perspective. By Mr J. C. W. E tis. (2) On a Method of Levelling (communicated by Mr Euuis) proposed by Mr W. H. Sranuey. 1. On methods of drawing in. Perspective. The problem attempted to be solved was this: ‘Given the Plan of a building drawn to scale and in any given position with regard to the eye, to cause a pencil by a mechanical arrangement to trace out the corresponding Perspective of the Plan, whilst the operator causes another pencil to follow the outline of the Plan.’ The Mechanical difficulties due to friction, jamming &c. were such, that no satisfactory result was obtainable by the methods employed. | The following step-by-step methods might however be of practical utility, especially in the case of complicated curves. The Perspective of any object with regard to any given position of the eye is obtained by joining the eye with every point of the object and cutting the cone so formed by a vertical plane. We draw;to scale the Plan (whose perspective we re- quire) on a horizontal plane. We assume the eye to be ata height HZ above this plane. We take any vertical plane be- tween the plan and the eye, at a perpendicular distance SH from the eye, so that if S be the position of the eye, # is its orthogonal projection on the vertical plane. : ABCD is a Drawing-Board, ZSX a bar of wood fixed to the centre of CD and flush with the Board’ M,H,TH,M, a : straight bar sliding on the Board through the fixed guides H, H,, : so that 7@Q a ruler, fixed at right angles to M, YW, at its central } point 7, sweeps over the Board and is always parallel to the : edges AC or BD. Sis a small ring fixed at any required point in ZX. E£ is another, fixed at any required point in the Board. Any point _@ in the ruler will trace out a line (as QZ) parallel to CD. ELS is perpendicular to QZ. EL (drawn to scale) is the height of the eye above the given Plan, ZS the distance from the plane of reference. 278 One end of a fine thread is fixed at Q, passes through the rings E, S, and is tightened by the hand so as to pass through the point P, where the ruler 7’Q meets the given Plan. The point p, where the portions of the thread QE, PS intersect, is the ‘perspective’ of P, or rather of P’ the oo in the actual Plan to which P corresponds. That p is the ‘perspective’ of P, or the iat of intersection of the straight line joining S and P made by a vertical plane at a distance HS from the eye, can be seen thus. To obtain the actual position of the eye, we must draw astraight line HS’ from E towards us and perpendicular to the paper so that HS’=ES. S’ will then be the actual position of the eye. Again, the actual position of P (on the assumed scale) is obtained by drawing QP’ perpendicular to the paper and on the other side, so that QP'=YP. Now if we join S’P’, the point when this line meets the paper is the perspective of P’. This point manifestly lies in the intersection of the paper with a plane containing the parallels QP’, HS’, 2.e. it lies in the straight line EQ, and divides HQ in the ratio of HS’ to QP, p fulfils these conditions, and is therefore the perspective of P’ required. By sliding the ruler, the perspective of every point in the plan, however complicated, may be arrived at; or, in other words, the section of any cone with any vertex and any base may be obtained. To determine the perspective of any point in an elevation, say » feet above P, draw a straight line through p parallel to the ruler, and where this cuts the thread from S through a point corresponding by scale (as marked off on the ruler) to n feet above P, is the required perspective of the point. There is a little difficulty, especially in some positions, in marking accurately the point p with a pencil however fine, owing to the effect of Parallax, as both threads cannot lie exactly in the plane of the Board, and also because they must be slightly pushed aside in order to mark with the pencil. 279 _. These difficulties may be overcome by a metbod of shadows in the following manner. _ Two rods R,M,, R, M, are screwed into the middle points of the, opposite sides CD, AB of the Board, and their ends are connected with the rod R,R, so as to form the three sides of a parallelogram. ‘The plane of this parallelogram by revolving about MM, as an axis may make any angle with the plane of the Board. F¥, isa thread stretched parallel to R,R,. E,S are fixed points in the Board in the straight line U,M,. B The thread R,E meets FF, in q; SS, is a thread tightened by hand. x is a candle placed in a vertical plane through g. The shadow of g, namely Q, will trace out a straight line, QJ, as in the first method parallel to CD. The shadow of FF, will be QP parallel to AC, and the intersection of the shadows of R, E and SS,, t.e. p will be the required Perspective of P. Another method is represented below. The Board is sup- ported upon a stand which may be slid to any position by run- 280 ning it in the horizontal groove. It. may be raised to any required height by turning the milled head W, and the height is read off on a scale. Mis a sheet of transparent glass, which is hinged at its lower end, and may be fixed at any required angle with the horizontal plane. In ordinary perspective this angle HT EEEPTTTTM is 45°, The eye EF on looking through a small hole in an up- right attached to the upper edge of the glass sees by reflexion any point P of the Plan transferred to p, which is the per- spective of P corresponding to the eye in the position # with regard to the given Plan. To obtain the elevation of the building, say at a height of 20 feet, we have only got to elevate the Plan by turning the milled head W. The Perspective of the Plan at any elevation above the eye may be obtained by drawing the Plan for the same eleva- tion below the eye, and then transposing it, by copying it through the sheet of glass held vertical. It is manifest by giving various inclinations to the glass ‘ve may obtain the corresponding plane sections of a cone of bey form. —— 4a 281 2. On a Method of Levelling. A gauge, similar to a steam-gauge, is attached to a fine jindia-rubber pipe 22yds. long. At the other end is a glass tube a few inches long with a fixed mark upon it, and just below the glass tube the pipe expands into a small india- rubber bulb. The observer holds the gauge and advances 22 yds., his assistant remains behind and holds the other ex- tremity of the tube, the tube being filled with water. The gauge indicates the pressure due to the difference of Level. By squeezing the bulb the water can always be brought up to the fixed mark in the glass tube G. The observer and his assistant always hold the gauge and the glass tube, say at the height of the eye in each observation. Each lb. of pressure corresponds to a difference of Level of nearly 2 ft. The number of feet corresponding to the pressures are marked upon the gauge, from 0 up to + 60ft. on the right hand and down to —60 ft. on the left hand. In the figure the gauge points to + 20, indicating that the observer is 20 ft. below the position of his assistant; had it been — 20, he would have been 20 ft, above him. 282 As soon as the observer has entered this + 20, in his field- book, the assistant comes on and stands where the observer did, whilst the observer goes on and takes another observation. In order to determine the difference of Level between any two stations he. has only to take the algebraical sum i the No. of feet recorded in his field-book. Instead of the glass tube, a second gauge might be em- ployed, and the difference of readings of the gauges entered as the differences of level at each step. In this case the gauges need not be constructed to read below zero. This method of Levelling would be extremely rapid and sufficiently accurate for most practical purposes. The instru- ment would be inexpensive, and could be used by an un- educated observer. It could also be employed at night, which would render it useful in military work before an enemy. By using mercury instead of water, and a very fine flexible tube of copper wire, the readings might be rendered extremely accurate, and the whole weight of the instrument reduced to .a very few lbs. Professor LIvEING communicated a note on Mr Paley’s paper, “On effects of light on Portland stone,” read at a former meeting. During the summer he had visited Portland, and had found that all the stone there became black on all old exposed surfaces. He had sent specimens of the black part to Mr Berkeley, from the Portland quarries and from King’s College Chapel; who said that they were the early stage of some lichen. Professor Liveing had found this blackness on St George’s Church on the top of Portland, but only on surfaces exposed to rain drip—the tombstones also were discoloured. Mr Berkeley had informed him that the lichens in the two cases named above were not the same species. The stones at Portland also occasionally had a red lichen growing on them. New Fellow elected: H. W. Witson, Hon. M.A. 283 ! November 11, 1872. The Presipent (PRorEssoR HuMmpPHRY) in the Chair. Communications were made to the Society : (1) On the advantages of Denison’s Gravity Escape- ment for recording time by Electricity. By Mr W. Kuinecstey. The effects of corrosion and irregular impulse are avoided in this escapement by making the current pass through a por- tion of the pendulum and gravity arm; the pivot on which __ the pendulum is hung, the gravity arms and the banking pins being insulated, and the terminals of the battery being con- nected one with the top of the pendulum and the other with the banking pins, the current is made and broken at the end of each impulse ; the wear and tear being confined to unim- portant parts. (2) Description of a form of Remontoir Clock invented by M. Groux. By Mr W. Kuines ey. The principle of this remontoir is the making the fly fan of the remontoir lock on an intermediate detent, so that the usual concussion and friction are avoided. Professor MAXWELL commented with approval on the prin- ciple of the remontoir clock, mentioning one or two conspicuous defects in existing clocks, and alluding to the difficulty under which a clockmaker laboured in devising any improvement on existing patterns. (3) On certain facts connected with the wasting and final disappearance of the Glaciers of North Wales, By Mr W. Kinastey. The object of this paper was to shew that the Glaciers of North Wales were much larger than is commonly supposed, 284 and more like those of Greenland than of the Alps; filling, as they did, extensive basins, and having little motion except- ing at certain points of escape. — | That these icefields during their decrease dropped their moraines over large slopes and the hill-sides in such a way as to make it difficult to distinguish the moraine deposit from drift. Thirdly, shewing in certain cases the manner in which moraines were deposited as the glaciers broke up into smaller ones in the higher recesses of the mountains. Fourthly, drawing attention to the marks left as evidence of great floods in the valleys during the periods of the wasting _ of the ice. Lastly, giving an account of large deposits of fresh-water diatoms in the lakes; these deposits having been made since the glaciers disappeared, but during a cold epoch; and proving that no sea had reached these lakes since that epoch, but giving a means of estimating the time that has elapsed since the glaciers thawed. The whole paper was intended to draw attention to these facts.in order that persons might be induced to pursue these investigations to a much greater extent than has hitherto been done. Professor C. C. BABINGTON spoke very highly of the value of Mr Kingsley’s paper, which, he said, had explained to him several things which he had always found much difficulty in understanding. Mr Bonney said that Mr Kingsley’s paper was a most interesting one, dealing with a particular case of a general problem, the condition of the northern hemisphere in the glacial period. The author appeared to him to have proved that in the earlier part of that period Wales, like Scotland and Scandinavia, had been covered by.an ice-sheet—differing thus from Switzerland, where separate glaciers seemed rather to have existed. He was disposed to refer the drift-like scattered 285 moraine matter, the like of which he had seen in the Alps, not to the retreat of the glaciers after the above period, but to their retreat when, after the great submergence following the ice-sheet period, glaciers had formed in the valleys. The enlarged river-channels mentioned by Mr Kingsley were, in his opinion, not due to floods in the ordinary sense of the word, but to the rivers themselves having once been much greater than now, as they might be expected to have been towards the end of the glacial epoch. He had investigated these large river channels in most districts of Great Britain, and in a considerable part of Northern and Central Europe. Mr O. FIsHER was disposed to think that evidence of the ice-sheet period might be obtained even in so flat a country as Cambridgeshire, in certain singular contortions and dis- turbances of the drift, and superficial deposits, which he could only explain by the pressure of a great mass of ice. He thought that the absence of shells in many of the drifts ren- dered it unlikely that they were marine, and was disposed to consider that the numerous ice-marks which he had seen in the East of England belonged to an earlier period than those in the West. New Fellow elected: J. B. Lez, B.A., Sidney College. November 25, 1872. The PRESIDENT (PROFESSOR HuMpPHRY) in the Chair. Communications were made to the Society : (1) On the appearance of an extra digit on the hind limbs and then on both fore and hind limbs in two successive generations ; and its bearing on the theory of Pangenesis. By Mr N. Goopmay. The facts which had come within his personal knowledge, and on which he submitted some remarks, were the following : 286 Mr Daintree, of Fenton, Huntingdonshire, bought a cow with three well-developed toes-on each hind limb besides the two ordinary rudiments which hang behind the foot. This cow was without a pedigree or history. She had a cow calf with the same peculiarity as its dam, which was as well de- veloped as in her case, notwithstanding that the other parent was a normal bull. This cow has had two calves by normal bulls. The first was a cow calf with three toes on each hind limb, but somewhat less developed and less functionally in- sistent on the ground than in the case of its mother and grandmother ; the second was a bull calf, which had three toes on all four feet. All the toes assumed to be the extra ones have a similar attachment, viz. on the inside of the foot be- tween the internal functional toe and the rudimentary toe on the same side. Mr Goodman gave a short account of Mr Charles Darwin’s theory of Pangenesis, whose main feature is that each indi- vidual is made up of organic units, all of which are constantly giving off minute gemmules which float freely through the organism and are transmitted in the generation products to the next offspring, and which are so much in excess of what: are required for the building up of the body of the immediate progeny, as to be handed down, many of them, through a great many generations in a latent undeveloped condition. He then applied this theory to explain the facts. The extra toe might be due, in accordance with this theory, to one or more of the following causes :— I. Atavism or reversion to an ancestral type. II. The modification of the proliferous function of certain of the organic units produced by external causes. III. Correlation of growth, supplementing the last-named cause. He argued that the abnormality was not due to atavism— 287 (1) Because it was necessary to travel so far away from the species before a three-toed ungulate was found. (2) Because the toe did not appear in the position where it must have appeared, on this hypothesis, if received. homo- logies of the toes were reliable. (3) Because precisely analogous. cases had been found in the human subject. Six-fingered and six-toed men were not very uncommon; and as no beast, bird, or reptile had more than five digits on each limb, and yet the extra fingers and toes were definitely human, this evidence was conclusive against the abnormality being due to atavism. The second cause might operate in two ways. The organic units might be stimulated to throw off more gemmules, or these gemmules might have enhanced affinities. In either case they might attach themselves to any nearly allied growing part between which and themselves there was an affinity resembling their natural one, and so by effecting a double attachment give rise to a double organ. Nor would the extension of the extra abnormal part to a fresh limb, as in the case of the young bull, be unaccountable. For the extra digit, being the direct de- scendant of a normal one whose organic units had been exces- sively proliferous, would resemble the parent part not only in structure but in the vigour of its budding function. Thus we should have in the desired animal two energetic manufactures of gemmules instead of one, and in the third generation a still greater excess or avidity in the transmitted gemmules which would manifest itself in a fresh attachment. But if this were the true explanation of the. peculiarity, it would follow that the extra digit, though it had the attach- ment of a finger, would be in reality, as in structure, a toe. It is difficult to draw a distinction between the toe and finger of an ox; but as a precisely analogous case had occurred in the human subject, they might safely reason from that. In the case referred to, the extra part had originated in the hands 288 and been extended to the feet, and in that case the digit was decidedly not a finger but a toe: | They were then driven to the last cause— III. The correlation of organs. This phrase expressed our ignorance rather than explained it away. Nevertheless some distinctions might be drawn as to the nature of the correlation, which was exhibited by the extension of the extra part to the fore limb. 1. It was not a teleological correlation, like that which associated the carnivorous teeth with the transverse glenoid cavity for the reception of the hinge of the lower jaw. 2. It was not a correlation such as that by which the single occipital condyle was found associated with the seg- mented jaw in the sauroids. 3. It was not a physiological correlation, since it affected parts containing various structures with various functions—as nerves, bones, &c. 4. It was not an entirely arbitrary (which meant an entirely lawless and unexplained) correlation, as that which determined that white cats with blue eyes should be deaf. 5. It was a correlation which obeyed laws and was capable of analogical illustration. It would seem to be due to a force: operating in the organism which is best expressed by calling it. “polarity,” and finds its best analogy in the force which determines that a crystal shall be built up symmetrically around its axes, and that no molecule can be added on one side of these axes without a corresponding one attaching itself to the other. Or perhaps the polarity exhibited in electrolysis affords even a closer analogy. It would be interesting to find that there was latent in the system of the highest vertebrates a force so masked by other dominant forces and exacting con- ditions as only to appear thus suddenly and abnormally, yet which was identical with that which determines the shape of the simplest forms of inorganic matter. 289 Mr Goodman, during the reading of the paper, referred to many abnormalities besides the one which formed the subject of his paper, and many of his deductions were derived from these. The PRESIDENT made some remarks on the theory of Pangenesis, and read an extract from Darwin, wherein he stated his hypothesis of Pangenesis, and pointed out’ some of the difficulties attendant upon it, as the difficulty of under- standing how such an immense number of gemmules could be contained in a minute ovum, and how they could be trans- mitted unchanged through successive generations. Professor MAXWELL spoke of the difficulty of conceiving of chemical molecules in sufficient quantity being packed in these small gemmules. Dr M. Foster spoke of the difficulties in the theory of Pangenesis, especially that mentioned by Prof. Maxwell; still on that point we have limited positive information as to the size of those chemical molecules, nor do we know how many gemmules were required to be contained in an ovum. It was a morphological limit Mr Darwin was really rather seeking— such a one he (Dr Foster) did not think to hold good, rather was there a physiological one. The “cell” did not always exist as in some low forms of Protozoa. We cannot detect a structure in Protoplasm, still there must be some. approach to structure. If you go on dividing it there must be a limit of division, beyond which it cannot continue to live; this there- fore suggests a physiological limit. He thought that to the “primary ” affinities of gemmules secondary affinities must be added, these must influence also its future fate. With regard to this special problem described by. Mr Goodman we must consider (1) How this abnormality first made its appearance ; this he regarded as a case of re-duplication, a tendency to which as to fusion was well-known: this he thought did not disa- gree with Pangenesis, but that the secondary affinities were 22 290 acted upon. (2) How it was reproduced in the offspring: he thought that the transference of the digit could be explained as Darwin had done, by the correlation of homologous parts ; he thought that these changes were a strong confirmation of . the doctrine of homology. He preferred then to regard this result as coming from the action of secondary affinities (he objected to the term polarity) causing re-duplication, and from the homologies of the members. . Prof. PaGET asked how on Darwin’s hypothesis it happened that parental defects were often not transmitted. Dr Campion asked whether there was a tendency to re- duplicate on the outside, especially in the thumb of the hand and the little toe. Mr GoopMAN, in reply, said that he had, in.a portion of his paper which he had omitted for curtailment, called attention to the action of primary affinities which determined the asso- ciation of the gemmules in the generative products, and the secondary affinities by which they were built up in the derived organism ; he shewed that according to Mr Darwin when re- sults were produced by defect of gemmules this could be made up in the next generation by fission of the gemmules. He shewed from instances that the increase was not always on the inside. He gave also some explanation of the muscular system of the abnormality. Prof. PAGET objected that defects were often made up while excesses were very rarely, so that the chances seemed against the explanation offered by Mr Goodman. Mr GoopMAN pointed out that it was not only the gem- mules from the last ancestors that were transmitted, but from many previous ancestors. Some further conversation took place on the- subject: of Pangenesis. eee, 291 (2) A Pneumatical Design for saving life at sea. By _ MrW. M. Srantzy. (Communicated by Mr J. C. W. Ents.) Reservoirs of condensed air communicate by means of pipes (similar to gas-pipes) laid throughout the ship. These pipes serve to lead the air into large flexible balloon-like bags stowed away against the ceilings of the various compartments. A handle being turned on deck allows the condensed air to escape from the reservoirs and to expand the bags. Hence in such a case as that of the “London,” where the waves filled the vessel, or in the case of a severe leak, the water would be expelled as no pumps could expel it, and a great additional buoyancy given to the vessel. Again, in the case of a fire, the vessel could be partly submerged by opening valves, and the water again driven out by turning on the condensed air. -A better method than that of condensed air would be, perhaps, were a reservoir of water used saturated with am- _monia. A steampipe leading to this reservoir would cause the water instantly to part with many times its volume of ammonia and to fill the bags. Or some gas, such as carbon dioxide, in a liquid form, might be employed. February 3, 1873. The PRESIDENT (PROFESSOR HuMmPHRY) in the Chair. _ A SPECIAL general meeting of the Cambridge Philosophical Society was held, when the following alterations were made in the bye-laws :— To substitute in bye-laws, Sec. vi. § 2 (hour of meeting), half-past eight for “half-past seven,” and half-past ten for “half-past nine.” 22—2 292 To make the following new bye-law— — Residents in Cambridge or the neighbourhood, not being graduates, may be elected Associates of the Society. Each one shall be proposed by three Fellows of the Society, nominated by the Council, and elected by the Society. An Associate shall be elected for a period of three years, and if not then a graduate, shall be eligible for re-election. Associates shall have the privilege of attending the meetings and consulting the books in the Library of the Society. After an eloquent address from the President on the loss sustained by the death of Professor Sedgwick, whom he justly described as really the founder as. well as the ardent promoter of the Society, it was moved by Professor MILLER, and seconded by Professor LIVEING, that an expression of the deep regret felt by the Society at the loss of Professor Sedgwick be recorded on the minutes. The following communications were made :— On the Proof of the Equations of Motion of a connected system. By Prof. Crunk MaxweEtt. | To deduce from the known motions of a system the forces which act on it is the primary aim of the science of Dynamics. The calculation of the motion when the forces are known, though a more difficult operation, is not so important, nor so capable of application to the analytical method of physical science. The expressions for the forces which act on the system in terms of the motion of the system were first given by Lagrange in the fourth section of the second part of his Mécanique Ana- lytique. Lagrange’s investigation may be -regarded from a mathematical point of view as a method of reducing the dy- namical equations, of which there are originally three for every 293 particle of the system, to a number equal to that of the degrees of freedom of the system. In other words it is a method of eliminating certain quantities called reactions from the equations. The aim of Lagrange was, as he tells us himself, to bring ; dynamics under the power of the calculus, and therefore he had to express dynamical relations in terms of the corresponding relations of numerical quantities. In the present day it is necessary for physical enquirers ‘to obtain clear ideas in dynamics that they may be able to ‘study dynamical theories of the physical sciences. We must therefore avail ourselves of the labours of the mathematician, and selecting from his symbols those which correspond to con- ceivable physical quantities, we must retranslate them into the language of dynamics, In this way our words will call up the mental image, not of certain operations of the calculus, but of certain character- istics of the motion of bodies, The nomenclature of dynamics has been greatly developed ‘by those who in recent times have expounded the doctrine of the Conservation of Energy, and it will be seen that most of the following statement is suggested by the investigations in Thomson and Tait’s Natural Philosophy, especially the method of beginning with the case of impulsive forces. I have applied this method in such a way as to get rid of the explicit consideration of the motion of any part of the _ system except the co-ordinates or variables on which the motion of the whole depends: It is important to the student to be able to trace the way in which the motion of each part is determined by that of the variables, but I think it desirable that the final equations should be obtained independently of this process. That this can be done is evident from the fact that the symbols by which the dependence of the motion of the parts on that of the variables was expressed, are not found in the final equations, 294 The whole theory of the equations of motion is no doubt familiar to mathematicians. It ought to be so, for it is the most important part of their science in its application to matter. But the importance of these equations does not depend on their being useful in solving problems in dynamics. A higher func- tion which they must discharge is that of presenting to the mind in the clearest and most general form the fundamental principles of dynamical reasoning. In forming dynamical theories of the physical sciences, it has been a too frequent practice to invent a particular dy- namical hypothesis and then by means of the equations of motion to deduce certain results. The agreement of these results with real phenomena has been supposed to furnish a certain amount of evidence in favour of the hypothesis. The true method of physical reasoning is to begin with the phenomena and to deduce the forces from them by a direct application of the equations of motion. The difficulty of doing so has hitherto been that we artive, at least during the first stages of the investigation, at results which are so indefinite that we have no terms sufficiently general to express them without introducing some notion not strictly deducible from our premisses. It is therefore very desirable that men of science should invent some method of statement by which ideas, precise so far as they go, may be conveyed to the mind, and yet suffi- ciently general to avoid the introduction of unwarrantable details. For instance, such a method of statement is greatly needed in order to express exactly what is known about the undulatory theory of light. (2) On a problem in the Calculus of Variations in which the solution is discontinuous. By Prof. CumrK MAxwELL. The rider on the third question in the Senate-House paper 295 of Wednesday, January 15, 14 to 4, was set as an example of discontinuity introduced into a problem in a way somewhat different, I think, from any of those discussed in Mr Todhunter’s essay’. In some of Mr Todhunter’s cases the discontinuity was involved or its possibility implied in the statement of the problem, as when a curve is precluded from transgressing the boundary of a given region, or where its curvature must not be negative. In the case of figures of revolution considered as generated by a plane curve revolving about a line in its plane, this forms a boundary of the region within which the curve must lie, and therefore often forms part of the curve required for the solution. In the problem now before us there is no discontinuity in the statement, and it is introduced into the problem by the continuous change of the co-efficients of a certain equation as we pass along the curve. At a certain point the two roots of this equation which satisfy the minimum condition coalesce with each other and with a maximum root. Beyond this point the root which formerly indicated a maximum indicates a minimum, and the other two roots become impossible. New Fellow elected: A. FREEMAN, M.A., St John’s College. February 17, 1873. The PRESIDENT (PROFESSOR HumpuHRyY) in: the Chair. (1) On the name “ Odusseus” signifying ‘‘ setting sun,” and the Odyssey as a Solar Myth. By Mr Patey. This shewed that the name of Odysseus or Ulysses was more probably connected with Sduvouevos jdsos, “setting sun,” than with ¢dddyos, “dwarf.” It was shewn that all the details 1 Researches tn the Calculus of Variations, &c. 296 of the Odyssey were easily interpreted as connected parts of a solar myth, describing the journey of the sun to the west, and his return, after many struggles and adventures, to his ever- young bride in the east, Penelope, the “spinstress,” 2.¢. cloud- weaver. The general geography of the Odyssey was noticed, as pertaining rather to Magna Grecia, while the Iliad is essentially Asiatic in its scenery and deseription. The Cyclops was shewn to be the Sun’s eye, extinguished by Ulysses, z.e. lost by the Sun when he sinks into the west. The sorceress Circe, and the nymph Calypso, “the coverer,” were mterpreted as exercising that weird influence over the Sun that is still, by rude races, attributed to magic or to the evileye. The guidance of Athena, the goddess of dawn, was shewn to be a fitting companion and guide to the Sun in his return. The wreck of Ulysses, and his narrow escape from drowning, was shewn to represent the Sun ' sinking in the western ocean. Finally, the killing of the suitors with the bow was shewn to be consonant to the usual represen- tation of Apollo and Diana, the sun-god and moon-goddess, who were thought to slay mortals with their deadly arrows. The old Laertes, the father of Ulysses, was compared with old Ti- thonus, the bride of Aurora, and the symbolism explained by the union of the ever old with the ever new. _ | Prof. SELWYN remarked with reference to the small stature of Odysseus that the sun became larger on approaching the horizon. A passage, however, in the eleventh book shewed a connexion with a solar myth, where Circe tells Odysseus: he will feed the herds and flocks of the sun, which were tended by two shepherds (Mercury and Venus). Also a passage in the Tliad, where Jupiter says that all the gods together could not draw him from his seat, but he could lift them with his left hand, the centre of gravity of the system being within the sun. Mr Patsy said that loss of strength might be meant, and that Plato had referred the myth above named to a solar origin. Mr JepB asked (1) how far the allegory was conscious or 297 unconscious. (2) Concerning the etymology. He thought that the first idea of a great journey might be taken from that of the sun, but he doubted whether the human interest, intro- duced into the Odyssey, allowed us to bring it into the same category, except so far as that one archetypal idea might under- lie it. Therefore he doubted whether the allegory was at all ~ present to the mind of the writer; and so he thought that the journey and its incidents were essentially human; the framework indeed might be supposed suggested by the solar journey, but in the incidents the writer was dwelling on the human side, and so he doubted whether it could be called a solar myth. (3) As to the etymology, the derivation from ddvecero, as used by Ino, “he with whom the gods were angry,’ was generally accepted in the best and most critical times of antiquity; and still has the sanction of Curtius, who refers it to oddvccoua, explaining the 6 as prosthetic (of oBeror, &c.), thus it would mean the wrathful one, and express the majesty of anger. Mr PALEY said he thought that the author viewed Odysses simply as a man, but unconsciously followed the tradition. Mr Jess asked how far Mr Paley regarded the details, e.g. those concerning Circe, Calypso, as supplied by the earlier allegory, or as arising from the mind of the poet when writing on a very simple framework. Mr Pavey said it was very difficult to say, but he thought that the vitality of the myth would affect them. Mr FENNELL thought the human interests attached to a myth would cause a very anthropomorphic form to be given to it; and so a hero would grow out of the myth, and action be grouped around him, in accordance with a tendency common in ancient times. He thought Curtius might be wrong about the prosthetic omicron, and that it might be a relic of an old preposition, which still remained in Sanscrit. 298 (2) On the identity of the-modern Hindu with the ancient Greek ship. (A model of a Bengalee ship was exhibited.) 7 This communication explained, by reference to the model of a Hindu (Bengalee) boat, the minute identity in all the details of the mast, sails, tackle, and rudder (or stern-paddle) between the old Greek ship and the modern Indian river-boat. The lowering of the mast, the working of the yard-arm by a man who watches and co-operates with the steersman, and the manner of bringing to shore and fastening the boat, stern ashore and prow to the sea, were illustrated by quotations from Greek poets. The mechanism of the rudder, or paddle, with its “ rudder-bands,” was explained, with the motion ofits own axis produced by the tiller or handle, ota&, the rudder itself being called mndaduov. It is a large and heavy timber requiring many men to lift and carry it. Many technical terms in the Greek writers were identified and explained by the model, which appeared to represent the unchanged model that has prevailed for above 2000 years. Professor MILLER said that the boats on the Boden See were exact models of ancient boats, and described some peculia- rity of their rudders. Professor LIVEING said, with reference to a point Mr Paley had discussed, that he believed the larger junks in the East still had two rudders, Mr Pearson (Emmanuel) thought that at any rate in classical times the rudder had been doubled. 299 March 3, 1873. The VicE-PRESIDENT (PROFESSOR ADAMS) in the Chair. Notes on the Hippopotamus. By Mr J. W. Crarx. Mr J. W. Clark exhibited the mounted skeleton, and some portions of the visceral anatomy (preserved in spirit) of the young female hippopotamus, which was born in the Gardens of the London Zoological Society, on January 7, 1872, and died on the following Wednesday. The remarks he made in illustration of the specimen were in substance a réswmé of the paper read by him before the Zoological Society on Feb. 20, 1872, and printed in their Proceedings. On the Foraminifera and Sponges of the Cambridge Upper Green Sand. By Mr W. J. Sotuas. After a description of the Green Sand, and an enumeration of its characteristic foraminifera, the author discussed the origin of its abundant green grains, and indicated that to a large extent these bodies consist of the casts of foraminifera. The included coprolites of the formation ‘were next investigated. Their marked connection with previously existing organic matter was noticed, and it was shewn that this connection characterised the coprolites of various other deposits. Hence was derived a definition for the word “coprolite:” coprolites being defined as “those bodies which have been produced by the phosphatic fossilisation of organic matter, or of the imme- diate products of its decomposition.” The nature of the organ- isms which furnished this organic matter was shewn, in the case of nodules of hitherto obscure origin, to be spongeous. The sponge-like form of these nodules, the characters and arrangement of their well-marked oscules, and the forms and 300 disposition of their siliceous spicules, seemed to leave no doubt upon this point. A comparison was instituted between the coprolites of the Green Sand and those of other formations; in the Lower Silurian of Canada, sponge-like coprolites had been met with. The chalk flints only differed from coprolites in being silicified instead of phosphatised sponges. Ventriculites accompanied both, and in both xanthidia and foraminifera were found. The phosphate of lime which fossilised the green sand sponges might have been derived from the volcanic rocks of Lammermuir, and conveyed to them by the cold current which afterwards eroded the gault and supplied the silicates to in- filtrate the foraminiferal casts. Finally, the formation of coprolites appeared to be a in the Chincha Islands at the present day. Mr Patey asked how the occurrence of phosphate in Tere- bratula was to be explained if the coprolites were to be attributed to sponges. Mr Sous explained that he did not refer all these phos- phate nodules to sponges, but to the phosphatization of animal matter. Professor LivEING asked what was meant by the nodules being derived from the Gault, for he thought that there was no evidence of the nodules occurring in the Gault. Mr Bonney stated that nodules corresponding very closely with those of the Upper Green Sand did occur in the Gault, as for example in the Barnwell pits and in Roslyn pit at Ely. They also occurred high up in the Gault at Folkestone, and were not confined to the base of that formation, although layers existed there, as at Upware and other localities. He con- gratulated Mr Sollas on the excellent work which he had done with these obscure organisms, and agreed with his results. Mr PALEY said that the form of the green grains was too regular to make it probable that they were fragments from voleanic rock, 301 | On a Boulder in a Coal Seam, South Staffordshire. he By Mr Bonney. This boulder was found in the 13th coal of the Cannock and Rugeley Colliery—which seam -is about three yards thick, and probably about 200 feet above the base of the -coal-field, which in South Staffordshire rests on upper Silurian rocks. It weighs 13 lbs. 1340z., and is about 19 inches in girth either way, and about 44 thick. The rock is a very compact grey quartzite, which exactly resembles that of the pebbles in the Bunter conglomerate of Staffordshire. He thought it had been brought entangled in the root of a tree. The difficulty was to find out whence it came. The Bunter pebbles were supposed to have chiefly come from Old Red Sandstone rocks of east Scotland, and to have been originally derived from much older highly altered rocks, probably rather in the north-west of Scotland. The general course of the sediment in both the Bunter and Carboniferous times was from the north-west, and it was probable that the pebble too came from that quarter. The principal difficulty in that supposition was that all the known beds containing similar pebbles to the north-west did not appear likely to have been undergoing denudation in Carboniferous times. Hence the author thought that with our present knowledge the problem could only be stated and not solved. New Fellows elected : C. W. Moutg, M.A., Corpus Christi College. C. W. Hrrcutns, B.A., Sidney Sussex College. - 802 Monnay, March 17, 1873. The VicE-PRESIDENT (Professor LIVEING) in the Chair. On an improved Camera Lucida invented by Professor Govt of Turin. By Prof. Miuuzr. The peculiarity of the instrument consisted in the use of a transparent film of gold leaf, by means of which the view of the object was improved; but a description cannot be given without diagrams. Professor LIVEING spoke of the advantage of a methylic aldehyd for depositing. silver, and enquired why gold was selected for the above process in preference to other metals. Professor MILLER said it. was, he believed, because of the thinness and durability of the film that could be obtained. Professor CLERK MAXWELL spoke of some of the advantages of using the films of gold, and commented upon the instru- ment. The following were elected Fellows of the Society :— V. H. Stanton, B.A., Trinity College. A, P. Humphry, B.A., Trinity College. ‘J. Dew-Smith, B.A., Trinity College. C. Yule, B.A., St John’s College. The following were elected Associates of the Society :-— Mr J. Carter, Cambridge. Mr A. Graham, Cambridge. Dr Bacon, Fulbourn. Mr W. J. Sollas, St John’s College. Mr H. N. Martin, Christ’s College. Mr T. Bridge, Non-Collegiate Student. Mr A. J. Jukes-Browne, St John’s College. Mr P. H. Carpenter, Trinity College. . Mr W. Marshall, Ely. Mr W. E. Pain, Cambridge. Mr F. M. Balfour, Trinity College. ———— 303 April 28, 1873. The Vicr-PrEsIDENT (Professor LIVEING) in the Chair. On some so-called “ Horite” caves at Beit Jibrin (Eleu- theropolis). By Prof. Parr. Beit Jibrin is the ancient Bethogabra or Eleutheropolis, but the modern name is much older than the Greek appellation and represents the Hebrew Beth Giborim, the House of Giants, a name suggestive of the gigantic Philistine inhabitants of Gath: indeed Beit Jibrin is without question the site of Gath, and not only does it fulfil the topographical conditions, but amongst its ruins is one bearing the name Khirbet Jat, i.e. the the ruins of Gath. . Here are some curious excavations which nearly all travel- lers who have visited them assume to be Horite, and of great antiquity. Dr Robinson, in his Biblical Researches, says some contain inscriptions which are the work of casual visitors and do not throw any light on the age or object of their construc- tion: as many of them are written on the domed roof at a height of about 30 feet, and in a totally inaccessible position, it is hard to imagine how this could have been the case. Professor Palmer read some extracts from his diary, written during a visit to the caves in question, in which he stated that :— The caves at Beit Jibrin are evidently quarries, though afterwards wrought into their present shape with some ulterior object, such as the formation of granaries, stalls for cattle, &e. The stone is much better at the bottom than at the top, and the method pursued in excavating seems to have been to work downwards, leaving a hole in the roof to give light, and smooth- ing off the walls as they went on. The last touch of smooth- ness in some of the walls appears to have been given by cutting out little niches or pigeon-holes and then knocking out the 304 interstices, by which means a good deal of labour was saved. One cavern is completely covered with blocks which have not been removed in this manner—but the walls of others shew ‘frequent traces of them. The caverns are certainly not earlier than the Christian era, as there are numerous crosses and figures and Cufic inscriptions, the last apparently not earlier than the 4th or 5th century. One of these inscriptions is a personal prayer for the writer as one of the labourers in the cave. Nor could these inscriptions have been done at any other time than during the construction of the chamber. That just mentioned for instance is at a height of 30 feet, and on the arch of the domed roof, so that it could not have been written from a ladder (even if such a thing could have been obtained, and even now tall ladders are unknown in native oriental building), for the writer would have been leaning back in an impossible position, and had to stretch out and carve with mason’s tools a distance of 3 or 4 feet on either side. Nor could a scaffold have been used, as it would have been impossible to sling it from the walls, and it must therefore have been built up from the ground, which is absurd for such a mere private and idle inscription. The figures and some of the crosses, especially the geometrical figure (which is done with mason’s compasses), imply the same difficulties and, as well as the inscription just mentioned, would require hours of idle work ; just such as might have been done while the steps were still remaining. The inscriptions on the unfinished and finished parts are of the same date and character. The inscriptions consist of formule like those of Islam, but without mention of Mohammed; there is, however, no reason to suppose that the well-known formule of Mohammed were other than borrowed from older rites. The Cufic, although posterior to Christianity, is probably anterior to Islam for paleographic reasons, as for instance the word puke 3), the form > being found for the affixed pronoun as in the Hi- myaritic writing. The cave in the flat-topped hill is not of any 305 great extent, and was probably a cistern. Its name Sendahanna suggests the Christian title St Anna. The caves are certainly not Troglodyte, and as Ptolemy i in the 2nd century mentions nothing of them there is an additional argument, if such were required, for placing them subsequent to the Christian era. One or two appear to have been inhabited, and are covered with black from the smoke of fires, but here the crosses are as much stained as the rest of the wall, neither more nor less. As the floor shews that it is now used for a sheepfold, the smoke is probably an accumulation of years, this particular cave being for some reason more convenient for the herds. Mr PALEY instanced the Royston cave cut in the chalk, with figures of saints on the sides, but reached by a descending pas- sage, as a somewhat parallel instance. Mr Lewis stated that the ear of Dionysius furnished a parallel, but the inscriptions there were now illegible. Mr Bonney asked whether there was any resemblance be- tween these caves and those under the Dome of the Rock, and instanced the Royal caverns at Jerusalem as a case of sub- terranean quarrying. Professor LIVEING mentioned that the dome-shaped method of quarrying was not very uncommon, as it was often the most economical; and instanced a case in the neighbourhood of Cambridge where the stone had been thus quarried. Prof. PALMER replied, by briefly sketching out the different kinds of caves in Palestine; he considered that these at Beit Jibrin had no relation with the Dome of the Rock at Jerusalem, for it was quite small. Some caves now used as granaries had probably once been quarries, but many had been excavated for the purpose. He was glad to find that other instances of quar- rying in this method could be produced, for he had found that some students of Palestine antiquities had met with diffi- culties in accepting his views. 306 On the English sounds of the vowel-letters of the alpha- bet, on their production by instruments, and on the natural musical sequence of the vowel-sounds. By Mr Porrer. This subject falls under the consideration of the gram- marians in their studies of the rules which connect spoken with written language, under the investigations of the physiologists in their discussions of the structure and functions of the organs of the voice and articulation, and under those of the natural philosophers again in their studies of the science of acoustics é as the general theory of sound. The vowel-sounds are shewn by instruments as well as in the voice to be infinitely numerous as they slide or glide. gradually from one to. another through the whole series or sequence IEAOU from I (i) to U (u), without breaks or dis- continuity. Certain sounds of the series are however considered normal sounds, and are supposed to be represented by the vowel- letters of the alphabet; though with little unanimity amongst our grammarians. The comparison of the speech of different countries is of course a distinct study, and does not fall generally under the present subject of discussion. The author of the paper having had to lecture through many years on acoustics in the general course of experimental natural philosophy, using, amongst other acoustical apparatus, Kempe- len’s funnel-shaped instrument and Professor Willis’s sliding tubes producing the vowel-sounds, found the sound of the English vowel I (i) not to be given by them, though he was convinced that it was a simple vowel-sound as now used in all. large towns, and no diphthong, as many assert it to be from. provincial pronunciations. Where opinions differ so much a reference to actual ex- periment is the only safe alternative, and a representation of 307 ‘the organs of the voice and articulation as near as can be made ‘in the ordinary materials at the service of the instrument maker _-is desirable. The author found the metallic free reeds as they are called; with sheet brass tongues fixed at one end, and vi- brating freely in rectangular apertures cut in sheet brass plates, ‘to be the most available substitutes for the chorde vocales (thyro-arytenoid ligaments) of the human larynx, and india- ‘rubber hollow spheres the best representation of the human mouth, as resonant cavities. The reeds being fitted by short tubes to apertures cut in the hollow spheres to represent the fauces of the posterior part of the mouth, and opposite apertures to represent the opening of the lips being cut, the vowel-sounds are produced by compressing the india-rubber shell when blow- ing through the reed, to make the shell take the same form as the mouth when producing a like sound. In this manner the vowel-sounds of I (i) as in the word pipe, of E (e) as in peep, of A (a) as in papa, of O (0) as in pope and of U (u) as (00) in poop, are readily obtained, and shew the English sequence of the vowel-sounds to be the most philo- sophical. The slender (a) as in paper, the open (a) as in father, and the broad (a).as in water, are also easily produced as varia- tions in sound of the first letter of the alphabet, and in English are rightly treated as such. The sound of [ (i) is produced when the idl aperture re- presenting the fauces is constricted and that representing the lips left open, and then the vibrating current of expired air is divergent within the mouth, becoming slightly rarefied from the first law of motion by which each particle tends to move ‘in a straight line and with a uniform velocity, and experiment also shews this rarefaction to take place. Such a change in the vibrating stream of air we know by the properties of organ pipes and wind instruments generally affects the tone of the sound escaping to the open air. On the other hand the sound of U (u) as 00 in poop, is pro- 23—2 308 duced when the front aperture representing the opening of the lips is constricted and the vibrating current of air is convergent within the mouth and slightly condensed, as shewn by experi- ments ; so that another character is given to the sound passing the lips, and the vowel-sound U (u) is produced. The sound E (e) as in peep is produced when the constriction is less than for I (i) and somewhat in front of the back aperture. The sound of A (a) open as in papa is produced when the front and back apertures are both open. The sound of O (0) as in pope is produced when the front aperture and part of the shell near it is somewhat constricted. The sounds A (a) slender and broad are given by slight compressions of the hinder hemisphere for the first and of the front hemisphere for the latter. | The whole of the vowel-sounds in their infinite variations are thus communicated to a vocal note produced in the larynx by the state of the expired vibrating air as it passes through the mouth to the external air, Professor MAXWELL thought that these experiments could hardly be connected with those of Helmholz and Donders, as the vowel-sounds differed in different nations. Mr Trotrer shewed that what might be called pure vowels were very numerous indeed, but that in his opinion 7 was not a pure vowel-sound, and he commented upon Helmholz’s in- vestigations in the analysis of sounds. Mr SHILLETO observed that he thought that ¢ in English was always a diphthongal sound. Mr FENNELL asked if it was known which of the vowels had the greatest and which the least intensity, supposing the funda- mental note constant. Mr Potter said he thought a (as in father) had the greatest intensity, w (like oo in book) the least, and made some further remarks on the subject of the paper. Mr H. GorosBED was elected an Associate of the Society. 309 May 12, 1873. The PRESIDENT (PROFESSOR HumpuHRy) in the Chair. On some conditions of reflex action. By Dr M. Foster. Goltz observed that, while an uninjured frog, placed in a vessel of water the temperature of which was very gradually raised, made efforts to escape as soon as the water became warm, a brainless frog exhibited no movements, and eventually became rigid in the position in which it was first placed. Yet when a brainless frog was so suspended that the toes or feet only dipped into a vessel of water the temperature of which was gradually raised, the feet were always withdrawn by reflex action when the temperature reached 30° C. or there- abouts. The slower the rise in temperature the longer was the withdrawal deferred, but eventually the feet were always with- drawn however gradual the heating of the water. When the whole of both legs was immersed, no withdrawal took place on gradual heating, and the legs became rigid without any attempt to escape having been made. When the legs were immersed up to the knees only, the results were more or less uncertain. It thus appeared that when a sufficiently large surface of the animal was subjected to a gradual heating, reflex actions which otherwise would have taken place were prevented, though the stimulus was by the increase of surface affected largely in- creased. The absence of reflex actions in these cases cannot be attributed to diminution of conductivity in the motor or sensor nerves, or of irritability in the muscles, as these are not di- minished at the temperatures in question. The author was at first inclined to regard the facts as an example of the more general law of sensation that when a surface of skin is affected 310 by a stimulus the sensation is most intense at the junction of the affected and unaffected parts (as when the foot is dipped into hot water). But all attempts to get any similar results with. other stimuli than heat failed; and an experiment in which the upper part of the body was raised in temperature while the legs were not affected, shewed a great diminution of reflex action in the spinal cord. ‘Raising the temperature of the spinal cord would naturally ‘be expected to raise (for a time at least) rather than to lower the reflex excitability—but the author has been led by other experiments to conclude that one has to deal here not with simple rise of temperature but with effects of supplying the spinal cord with blood heated above the normal oT therefore possibly carrying in it abnormal products). Yipeiiess: The lowering effect of heated blood is shewn by immersing -brainless frogs tetanized with strychnia in water at from 30° to 35°C. Ina short time all tetanus disappears, and the animal -becomes perfectly flaccid, though both muscles and nerves are thoroughly irritable. On removal the tetanus speedily returns, and may be again removed by re-immersion. The absence of reflex action of the brainless frog immersed in gradually heated water is due to the fact that the gradually heated water is but a comparatively feeble stimulus for the production of reflex action, and before the skin has become _ sufficiently affected to call forth a reflex action, the spinal cord has become so lowered by the heated blood that it fails to respond by any movement to the stimulus coming from the skin. But inasmuch as a feebler stimulus is needed to awaken consciousness than to produce a mechanical reflex action, the frog possessing a brain begins to move in the heated water at a very early period; and as each movement increases the stimulating effect of the heated water, the movements soon. become very general. The PRESIDENT in proposing a vote of thanks said that the dll subject was one of great interest, and that Dr Foster appeared to have clearly shewn that the raising the temperature of the blood had affected the reflex excitability of the spinal chord, and regretted that Dr Foster had not carried his experiments further and applied them tomammals. For example reflex action . in human beings in cases of fainting was increased by sudden cold—again, increase of temperature of blood (as in fever) lowered the nerve power—this seemed to correspond with the result obtained by Dr Foster. This depression might. result from wear and tear of system, as had often been suggested ; but it seemed possible to connect it with the results of Dr Foster's experiménts. Again, it might be possible to discover. in this way something with reference to the treatment of - tetanus, at present so difficult and inscrutable a malady. He thought it would be well to see how far the nerve power in _ frogs with brains was affected by raising the temperature of blood. Mr TROTTER enquired if Dr F oster had estimated the amount of heating produced on the spinal chord when one. leg only was immersed, and whether that required raising to a higher temperature to produce reflex action. _ PrRorEssoR MAXWELL mentioned that the effect of cooling certain nerves had been to quicken the circulation. Dr Foster said that these experiments belonged to a different class of facts to those which he had described. With regard to Mr Trotter’s question, he had not been able after several experiments to arrive at any very satisfactory results. It was very difficult to get the frog properly placed. He did not think that any very practical result would come with regard to tetanus, for a bath of high enough temperature to affect the spinal cord would probably affect the respiratory functions also. The cause of the lowering of nerve power in fever had yet to be explained, 312 On the Rete mirabile of the Narwhal. By Dr H. 8. Wutsov. He divided his remarks into three parts. The first portion consisted simply of the anatomical facts derived from his dis- sections made of a foetal Narwhal. The second contained remarks on these dissections as far as they differed from man and from the statements of authors who had investigated the ‘subject. The third part embraced the teleological deductions derived from the facts recorded. In his first portion, after describing the principal sources whence the thoracic rete of the Narwhal derived its constituent vessels, and after pointing out wherein they differed from the arrangement of the same vessels in Man, Dr Wilson proceeded to give a minute account of the position, relations, and structure of the rete itself. He shewed that the rete was divisible into halves, each of which derived its constituent vessels from two sources, that these vessels were peculiar in presenting at their origin the same calibre, being very minute, and consisting of great numbers; that they arose from trunks of the aorta of primary or next to primary calibre, and that, thus, in position, the rete was central to the arterial system. He divided the constituent vessels of the rete imto _ three sets, vasa maxima, v. media, and v. minima, giving their distinguishing characters, and concluded this part with the enumeration of the various structures found imbedded in the substance of the rete. The second portion of his paper had reference chiefly to the discrepancies existing between the notes from his dissections and the statements given by Hunter, Breschet, and Owen, on the thoracic rete of Cetacea in general. In his teleological deductions he attempted to bring the arterial retia mirabilia under headings by dividing them into two great classes, bilateral and axial. The axial he further subdivided into terminal and mediate, and each of these again into com- plete and incomplete. In commenting on the axial system he 313 remarked that their. probable function was threefold, in some eases to supply a large amount of blood to parts, in others to avoid injury from compression of the vessels, and, in many instances, to check the sudden pressure on nerve centres. In considering the bilateral, after stating that it was found only in Cetacea, he proceeded to give not only Breschet’s view of its function as a diverticulum for the storing up of oxygenated blood to be supplied to the circulation during the suspension of respiration, but also the more commonly accepted theory that it is a diverticulum protective against over pressure of blood in the circulation. He inclined to believe that its function em- braced both theories. After noticing some deductions derived - from the peculiarities in the origin, size, and relations of the constituent vessels of the rete, he concluded with a tabular view of the vessels of the thoracic rete, and of the divisions he had proposed for the arrangement of the various forms of retia mirabilia. The PRESIDENT, in inviting remarks, enquired what became of the outgoing currents of blood from the rete mirabile, as to how it was distributed. Dr Foster spoke in praise of the paper, and regretted that the specimen was not in a more favourable state for examina- tion ; he doubted whether the vasa could be used as reservoirs for oxygen, for the blood at the temperature of the body would speedily oxidate itself with its own oxygen. Mr Trotrer did not quite see the force of Mr Foster’s objection, for this process was the function of the oxygen. Dr WILSON said the blood returned by the same way as it eame. He thought that the storing oxygenated blood was not the sole function of the rete; there must, in his opinion, be some other function, for the seal did not present a rete. What this function was could not yet be settled. New Fellow elected: C. Taytor, M.A., St John’s College. 24 314 May 26, 1873. The VICE-PRESIDENT (PROFESSOR CAYLEY) in the Chair. On curves of the fourth degree. By F. W. Newman (communicated by Mr J. Stuarr). The curves spoken of in this communication were classified according to their symmetry and the number of their axes. Prof. CAYLEY said that he considered that the best method of classifying quartic curves was according to the quartic cones of which they were sections. Quartic cones might be divided into singular and non-singular forms; and non-singular forms might be considered to belong to the same species if they could be transformed continuously into one another without passing through a singular form, Pl PRINTED October 20, 1873. The PRESIDENT (PROFEssoR Humpury) in the Chair. Communications were made to the Society : (1) On the Mechanical Means for obtaining the real roots of Algebraical Equations. By J.C. W. Exus. The general equation A 2" + Ao" + &e.+ A,_ 2+ A"=0 having been converted (by putting cos @ or aaa for x) into B,+ B,_, cos + B,_, cos 26... + B, cos nO =0, various methods were shewn and illustrated by means of models for finding the values of 0. I. Wheels whose diameters were as 1 : 2:3, &c., were con- nected together by cogs or straps. Long arms were fixed diametrically on their faces, which carried sliding weights. The values of @ corresponding to positions of equilibrium were read off on a dial and gave the roots of the second equation. II. Another method was by hingeing a number of rods together of variable lengths and causing them to revolve through angles in the ratio of 1: 2:38, &c., round a fixed point in the first rod. This was effected by means of an arrangement of fixed and moveable pullies, when a pencil in the last rod passed through a fixed line determined by the constant in the given equation. The angle revolved through by the first rod was a root of the equation. In the discussion that followed, Mr GLAIsHER, of Trinity College, went briefly into the history of this class of mechanical 25—2 318 inventions, stating that he had met with descriptions of several in the Philosophical Transactions of the Royal Society, and elsewhere. He then put some questions to Mr Ellis respecting the degree of accuracy possible to be attained by such a machine. Professor MAXWELL described a machine which he had seen at the meeting of the British Association this year at Bradford for shewing the extent and action of the tides. . (2) Graphic representation by aid of a series of Hyper- bolas of some Economic Problems having reference to Monopolies. By Mr A, MarsHatt. The price at which a given amount of any commodity can be disposed of in any market is determined by the cireumstances of the buyers. If this amount be measured along Oz and this price along Oy, there is thus determined a value of y corre- sponding to each value of x; and the locus of the points so obtained may be called the demand curve: let its equation be y= (a). So if y be the price at which an amount a of the commodity can be produced for the market (a, y) is found, the locus of which may be called the supply curve : let its equation be y=f (x). This method of expressing the problem of value has been known certainly for 85 years: an intersection of the two curves has been explained as giving the ‘average price” about which Adam Smith proved that the “market price” will oscillate. But it has not been pointed out that, under some circumstances, there may be more than one point of intersection, and that Adam Smith’s arguments apply only to the cireum- stances of every alternate point. Only at every alternate point of intersection can the exchange value remain in stable equili- brium : at the other points it is in unstable equilibrium, 319 If an individual has the monopoly of the supply of the commodity in the market, his immediate interest will, of course, lead him to determine z so that x{F(x)—f(a)} shall be a maximum. Let the curve y= /'(x)—f(x) be traced, whether by direct inductions or otherwise, on a paper on which are already lithographed a series of rectangular hyperbolas having Ox and Oy for asymptotes. It will then be obvious by inspec- tion for which of two amounts that the monopolist may throw upon the market—or, which is the same thing, for which of two prices that he may demand—he will obtain the greatest total nett profit. Many striking results can thus be obtained in cases in which the curves cut one another more than once. This mode of representation of the problem of monopolies is elastic, and lends itself to the treatment of some complex hypotheses. Specially important results will present themselves, if the assumption be introduced that the monopolist is willing to undergo some abatement of his claims, when, by so doing, he ean confer great benefit on the consumers. (3) A Machine for constructing a series of Rectangular Hyperbolas with the same Asymptotes. By Mr H, H. Cunyyename. This machine was intended for the purpose indicated in the last paper. 320 ANNUAL.GENERAL MEETING, October 27, 1873. The PRESIDENT (PROFESSOR HuMPHRY) in the Chair. The following officers were elected : President. Professor C. C. BABINGTON. Vice-Presidents. Professor LIVEING. Professor MAXWELL. Mr PALEY. Treasurer. Dr CAMPION. Secretaries. Mr J. W. CLARK. Mr TROTTER. Mr J: BATTERIDGE PEARSON. New Members of the Council. Professor HUMPHRY. Professor CAYLEY. Professor HUGHES. Mr ELLIs. Mr GooDMAN. Mr §. S. Lewis. 321 PRoFEsSOR HUMPHRY made a communication on certain depressions in the parietal bones of the skull of an Orang and in Man. He showed the skull of an Orang which had been lately presented to the Anatomical Museum by Mr Vores of Caius College, in which these depressions exist. They look as if the bone had been indented on either side of the sagittal suture by the pressure of the finger, the surface being quite smooth and the edges of the depressions bevelled. There was no corre- sponding alteration in the contour of the interior, the bones being simply thinned at the part. He had not met with a similar abnormity in any other instance of an animal, but had seen it a few times in man, and showed two skulls from the Museum in which it was present, the outer table of the skull ‘being depressed in a considerable area of each parietal bone, and the skull at the parts being quite thin. The remaining bone-structure was healthy, and there was no reason to attribute the condition to disease of any kind or to accident. The appearance and the symmetrical position of the depressions were against both these suppositions. Neither did it seem possible to account for the depressions by any kind of pressure that was likely to occur. Professor Humphry thought they were probably due to a deficiency in the early formative pro- cesses in consequence of which the bone had not been produced of proper thickness at these parts, but he could not in the least explain why such deficiency should occur. A paper was read by C. Yuus, B.A., late of St John’s, Fellow of Magdalen College, Oxford, “On the Mechanism of opening and closing the Eustachian Tube.” In the first part of this paper the arguments in favour of the Eustachian tube being normally closed were reviewed, in consequence of the contrary view having been again revived by Dr Cleland, and some new ones added. The chief point brought forward, however, was the undoubted voluntary power possessed by Mr Yule over his 322 own Eustachian tube, by which he was able to open and close it at pleasure. When the tube is thus opened, he described the noises made in singing, breathing, &c., as being much intensified, and during loud singing quite unbearable. In order to complete the reasoning logically, the sensations heard in the ordinary ear - with a closed Eustachian tube were compared with those heard in the other ear when the tube was kept patent by means of a catheter adapted to the purpose; and in the latter case the modification of hearing was exactly the same as when the tube was voluntarily opened. The second part of the paper was devoted to examining the mechanism of the tube. During the opening of the tube the following points were observed. The ‘soft palate was unchanged in position and form, and hung flaccid ; this was important, showing that the tensor and levator palati did not participate in the action; the tongue was not raised, but the only observable change was the approximation of the posterior pillars of the fauces. The reason of this is as follows: The Eustachian tube presents at its inner margin a bluff mass of cartilage which ordinarily occludes the tube; to this is attached the tendon of the salpingo-pharyngeus; below the latter muscle is attached to the palato-pharyngeus, which above arches over to meet its fellow of the other side. At rest the direction of the salpingo-pharyngeus is such as to press the mass of cartilage into the tube, but when the palato-pharyngei contract, the insertions of the salpingo-pharyngei are carried inward and a new direction given to these muscles, such that when they contract they tend to draw the lobe of cartilage out of the lumen. As the posterior pillars of the pharynx are chiefly made up of the palato-pharyngei, this explains their approxima- tion when an effort is made to open the tube. The clicking sound heard at the commencement of the act of swallowing was pointed out to be due to the separation of the walls of the Eustachian tube. 323 November 8, 1878. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. Mr F. J. Canpy, M.A., Professor of Mathematics, &e. in the University of Bombay, read a paper containing a description of a new Physiological alphabet, devised by himself, to represent the various consonant- and vowel-sounds of the human voice by a series of symbols formed so as to be analogous to the different positions assumed by the mouth, palate, &c. in ex- pressing them. By means of eleven consonant- and three vowel-forms, each admitting of several small modifications, Mr Candy stated his belief that he had included all possible sounds of which any language is susceptible: and illustrated his posi- tion by examples taken from the dialects of Hindostan. The paper was intended as a sequel to one read by him before the Society, on the same subject, May 25, 1857: an ana- lysis of which is given in the “ Proceedings” of that date. No discussion on the merits of the invention followed, it being thought that the system was in too crude a state to be of practical utility. November 17, 1873. The PrestDENT (PROFESSOR BaBINGTON) in the Chair. Mr SepLey TAYLOR read a paper “On a suspected forgery in the Vatican manuscript record of the trial of Galileo before the Inquisition.” The preamble of the sentence pronounced in 1633 contains an enumeration of the grounds on which the Inquisition based their verdict of guilty. The existence of unquestionable dis- crepancies in the face of this document points to a conflict of 324 evidence only superficially smoothed down—not brought to a definite issue—by the Tribunal. The most serious of these discrepancies relates to an inhibition which the Court asserted had been formally delivered to Galileo by the Commissary of the Holy Office in 1616. From a detailed comparison of Galileo's letters and published works with the contemporary records of the Inquisition, Mr Taylor argued that the documentary evidence on which the judges relied as establishing this point was a fabrication designed to insure the conviction of the accused. He pointed out in conclusion that, if this view be admitted, Galileo must be held entitled, even on the severest view of his legal obligations towards the ecclesiastical authorities, to an absolute acquittal on all the charges, After a few remarks from Mr Gorosep, the discussion closed. , , Professor SELWYN exhibited a combination of two hoops, united by a straight rod, on which the inner hoop moved, the rod carrying a ball in the centre: the whole being designed to represent the Sun, the orbits of the Earth and Venus, and the conditions under which a transit of Venus between the Earth and the Sun becomes possible. December. 1, 1873. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. On the Inequalities of the Earth's Surface viewed in connection with the secular cooling. By Mr O. F'IsHER. This paper assumes that the elevations and depressions out of which the inequalities of the earth’s surface have arisen, are 325 due to lateral pressure owing to the contraction of the heated interior and consequent wrinkling of the crust to accommodate it to the diminished nucleus. Let ABCD be a layer of rock of unit of width, length J, and depth & And suppose the abutments at AC and BD to approach each other through the space le, where e is a small Ax aos ee eek Fe et D ze a BE NM c 2 BOP BD ad @\ [FP NE 2 SP" Biase i » ill Ts A: fraction. Then the layer of rock in question would take some new form, as one of those given in the figure, or any other whatsoever possible. Call AB “The datum level.” Let a, a, &., be the areas formed by the upper curved line above AJ, and J, b, &c., the areas formed by the same line below AB. In like manner let a, 8 be similar areas for the lower datum level CD. Then the space included between the curved lines must be equal to AbCd = kl (1 + e). It is also evidently equal to ABCD +a+a+&e.+8+8 + &e. —b—b-—&e.—a—a-— Ke. or, denoting the sums of the quantities similarly situated by the symbol =, we get kl (1 + e) =kl+ & (a) —= (6) += (8) —-E (2). wkle=% (a) — = (6) +2 (8)-Z (a). (A). 326 Since the pressure is supposed to take place in a horizontal direction, it will not have any direct effect to raise the centre of gravity of the portion of the crust under consideration ; so that, if the layer in question rest upon a liquid substratum, we may expect some portions of the disturbed crust to dip into. the super-heated rocks. But in that case a corresponding volume of such subjacent rock must rise into the anticlinals. Hence, = (a) = = (6). And the equation becomes kle = & (a) — & (0). Extending the inquiry to any area of the surface of length / and width w, the equation becomes 2 klwe == (A) — 3 (B), where > (A) and & (B) are the volumes of the elevations above, and of the depressions below, the “ datum level.” The whole surface of the globe being next taken into account, the relation becomes, Area of the Globe x 2 ke == (A) —3 (B). It is important to understand what is meant by the “ datum level.” It is an imaginary surface, which occupies the position which the surface of the crust would occupy at the present time, if it had been perfectly compressible, so that no corruga- tions would have been formed in it by lateral compression. For it would in that case have become simply more dense, without being disturbed in position. The above relation is applicable to the earth’s surface, although that is not strictly regular in its general form, and may contain local elevations and depressions affecting its mean figure,—that is, its mean figure as uninfluenced by lateral compression. For these inequalities, though of small amount ‘as compared with the dimensions of the globe, may be large in 327 comparison with the quantities of which we have to take cognizance in this investigation. Its truth in no way depends upon the arrangement of the disturbed rocks, nor upon the time at which successive movements have taken place, nor upon the alternate elevations and depressions which have at different times affected any given region. It includes every effect of subsequent denudation, from whatever cause, and to whatever amount. In short, it is perfectly general, so long as it is strictly interpreted. But it does not take account of elevations or depressions of regions of the surface arising from unequal contraction in a radial direction, if their result should be to cause a defect of parallelism between the datum level and the surface of the ocean, to which all our measurements must be in practice referred. However, it does not necessarily follow that contractions in the radial direction will cause depressions in the ocean-bed accompanied with a corresponding increased depth of water. For instance, the defect from a true circular form in the equator affects the surface of the ocean, to which the measurements of geodesy are always referred, so that we do not get an additional mile depth of ocean at the extremity of the shorter radius. If the earth had cooled as a solid body, the outer layers at any epoch having attained their complete amount of contraction sooner than the interior, would have been too large to fit the interior after the cooling had proceeded further. They would therefore have become corrugated. But in this case the corru- gation would have necessarily taken place wholly in an upward direction ; and there could be no places where any portion of the surface could have become depressed below the datum level. Hence upon this hypothesis we may introduce into our datum-level equation the supposition that }(B)=0. And it becomes Area of the Globe x 2 ke= = (A). A little consideration will give the following geometrical relation : 328 The volume of the Sea above the datum level = the area of the whole surface of the globe x the depth of the datum level below the sea level—the volume of rock displacing water between those levels. Assuming then that the continents have been shaped out of the master elevations, and that the oceans indicate the positions of the master depressions, and that both are ultimately due to lateral pressure, an estimate of 2ke for the whole globe is obtained from the above relation upon the following data :— (1) The area of the ocean is 146 millions of square miles. (2) That of the land is 51 millions. (8) The mean depth of the ocean is three miles. (4) Its deepest parts are about four miles. (5) The mean height of the land is 900 feet (as shewn by Mr Carrick Moore)’. From these data, asa probable value, 2 ke = 9504 feet, which appears more likely to be too small than too large. The meaning of this in plain language is, that if all the inequalities of the earth’s surface were levelled down, they would form a coating 9504 feet thick over the whole globe above the datum level; the datum level being such a surface as has been already defined. Having thus obtained a value for the thickness of the coat- ing which all the inequalities of the earth’s surface would form, if levelled down, a measure of the same thing is sought on physical grounds. For this purpose Sir W. Thomson’s paper “On the Secular Cooling of the Earth,” is used as a basis to work. from’. From Mr R. Mallet’s late investigations on the contraction of slag from an iron furnace’, a probable coefficient of contraction for melted rock is deduced, viz. 0°0000217 for 1° Fahr.; and with this is obtained a value for 2 ke, or the thickness 1 Nature, 1872, Vol. v. p. 479. 2 Edin. Trans. 1862; and Natural Philosophy, p. 711. 3 Royal Soc. Trans. 1873, 329 of the coating above defined. Sir W. Thomson’s investigation proceeds upon the supposition, founded upon Bischoff’s experi- ments upon the contraction of melted rocks in cooling, that, if the earth, or an outer coating of it, were once in a molten state, then, as soon as a crust began to form, it would break up and sink, and thus the whole would be reduced to the tempera- ture of incipient solidification before it could be permanently crusted over. From the time of such incipient solidification it has gone on cooling, subject to the laws of cooling of a solid. He then proves that upon this supposition the temperature would increase from the surface downwards, at first at a nearly uniform rate, but at a greater depth much more slowly, until at a certain point such a temperature would be arrived at, as would be about sufficient to induce fusion under the pressure existing at that depth. Now the rate at which the tempera- ture first begins to increase is known to be about 1° Fahr. for 51 feet. Sir W. Thomson has determined, by observation on the rocks at Edinburgh, that their conductivity on an average is 400. With these data he proves that if, for the sake of illus- tration, the temperature at which the crust began to solidify be taken at 7000° Fahr., then the time since such solidification commenced will have been about one hundred millions of years, and that at about 100 miles below the surface the melting temperature would be reached. Proceeding upon these assumptions, with the coefficient of contraction for rock above mentioned, the value of 2 ke is calculated, or the thickness of the coating which all the eleva- tions would form if they were levelled down, and it is found to come out less than 800 feet. Still further, if instead of 7000° Fahr. 4000° is assumed to be the temperature for melting rock, which seems to be justified by Mr Mallet’s experiments, then the value of 2ke, or the thickness of the coating referred to, would be less than 150 feet. In the latter case the time since solidification commenced would be about thirty-three millions of years, 330 If we compare the values thus found upon two different suppositions respecting the temperature of melting rock (one of them being extravagantly large) with the value for the same measurement as determined by estimating the actually existing inequalities of the earth’s surface, we cannot but be struck with the immense discrepancy between them, the latter being from 12 to 66 times as large as the former. The author is con- sequently led to doubt the necessity for accepting Sir W. Thomson’s restrictions upon the manner in which the earth has come into its present state, especially since it seems now generally admitted that Bischoff’s results concerning the con- traction of melted rock cannot be relied upon. This was ,pointed out in 1868 by Mr David Forbes, and quite recently by Mr Mallet, who has determined the contraction in passing from a molten to a solid state to be scarcely 6 per cent., instead of 25 per cent., as stated by Bischoff. Probably, therefore, when we take into account the intermediate condition of viscosity, we need not assume the breaking up and sinking of a crust formed over a molten globe. This view is supported by what Mr Scrope tells us about a lava stream remaining liquid, and even more or less in motion in its central and lower portion for years’. Indeed, Sir W. Thomson is careful not to exclude as impossible “the case of a liquid globe gradually solidifying from without inwards, in consequence of heat ‘conducted through the solid crust to a cold external medium.” If this has been what has happened, there may have been a much larger nucleus inclosed within the crust in early times than we have at present, and thus the corrugations formed would have been larger. And a great portion of that nucleus consisting of superheated rocks in a state of igneo-aqueous fusion, much of the water may have escaped in steam during the frequent volcanic outbursts of pristine ages, so that a large portion, at any rate, of the oceans now above the crust may have 1 Volcanos, 2nd ed. p, 84. ae ene ae Qe 331 been originally confined beneath it; and thus a much greater amount of contraction may have taken place than mere cooling” would account for. It is obvious that this reasoning will apply equally well to the case of a solid globe originally covered with a sufficiently deep layer of molten rock, which is the condition supposed by Sir W. Thomson to be the most probable, a view strongly supported by Dr Sterry Hunt’, and more in consonance with the rigidity considered requisite to obviate the production of internal tides. But at the same time it is to be remarked, that a highly fluid original condition of the interior may have lasted long after mountains commenced to be formed, and yet its condition need not continue such at the present time. February 2, 1874. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. Mr F. A. PALEY gave a summary of a paper intended to shew that Thucydides must have been mistaken in describing what was really the city-wall of the Plateans, with its battle- ments and towers, as a temporary wall erected in three months by the besiegers. The paper contended that the Spartan army had got possession of and manned the city-wall, wishing to reduce the Platzans to the necessity of capitulating; and for this a political reason was given. Doubts were thrown on the account of a double wall and double moat, since the researches of modern travellers, which were quoted, did not bear out the statement, and no traces of either existed, though the ruins of the city-walls still remain in great part. It was shewn that ancient Greek cities had precisely such walls as Thucydides describes ; and his veracity in the account was impugned, on the supposition 1 American Journal of Science, Vol. v. p. 264. 26 332 that he sacrificed strict truth for the purpose of writing a ro- mantic and sensational story. Mr J. B. PEARson, of Emmanuel College, read a short paper on Eur. Pheen. 1115—1118, intended to establish its probable genuineness. He pointed out that the legend of Argus was an old and well-known one, and argued that the grammatical diffi- culties occurring in the passage were not insuperable. Admitting that the poet was desirous to introduce an elaborate and some- what novel scene out of the legend of Thebes, he suggested that anything uncouth or extravagant in the passage might well be ascribed to poetic licence. Mr Pearson also stated that the authority of the MSS. and Scholiasts was unanimous in recog- nizing it, as is not always the case with passages intrinsically questionable ; and that it was allowed by some, though not all, the best editors, especially Porson, who here dissents from the opinion of Valckenaer whom he generally follows. February 16, 1874. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. - (1) On the geometrical representation of Cauchy's theorems of Root-limitation. By Professor Cayuey. There is contained in Cauchy’s Memoir “Calcul des Indices des Fonctions,” Jour. de [ Ec. Polyt. t. xv. (1837) a fundamental theorem, which, though including a well-known theorem in regard to the imaginary roots of a numerical equation, seems itself to have been almost lost sight of. In the general theorem (say Cauchy’s two-curve theorem) we have in a plane two curves P=0, Q=0, and the real intersections of these two curves, or say the “roots,” are divided into two sets according as the Jacobian d,P.d,Q—d,Q.d,P 383 is positive or negative; say these are the Jacobian-positive and the Jacobian-negative roots, and the question is to determine for the roots within a given contour or circuit, the difference of the numbers of the roots belonging to the two sets respectively. | In the particular theorem (say Cauchy’s rhizic theorem) P and Q are the real part and the coefficient of 7 in the imaginary part of a function of «+7y with in general imaginary coefficients (or what is the same thing, we have P+iQ=f (x+ty) + ib (a + ty), where f, ¢ are real functions of a +7y): the roots of necessity are of the same class: and the question is to determine the number of roots within a given circuit. In each case the required number is theoretically given by the same rule, viz. considering the fraction z it is the excess Q of the number of times that the fraction changes from + to— over the number of times that it changes from —to +, as the point (#, y) travels round the circuit, attending only to the changes which take place on a passage through a point for which P is=0. In the case where the circuit is a polygon, and most easily when it is a rectangle, the sides of which are parallel to the two axes respectively, the excess in question can be actually determined by means of an application of Sturm’s theorem successively to each side of the polygon, or rectangle. In the present memoir I reproduce the whole theory, pre- senting it under a completely geometrical form, viz. I establish between the two sets of roots the distinction of right- and left-handed: and (availing myself of a notion due to Prof, Sylvester) I give a geometrical form to the theoretic rule, making it depend on the “intercalation” of the intersections of the two curves with the circuit: I also complete the Sturmian process in regard to the sides of the rectangle: the memoir 26—2 334 contains further researches in regard to the curves in the case of the particular theorem, or say, as to the rhezice curves P=0,Q=0. A communication was also read by Professor Cayley (2) On Peaucillier’s Parallel Motion. (3) On some models of Peaucillier’s and other Parallel Motions. By Mr Enus. Do } ic! DK is a fixed guide upon which A slides. ABC is a rod moveable round a pin at A and bisected in B. BD is a rod whose length is 44C and moveable round pins at B and D. It is manifest that C will trace out a straight line CDC’ per- pendicular to KD, Supposing CAD (= C’AD) when greatest to equal 45°, the space described by C to space described by A ::2: /2-—l,oras5 : 1 nearly. This method may therefore often be employed with advantage to reduce friction instead of employing a guide for C. The friction may be again reduced almost to any extent by the following arrangement. The point A instead of sliding is attached by a pin to the a ia ia ila 2 - ma 335 extremity of the rod AF. FSL is a fixed guide at right angles to AD upon which the extremity F of AF slides. ZL >! ae a . S —)2—D | £# is the middle point of AF, and ZS is a rod equal in length to AF, and moveable round pins at ES. The motion of F may be made as small as we please by increasing the length of AF. Hence we may approximate as closely as we like to a case of linkwork where the friction is entirely reduced to that round pivots. For example, if C were attached to a piston-rod whose travel was 10”, the travel of A would be 2", and the travel of -F would be less than ;.”, if the rod AF equalled AC. We might moreover do away with the sliding of F by making it the extremity of a third rod, and so on. Looking back at fig. I. we see that C would describe a straight line very nearly if KD instead of a straight line were the arc of a very large circle; and this reasoning may have originally have suggested to Watt his parallel motion. 336 Co WK Thus if A instead of sliding on a guide be attached by a pin to the extremity of a long link AK, whereof the other extremity K is moveable round a fixed pin k, A will describe a small portion of the arc of a large circle, and therefore move approximately in a straight line. This is in fact a case of Watt’s parallel motion. The fixed point % might have been above the line 4D. It is to be observed that CO will describe an approximate straight line for a considerable space, if A for a short space describes an approximate straight line. Hence we have only to make A move in a straight line for a short space by any means we can: for instance, by means of Watt’s parallel motion. The above remarks will explain the rationale of the follow- ing model in linkwork in the late Prof. Willis’ collection. ACCA’ is a parallelogram hinged at A, C, C’, A’; B bisects — a ee 337 AC and B, A’0’; BD, B'D,, cach equal in length to 44C, are rods moveable about the fixed pins DD’, TV, VW, wy are equal rods connected by pins at V and W. VWiéis attached to 4A’ by a pin through its middle point R. Tand Y¥ are pins fixed in the framework. JR therefore by a Watt's motion describes an approximate straight line per- pendicular to 44’. Now AJ’ is parallel to BB, and therefore to DD’, which is a fixed straight line. Hence AA’ moves parallel to itself. .. A, R, A’ describe parts of straight lines perpendicular to AA’. Hence by what precedes CC’ moves in a fixed straight line, 338 Peaucillier’s motion. BECD is a rhombus formed of equal rods jointed by pins at B, D, C, #. It is jointed by pins at Band C to two equal rods AB, AC, which are moveable about a fixed pin at A. It is manifest that the straight line A # makes equal angles with ZB and EC, and will therefore if produced pass through D. It is also manifest that AD x AE= AM’ — EM’? = AC? ~ EC’ =a constant, Hence whatever curve is traced out by the point Z, say p= F(6), the curve traced out by D will be 57 FO) when p=AE and p= AD. Hence, if # is made to describe a circle passing through A, D will describe a straight line. Z is made to describe this circle by means of a bridle-rod KH (= KA) moveable round a fixed pin at K. March 2, 1874. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. On the Relation of Geometrical Optics to other parts of Mathematics and Physics. By Prof. CrerK MAxwELL. ) The study of geometrical optics may be made more in- teresting to the mathematician by treating the relation between 339 the object and the image by the methods used in the geometry of homographic figures. The whole theory of images formed by simple or compound instruments when aberration is not considered is thus reduced to simple proportion, and this is found very convenient in the practical work of arranging lenses for an experiment, in order to produce a given effect. _As a preparation for physical optics the same elementary problems may be treated by Hamilton’s method of the Charac- teristic Function. This function expresses, in terms of the coordinates of two points, the time taken by light in travelling from the one to the other, or more accurately the distance through which light would travel in a vacuum during this time, which we may call the reduced path of the light between the two points. The relation between this reduced path and the quantity which occurs in Cotes’ celebrated but little known theorem, is called by Dr Smith the “apparent distance.” The relations between the “apparent distance” and the posi- tions of the foci conjugate to the two points, the principal foci and the principal focal lengths, were explained; and the general form of the characteristic function for a narrow pencil in the plane of zr was shewn to be Va Vit wrt Hs" ; 3 1 My (r, ial a,) a, + B, (r, = 9) x, za CA + fof) @,%, + &e. : (7,— %)(%2— @) — SS ; where 7,, 7, are measured from the instrument in opposite directions along the axis of the pencil in the media y,, p,, respectively, and ,, x, are perpendicular to the axis. a,, 4, are the values of r,, r,, for the principal foci, and Ff, fe, the principal focal lengths, and fu, = /,,. If oe. Ree ae Si T,—4, 0 ? ; the last term of V assumes the form 0° and an infinite number 340 of possible paths exist between the points (#,, r,), and Cr r,), which are therefore conjugate foci, Differentiating V with respect to x, and #, we obtain MVei1_ 1 fit thn dx, da, ee or. 3H a)(t— a,) File D is the quantity in Cotes’ Theorems which Dr Smith calls the Apparent Distance, or the distance at which the object must be placed that it may subtend the same angle as when viewed through the instrument. + &e., We have also dD dD Sis Gp. =a,— 1%, Cae hk aeat March 16, 1874. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. Communications were made to the Society : (1) On the use of the term Endothelium. By Dr Micuaru Foster. In this paper it was shown that the term “endothelium” has been recently introduced into histology: and the use of it has rapidly become common if not general. The speedy acceptance of a new term may in many cases, though not all, be taken as an indication that something of the kind was wanted: and the already frequent use of “endothelium,” both by Continental and English Histologists, would seem to shew the need of some other phrase besides “epithelium.” Nevertheless there are cogent reasons why the new term should not be allowed to take farther root. In the first place, its etymology is of the most grotesque kind. This is of course an objection of secondary value, but has still some weight. When a term has once come into daily 341 use with a well-defined meaning attached to it, it does not much matter what its etymology is, or how it is spelt, except on historical grounds. Many terms become so altered in their meanings, before they finally acquire a permanent application, that the chief interest in their etymology is confined to the light it throws on the ideas of the man who first introduced them. This is the chief reason why new terms should be etymologically correct, in order that future inquirers may read back through them into the minds of earlier observers. When a word is etymologically pure nonsense, this is apt to become impossible. Such is the case with “ endothelium.” It appears to have been first introduced by His to designate the kind of epithelium (“unachte Epithelien”) which is found lining the vascular, lymphatic, and serous cavities of the body, in contradistinction to the real epithelium of mucous mem- ‘branes (see Die Haute, &c. &c. Akad. Programm. Basel, 1865). “Sei es, dass man sie als undchte Epithelien den dchten gegen- iiber stellt, sei es dass man sie Endothelien nennt, um mit dem Wort ihre Bezeichnung zu den innern Korperflaichen auszu- driicken.” Endothelium is here contrasted with epithelium, so that the latter may be considered as the “thelium” of free surfaces (whether invaginated or not), and the former as the thelium of internal closed spaces, “thelium” being apparently taken to mean “a layer,” or “layers of cells.” Now what is the derivation of “epithelium”? Dr Sharpey gives the following account: he says, in a letter, “epithelium”, or rather “epithelida”, and especially “epithelia” (1st decl.), was first introduced by F. Ruysch. In describing a preparation of the face of a child finely injected, he refers to the cuticle over the red part of the lip (prolabium), and says, “I cannot call this ‘epidermis’, seeing that the subjacent tissue is not skin, but a different substance covered with sensitive papilla, which are finely injected red.” He then goes on to say that as 342 the cuticle lies on papilla, he will call it epithelhida or epithelia, from ézt and Od) “papille”, or “mammilla”, and he adds that for the same reason he calls the inside coating of the cheeks by the same name. (The original may be found, F. Ruysch, Thesaurus Anatomicus, 11. No. xxiii, p. 16, “Nulla subest, &c. &c. papillarum”); and again vi. No. cxv. p. 49, he says...“ Anterior pars prolabii anterioris—epithelia adhue est obducta...”. From this it is evident that “epithelia” (changed in course of time into epithelium, just as platina becomes platinum) means ‘that which covers or is upon a papilla’, and con- sequently “endothelium” means that which is inside a papilla. The extension of the phrase epithelium to the cellular covering of such parts of the corium as are destitute of papille may be easily allowed, but it seems a daring violation of propriety to apply the phrase “within the papilla” to the cells coating sur- faces of which one great characteristic is that they are devoid of papille! There seems to be something attractive about “thelium” that tempts writers to make use of it. Already “endothelium” has given rise to “ectothelium”, and probably “thelium” will soon become a kind of histological maid-of-all-work, with as many prefixes as there are kinds of cells. In ‘the second place, there are objections to the use of the term endothelium not etymological in their nature. Without considering the peculiar views of His on the connective tissues of the body, it still seems desirable to have some distinctive term to denote the epithelium which is formed out of the elements of the middle of the three layers of the germ (the mesoblast of ~ Mr Huxley and myself), the word epithelium being reserved for the nether layer (or hypoblast). If so the word endothelium cannot be employed with this meaning, for it would then in- clude structures still called epithelium, and differing in no essential characters from the epithelium derived directly from the hypoblast. 343 The cells lining the Wolffian duct and its derivative the ureter, with their branches, would then come under the heading endothelium. Whatever be the first formation of the Wolffian duct, whether by the central solution of a solid ridge, or by an infolding of the lining of the pleuro-peritoneal cavity, it is lined by cells which are clearly mesoblastic in origin, not hypoblastic, nor, as was once suggested, epiblastic. . The case of Miiller’s duct is still more clear. This undoubt- edly arises by an infolding of the lining of the pleuro-peritoneal cavity. Its epithelium is distinctly mesoblastic in origin. The germinal epithelium which gives rise to the ovaries is also essentially mesoblastic. If the word endothelium, then, be taken to denote an epi- thelium derived from the mesoblast, it must be extended to include the epithelium of the Wolffian and Miillerian ducts, and of the parts which are formed ultimately out of these structures. But if these be included, the phrase loses all its practical utility. If they are excluded, all the little meaning it ever had, vanishes. It may be urged that we need a word to denote the epithe- lium which is found in the vascular and lymphatic spaces, There does not however appear to be sufficient reason why the same term should be applied to the whole of this epithelium. As we have seen, its common mesoblastic origin will not justify this. From a structural point of view, three distinct varieties may be recognized in it, viz. the spindle-shaped cells of the blood-vessels and larger lymphatic vessels, the sinuous cells of the commencing lymphatics, and the polygonal cells of the large serous cavities. The fact that the epithelium of the peri- toneum is continuous with that of the lymphatics, affords no argument at all for classing them together. We find continuity everywhere, The epidermis is continuous with the alimentary epithelium, and with the urinary and generative epithelium ; and the generative epithelium is in turn continuous with the 344 peritoneal epithelium. In short, there is no reason why the cells spoken of as forming endothelium should have a common title, distinct from the general term epithelium. The introduction of the new term is really a step backwards from, instead of an advance beyond, the old classification given in Quain’s Elements of Anatomy, where epithelium is divided either physiologically into epidermic, mucous, glandular, vascu- lar, serous, &¢., &c., or structurally, into columnar, spheroidal, ciliated, tesselated, squamous, &c., &. Some such nomencla- ture as this satisfies all requirements, either morphological or physiological. The chief morphological importance, as far as our knowledge goes, attaches itself to the question, from which of the three primary layers, epiblast, hypoblast, or mesoblast, any given epithelium is derived; for physiological purposes, all we need is some system of phrases which shall clearly indicate the individual characters and the arrangement of any group of cells; and these requirements are met by the phrases enume- rated above. We do perhaps want easy terms denoting whether the epithelium in any spot consists of several layers, or of one pronounced layer only; monoderic may be proposed for the latter, polyderic for the former case. Epithelium itself would only mean cells lining a cavity or coating a free surface. (2) On some Problems on the Physiology of Nutrition, and the methods of solving them. By Dr MicHarn Foster. (3) On an Experiment of Galileo. By Mr Sepizy TAYLOR. Mr Sedley Taylor drew attention to an observation made by Galileo, and described by him in the first of his Dialoghi delle nuove scienze’. Galileo says that while scraping a brass 1 Vol. x111. pp. 104, 105, of the Florentine edition of Galileo’s complete works. — 345 plate with a chisel in order to remove some spots, he noticed that the passage of the instrument across the plate occasionally produced a powerful and distinct musical note, and that, when this happened, a long row of fine equidistant striations was left on its surface. These marks were closer together when the sound was acute than when it was grave. Having pro- duced by the above means two notes which differed by an exact Fifth, Galileo measured the distances between their re- spective striz, and found that three of those corresponding to the upper note occupied precisely the same space as two cor- responding to the lower. He hence inferred that the numbers of vibrations executed in the same time by any two notes forming this interval are in the ratio of 3:2; a conclusion which had previously been only conjectured from results ob- tained by the monochord. Galileo remarks further that the same principle applies to the case of any interval. Mr Taylor exhibited a brass plate with rows of strie upon it obtained by screwing the plate into a lathe, and, while it was rotating, holding the edge of a chisel against it in such a@ way as to produce a musical sound. The markings were in some cases extremely fine and regular. April 27, 1874. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. Communications were made to the Society: | (1) On the use of the “ Ligamentum Teres” of the hip- joint. By Mr Savory, F.R.S. This paper discussed the proper use of the “ligamentum teres,” which, though variously stated, has not, it was main- tained, been correctly given. The statement that the liga- ment is vertical and tight, when the person is erect, has been 346 challenged: but the author was satisfied ofits accuracy. It could be demonstrated by removing the bottom of the acetabulum with the trephine. The ligament is moderately tight when a person stands evenly upon both legs. It is tighter when the femur is slightly flexed, as it usually is. But when resting upon one leg, inasmuch as the pelvis is then raised on that side, which of course affects the ligament in the same way as adduction of the femur would do, then the ligament becomes extremely tense. In other words, it becomes tightest when the hip-joint has to sustain the greatest weight. When there- fore the pelvis is borne down upon the femur, or when the femur is forced upwards—that is when the pressure would be greatest between the upper part of the acetabulum and the opposite surface of the head of the femur—it is put directly on the stretch. More precisely, its great purpose is to prevent undue pressure between the upper portion of the acetabulum, just within the margin, and the corresponding part of the head of the femur. But for this ligament such undue pressure must inevitably occur. Suppose the ligamentum teres absent and the person standing upright, owing to the obliquity of the acetabulum and the head of the femur, pressure between the two could not be equally, or nearly equally, diffused over their opposing surfaces, but it would be concentrated on a spot in the upper part of the socket through which a line drawn down the body, through the joint into the leg, would pass. When the thigh is straight, when the femur is in a line with the body, as when .one stands upright, then is the ligamentum teres in the -same line too, and consequently any force which drives the femur and pelvis together must tell at once upon the ligament, and be directly checked by it. Owing therefore to the shape and obliquity of the hip-joint, and the weight of the body, the ligamentum teres is necessary to prevent con- centration of pressure at a particular point above it; The — line through which the weight or force acts between the upper 347 part of the acetabulum and the opposed surface of the head of the femur, forms, with the line of weight of force which passes through the ligamentum teres, an obtuse angle: and the resultant of these forces is in a line which passes through the long axis of the head of the femur. When the person is erect, the body partly hangs upon the ligamentum teres. This, he submitted, is the prime function of the ligamentum teres. Other purposes he did not deny, but would maintain that they only occasionally come into play, and are altogether subordinate to this one, which is especially called into action whenever the weight of the body is thrown upon one leg. He supported his view by reference to comparative anatomy, remarking that it is present when the acetabulum looks outward, and the head of the femur is inclined inward; in other words, when the hip-joint is placed obliquely, so that there would be otherwise undue pressure at a particular part; ‘and that it is absent in those animals in whom, although it is an instrument of regression, the posterior extremity does but little in supporting the weight of the body; e.g. seals, and the ourang-outang. In a discussion which followed, Prof: Humphry disputed, and Mr Savory still maintained, the tension of the ligament referred to in the paper. (2) On a Clepsydra. By Mr Etuis, (3) A Model shewing the mechanical arrangement of the Joints in the Limb of a Lobster. By Mr Enis. 27 348 May 11, 1874. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. The following communication was made to the Society : On the Bearing of the Distribution of the Portio Dura upon the Morphology of the Skull. By T. H. Hux- LEY, Sec, B.S, Tn the first place, the distribution of the seventh nerve or portio dura in Man was compared with that of the same nerve in the amphibia ; and it was shewn that, while the proper facial nerve, with the chorda tympani, corresponds in all essential respects with the posterior division of the seventh nerve in the Frog and other amphibia, the nervus petrosus superficialis major or vidian nerve, with its palatine branches and the nerve of Cotunnius, answers to the anterior division of the seventh, or so-called “palatine” nerve of the Frog. A branch which, in the Urodela, connects the portio dura with the Gasserian ganglion, appears to be the homologue of the nervus petrosus superficialis minor. The tympano-Eustachian passage, in both Man and the Frog, is included between the two main divisions of the portio dura.—The distribution of the seventh nerve in the Ray was next described. Its two divisions were shewn to have the same relation to the spiracle as they have to the tympano-Eustachian passage in the higher vertebrata. The anterior division, however, differed from that of the Frog and that of Man, in possessing no branch comparable to the nerve of Cotunnius. The place of this nerve appears to be taken by a large ‘palato-nasal’ branch of the fifth (as Bonadorff has already suggested), and it was suggested that the Cotunnian branches | of the palatine nerves in the Frog and in Man really belong to the Trigeminal. The distribution of the portio dura was then 349 compared with that of the glossopharyngeal and that of the branchial branches of the vagus, and the conclusion was drawn, that the portio dura is the nerve of the mandibulo-hyoid cleft (commonly called the first visceral cleft), and is distributed to the (morphologically) anterior and posterior walls of that cleft. As a corollary from this conclusion, it followed that the pte- rygoid arcade does not represent an independent visceral arch, but is a dependence of the mandibular arch, as Gegenbaur has already argued upon other grounds. . It was further shewn that the distribution of the second and third divisions of the fifth nerve is such as accords with the view that they represent the posterior division of the nerve of the trabeculo-mandibular cleft. The anterior division of that nerve was sought in the palato-nasal branch of the trigeminal—while the first division of the latter nerve appears to be the nerve of the (morphologically) anterior face of the trabecula. The sixth, third, and fourth nerves were regarded as special branches of the nerves of the mandibulo- hyoid, and trabeculo-mandibular clefts respectively, developed in relation with the special muscles of the eye. The author finally endeavoured to shew that the results thus obtained by the thorough investigation and comparison of the distribution of the cranial nerves were in entire accordance with those obtained by the study of development, and that the apparent anomalies in the distribution of the fifth and of the seventh nerves in the higher vertebrata are easily explained by the metamorphoses of the trabecular and mandibular and hyoidean arches in these animals. Professor Humpury expressed his thanks and the thanks of those present to Prof. Huxley for the careful and lucid account which he had given of a difficult piece of anatomy, and for the interesting and morphological inferences which he had deduced from them, and also for the illustration he had given of the fact that the dullest, most troublesome anatomical details may be brightened, and so rendered easy by an insight into their true 27—2 350 meaning. This was really the way to study anatomy, viz. to regard the various facts in connection with other facts, and so as the bases of scientific deductions. Prof. Humphry was glad to hear the nerves thus made the exponent of cranial morpho- logy, for he had attempted the same thing many years ago in a paper read at the British Association at Leeds, when he endea- voured to shew that the fore limb was not, as supposed by Prof. Owen, an appendage to the skull, but formed independently from it. He then shewed, from a consideration of the distribution of the cranial nerves, that the hyoid and not the scapula is the visceral arch of the occipital, and that the mandibular, the pterygo-maxillary and the ethmo-vomerine arches are the re- spective visceral arches of the post-sphenoidal, the pre-sphenoidal, and the ethmoidal parts of the skull. This view he believed to be in the main correct. The nerves respectively supplied to them are the ninth and the three divisions of the fifth. Each of the latter is very closely confined to its particular visceral arch, sending a special nerve to each bone of its arch, or nearly so, whereas the seventh pair of nerves is more promiscuous in its distribution, being supplied to muscles disposed upon all the four visceral arches, and having connecting links with the spinal nerves of those arches. It was to the orderly disposition of these connective links in relation to the visceral arches that Professor Huxley had now called their attention. Professor Humphry remarked that the communication between different nerves, — which is a means of establishing the harmonious action of the several muscles supplied by them, was effected in three ways, First, by junction of their terminal branches. This is most common in the lower animals. Secondly, by plexuses near their origin from the brain and spinal cord, which are found, to some extent, in the lower animals, but which are more numerous in the higher animals. Thirdly, by means of ganglia. This last, which may be regarded as the most perfect method, is almost confined to the higher animals. Accordingly the communicating 351 branches between the seventh and fifth, which formed the _ subject of the author’s paper, pass to Meckel’s ganglion, the otic _ ganglion, and the submaxillary ganglion in Mammals; whereas in Batrachians they do not pass to these ganglions, but their junctions are effected among the terminal ramifications of the nerves. He could not agree with Prof. Huxley that the fore part of the skull was not, like the hinder part, composed of ver- tebral elements. It was transversely segmented after the manner of the rest of the skeleton, and these segments are vertebre, whether the notochord exists at the part or not; and whether the segmentation takes place early or not, that is, in the cartilage or in the osseous nuclei developed in the cartilage, makes little difference. Sooner or later, in the higher animals at any rate, the segmentation occurs. The foremost elements derived from the trabecule had been designated as ribs by Prof. Huxley: and if they are so, they are components of those segments, of which the vertebree form the mesial elements. He could not quite accept the view of the homologies of the mandi- bular arch which had been given; but time failed to discuss these questions more fully. He concluded by again thanking Professor Huxley for this interesting communication. May 25, 1874. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. Mr PEARSON read a paper on some meridian observations of the Sun taken by him with a prism-circle and an artificial horizon, at Taormina in Sicily, on April Ist last. They were taken with the view of determining the latitude of the place. The watch was set to Greenwich time, but about 8m. 9s. ‘slow. 352 The observations were taken before and after noon, and, reduced, were as follows: Time. hb, ak S. Altitude. (1) 10 46 0 56° 41‘ 29". (2) 10 49380 ° 56° .48' 29%. (St 10 ae. 0 56° 44 59”, (4) 10 58 50 56° 44. 19%, (5) Fe ater Gare 56° 43° 9”, (6) 11 3 40 56° 41° 29”, (7) ll 12 40 56° 30° 19%. A comparative examination of these suggested 10h. 54m. 35s. as the probable time of apparent noon. (The method for ascertaining the time of noon given by Godfray, Ast. art. 150, was not available at the time the observations were worked out: by it, the times of app. noon on the mean of two separate observations are as follows: (1) and (2) 10h. 54m. Os., (3) and (4) 10h. 54m. 40s,, (4) and (5) 10h. 54m. 25s, (5) and (6) 10h. 54m, 15s....average 10h. 54m. 20s.) The method employed to find the latitude is that given in Raper’s “Navigation.” Tables are given containing a given series of numbers varying for all latitudes and declinations. The number in this particular case (Lat. 37°. 50. 50° N., Sun’s Decl. 4°. 34°. 6 N.) is 458. This is added to the sin. sq. of the time elapsing between the time of observation and that of apparent noon: the result is the log. of the sin. of the difference between the altitude of the sun at the time of observation and its meridian altitude. Employing this method we get these results: For ob- servation (1) Lat. 37°. 49'. 0“; for (2) 49°. 19; (8) 48°. 56"; (4) 49°. OY; (5) 49. 0"; (6) 48°. 53"; (7) 48. 43"; average 37°. 48°. 58*. 7, The methods given in Norie’s “Navigation,” and in Godfray’s Ast., art. 149, produce very nearly the same results; e.g. obs. ————— 353 (6) worked out (a) on Norie’s method gives Lat. 37°. 49°. 10", (b) on Godfray’s method gives Lat. 37°. 48°. 54", The hour-angle of the Sun, by obs. (7), on the theory (1) that the lat. of the chart is correct, and that the instru- ments were in adjustment, is 16m. 45s.—an error of 1m. 20s. ; (2) that there was an error of about 2‘ in one of the two, is 17m. 57s.—an error of 8s., which tends to prove the existence of some such error. The present Admiralty chart, issued as newly corrected, 1873, gives 37°. 50‘. 50" as the lat. of the spot where the obser- vations were taken. At first sight, this would seem to prove an error of nearly 2°, either in the instrument or the artificial horizon, as levelled at the time. But (1) Admiral Smyth, in his survey of Sicily, carried out in 1813—15, places Taormina in 37°. 48°. 40°. (2) He states in his book on the subject that he was remarkably well supplied with sextants and other surveying instruments. (3) The long. and lat. of the principal points on the coast given by him often agree with those now given in the charts. (4) His estimate of the height of Etna, obtained by triangulation from a base on the sea, viz. 10,874 ft., is very nearly accurate, that recently obtained by levelling being 10,840 ft. (5) The lat, of Taormina, as now ~ given in the charts, agrees exactly with, and may possibly be borrowed from, that given by Baron von Waltershausen, in his survey of Mt. Etna and its environs, executed from 1840 to 1850, and may therefore be not perfectly accurate. On these grounds it was argued that the latitude of Taor- mina, as given by Adm. Smyth, and (approximately) by this set of observations, may perhaps be more nearly accurate than that given in the present charts: at any rate they shew that it is perfectly feasible for a person, with simple instruments and merely arithmetical processes, to determine his latitude, in any part of the globe, with reasonable accuracy. Dr Campion said, that from practical experience, he was 354 aware of the uncertainty attending any set of observations made by a single individual. Prof. CAYLEY suggested a diagram, similar to those given in meteorological reports, indicating, by a curved line and dots, the altitude of the Sun at different times: from which the meridian altitude and its time might be approximately inferred. The Secretary then read a paper On the Temperature of the Earth in times anterior to the Eocene period. By Mr Ronrs. Mr Rours stated that geological evidence seemed to point to a warm and equable climate over a great part of the earth in preeocene days. He thought it probable that this high tem- perature was due to the internal heat of the earth, and that the amount of heat radiated by the sun and received by the earth may have been less than it is now—the solar atmosphere ob- structing radiation more than at present, although the energy and mean temperature of the sun were greater in early times. He referred the first great glacial period to a time when the internal heat of the earth was diminishing, and the solar ra- diation had not reached its present amount. Mr Sos said that there was clear evidence of ice-action at various epochs long anterior to the Eocene period, so that Mr Rohrs’ theory of a long period of uninterrupted high tem- perature was geologically untenable. 355 October 19, 1874. PROFESSOR CAYLEY in the Chair. On some Ice-hummocks in the Gorner Glacier. By Mr Trorrter. The speaker described some remarkable water-holes asso- ciated with hummocks of ice on parts of the Gorner Glacier which he had observed in 1863 and in 1874. The water-holes were oval in shape with their longer ones parallel and pointing east and west, the hummocks were on the south side of the water-holes. The direction was independent of that of the veined structure, and the whole was obviously a meridian phenomenon, but several points as to the origin of the hum- mocks and the shape of the water-holes were very obscure. An examination of them earlier in the season would pro- bably throw some light on their origin. 306 ANNUAL GENERAL MEETING, October 26, 1874. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. The following were elected officers for the ensuing year: President. Professor CHARLES C. Baprneton, F.R.S. Vice-~Presidents. Professor MAXWELL. Professor MILLER. Mr Munro. Treasurer. Dr CAMPION. Secretaries. Mr J. W. CLARK. Mr TROTTER. Mr PEARSON. New Members of Council. Professor LIVEING. Mr JACKSON. Mr GLAISHER. 357 On a nearly complete Skeleton of the Bos Primigenius found in Burwell Fen. By Mr J. W. Crarx. Mr J. W. Ciark exhibited and made some remarks upon a skeleton of the great extinct Ox (Bos primigenius). The bones had been found together in Burwell Fen early in the spring of 1874; and there could be no doubt that they be- longed to the same animal. The parts wanting are the right hind-leg, one lumbar vertebra, a few terminal vertebre of the tail, and a few bones of the carpus, tarsus and toes. The skeleton, after the bones had been properly treated with _ gelatine, had been mounted and placed in the Museum of Comparative Anatomy. It is the first skeleton found in England in a sufficiently perfect state to allow of its being articulated. Mr Clark briefly reeapitulated the history of the species, shewing, from the passages out of chronicles and other contemporary records collected by Mr Boyd Dawkins, that it had subsisted in a living state on the continent of Europe down to a much later date than had been supposed previous to his researches, November 2, 1874. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. On some further Observations with a Prism-circle. By Mr Pearson. This paper was intended as a sequel to one read on May 25th, the two being intended to establish by practical examples the facility with which a traveller may establish his latitude and longitude in any part of the globe. 358 The following observations were taken Sept. 5th a.m. in Lat. 52° 12'10"N.; Long. 30s. (7'30")E. The time given is local mean time, App. dist. of Sun from Moon. h. m 8. Sun’s Alt. i, eS. 7 42 27 21° 25° 40° 7 47 36 62° 54 40° 7 53 44 os st (Se) 62° 53° 40" The observation of the Sun at 7h. 42m. 27s. giving the local time obtained from the Philosophical Society’s clock accurately within 7s., the levelling of the art. horizon, and so the altitude of the sun may be taken as nearly correct. The above observations reduced give At 7h. 48m. 19s. (L. M. T.) alt. of Sun’s centre 21° 59° 42", True app. dist. of centres of Sun and Moon 63° 25° 54", The Moon being in 28° N. Dec. and having just passed the meridian (at 7h. 44m. 11s.), the altitude was obtained by calculating the Reduction to the meridian, instead of by the more difficult method of observation, the change in alt. in 4m. 8s. being only about 44". This gave the true altitude of the Moon’s centre 65° 47°50". Of course the altitude of the Moon could only have been approximately ascertained in this way, had not the longitude of the place of observation been accurately known. These data were worked out on four methods: Ist, on the plan given in Woodhouse’s Astronomy, which is believed to be based more or less nearly on the formula originally devised by the Chevalier Borda ; 2nd, on that given in the Introduction to Shortrede’s Logarithmic Tables, mainly identical with the first method; 3rd and 4th, on the two methods given in Arts. 700, 701 of Raper’s Navigation. No. 1 gives the longitude of the place of observation 1s. W. of Greenwich; error 31s. No. 2 gives it 3s. E. of Greenwich ; error 27s. No. 3 makes it 12s. E. of Greenwich ; error 18s. ae | Sh ———_s SL ee ee 359 No. 4, 3s. W. of Greenwich; error 33s. A second set of observations gave at 8h. 5m. 13s. local time, true alt. of Moon 65° 29° 20"; true alt. of Sun 24° 10° 0"; true app. dist. of their centres (mean of three observations) 63° 20° 27". | These data, when computed on the Ist method mentioned above, give the long. of the place 1m. 35s. E.; error Im. 5s: on the 4th method Im. 29s. E.; error 59s. It was thought unnecessary to make the calculations again on the 2nd method, because its form is nearly the same as that of the first; on the 3rd, because its result in the previous case differed considerably from the other three. The bar. and ther. were not taken into account, as neither of them were far from the point at which they do not affect the refraction. The index error of the instrument was too small to be ascertained by any one but a very good observer. The inaccuracy of the second set of observations might be due to a haze coming on at the time, and to the increasing brightness of the Sun. Several extracts were also given from the voyages of Cook and Krusenstern, bearing on the accuracy with which it was found practicable to use this method of fixing the long. at the end of the last and the beginning of the present century. For example, the long. of Santa Cruz (Teneriffe), where Cook is described (Voyage, Vol. 1.) as having met Borda in August, 1776, was given by the latter as 18° 35° 30" W. of Paris, Cook making it by his timepiece 16° 31' 0", by two sets of lunars 16° 30°45" W. of Greenwich. The true long. as given by the English Ad- miralty Chart (1873) being 16° 14° 56° W. of Greenwich, or 18° 35° 6" W. of Paris. 360 Nov. 16, 1874. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. Linear partial Differential Equations, and their Germ- integrals. By S. Earnsnaw. This paper will be found printed at length in the “Trans- actions of the Society.” [ Abstract. ] Long before it was discovered that A gut. way 2 ywa x Vx wT, is an integral of the equation = bala =u, it had been known that every linear partial differential equation with constant coef- ficients, whatever be the number of its variables, is susceptible of an integral of the form = Cemminyt., It thus appeared that the above equation admits of two integrals of essentially different types. The same was found to be the case with the equation Be = 53 of which both dx A PY w= Cemotm’y x2 and u= 7 ; ate are found to be integrals; and they are also of essentially different types as integrals. This discrepancy of types created in me a desire to ascertain the significance and true origin of each, and their mutual dependence if any existed. 361 I have worked out the case of two independent variables, but the method I have adopted is applicable to an equation of any number of independents. It is shewn that in every in- tegral certain constants, called germs, exist, or can be arbitrarily introduced into it if they do not already exist there; and by means of these a germ-integral can be found; and from this a series of sub-integrals; and the sum of these is the general integral of the proposed equation. This method depends for its success on the circumstance that the differential equation from which the sub-integrals are obtained contains fewer independent variables than the pro- posed equation. Hence when the equation to be integrated is the general differential equation of the second order- of two independent variables and constant coefficients, the sub-inte- grals are to be found from a differential equation of the second order of one independent variable ; and by its integration that of the general equation of the second order is accomplished. The same method is shewn to be successful in the inte- gration of certain other equations where the coefficients are not constants, but functions of the independent variables. Nov. 30, 1874. Proressor Humpury in the Chair. (1) On Lopsided Generation, or Right-handedness. By W. Arysut Hours, M.D. Cantab. This paper was read by Professor Humphry in the absence of Dr Hollis. The antiquity and universality of the preferential use of the right hand was shewn by reference to the Biblical and other 362 records, and to Egyptian, Assyrian and other monuments, as ‘well as to various members of the Semitic and Aryan groups of languages. All modern nations, with one or two questionable exceptions, are right-handed, and have words to signify “left- handed” corresponding with the French “gauche” and the Italian “mancino.” It appears to be a peculiarity of the human race, even the Apes using the right and left limbs indiscrimi- nately, and is associated with the higher and more elaborate muscular actions of the limbs in Man; and there being no other structural difference between Man and the lower animals to account for it, the cause of the peculiarity must be sought in that part of the system, viz. the brain, in which he excels other animals. The left side of the brain was stated to be the larger in Man; and it, through decussation of the nerve-fibres, pre- sides over the right side of the body, and seems from recent observations also to preside over the complex and delicate mus- ‘cular actions upon which articulation depends. The preponder- ance of the left side of the brain—the lopsidedness of the organ—thus engendered by the preferential use of the right hand, by the movements in speech and by much of subsidiary brain-work directly associated with speech, is not without its evil; and instances were adduced, including those of Johnson and Swift, in which the left side of the brain had suffered and paralysis of the right side of the body had been induced, appa- rently, as a consequence of this overwork. The inference was drawn that such result might have been avoided had a more equal duty been required of the two sides of the brain by a more equal use of the two limbs; and in these days of high pressure it is of especial importance to attend to such points, and by more equal education of the two sides of the body, to lead to a fairer distribution of work between the cerebral hemispheres. | Professor PAGET thought that more evidence should be ad- duced respecting the greater size of the left hemisphere of the 363 brain, and as to which side of the brain is likely to be affected when Aphasia occurs in left-handed persons. _ Mr ANNINGSON questioned whether lopsidedness was really a part of right-handedness, forasmuch as the left hand is em- ployed not only as a helpmate to the right, but for many purposes in which the right hand is less efficacious. Mr Carver thought the observations in the nursery shewed that right-handedness was acquired rather than innate; children having commonly a propensity to use the left hand, which it required some difficulty to counteract. Professor Humpury stated that the paper, which was one of much learning and interest, as well as suggestive, had been consigned to him for publication in the next number of the Journal of Anatomy and Physiology. In reply to various questions which had been asked, he said he believed an advan- tage gained by preferential use of the right hand, was a greater aptness and precision of movement requisite for delicate mani- pulations than could have been attained had both limbs been equally employed. Left-handed persons, being prevented by social custom from concentrating their attention on the left hand and being compelled to give a frequent preference to the right, are at some disadvantage in this respect. He could see no anatomical reason for the preference of the right limb, the slight advantage in circulation to the right arm through the innominate artery and vein applying, in nearly equal degree, to the right side of the brain. He agreed with Mr Carver that right-handedness was much a matter of education, and followed from the multifarious single-handed offices which are associated with the higher mental endowments, 28 364 (2) On the Peritoneum in Man and other Vertebrates. Dr WILSON made a communication on the disposition of the peritoneum im Man and other vertebrata. He gave a brief account from his own dissections, of the anatomy of the peri- toneum, and more particularly of its omental sac in Man and many Mammals, Reptiles, Amphibians and Fishes. He shewed that in many of these the omental sac is divided into two parts—a gastro-hepatic and a gastro-colic part—by a constric- tion corresponding with the upper border of the stomach. This he first observed in the dissection of a Narwhal, and had found it marked to a variable extent in Man, most evident in a young Hippopotamus, distinct in the Rat and in the human fetus about the 3rd month. In Reptiles and Amphibians the omen- tum does not extend below the level of the stomach. There is therefore only a more or less complete representative of the gastro-hepatic part of the omental pouch of Man. One or more of the hepatic lobes usually project into the gastro-hepatie part of the sac, In Man it is the lobulus Spigelei. He described the relation of the spleen to the omental pouch, and stated that his observations were, on the whole, in accordance with the old and commonly received view regarding the mode in which the colon is embraced by the two recurrent layers of the omentum which pass on to form the transverse meso-colon. Professor HUMPHRY remarked on the thorough manner in which Dr Wilson had investigated the anatomy of the omentum, which was of much interest with reference to the development of parts. The increasing size of the omental pouch in the higher animals and in Man must also be taken in connection with the recent investigations of Dr Klein respecting the rela- tions of the peritoneal cavity to the lymphatic system. 365 Feb. 8, 1875. The PRESIDENT (PROFESSOR BABINGTON) in the Chair. On the Centre of Motion of the Eye. By Pror, Cuzrx MAxwELL. | The series of positions which the eye assumes as it is rolled horizontally have been investigated by Donders (Donders and Doijer, Derde Jaarlijksch Verslag betr.-het Nederlandsch Gas- thuis voor Ooglijders. Utrecht, 1862), and recently by Mr J. L. Tupper (Proc. R. &, June 18, 1874). The chief difficulty in the investigation consists in fixing the head while the eyeball _moves. The only satisfactory method of obtaining a system of co-ordinates fixed with reference to the skull is that adopted by Helmholtz (Handbuch der Physiologischen Optik, p. 517), and described in his Croonian Lecture. A piece of wood, part of the upper surface of which is . covered with warm sealingwax, is placed between the teeth and bitten hard till the sealingwax sets and forms a cast of the upper teeth. By inserting the teeth into their proper holes in the sealingwax the piece of wood may at any time be placed in a determinate position relatively to the skull. By this device of Helmholtz the patient is relieved from the pressure of screws and clamps applied to the skin of his head, and he becomes free to move his head as he likes, pro- vided he keeps the piece of wood between his teeth. If we can now adjust another piece of wood so that it shall always have a determinate position with respéct to the eyeball, we may study the motion of the one piece of wood with respect to the other as the eye moves about. For this purpose a small mirror is fixed to a board, and a dot is marked on the mirror. If the eye, looking straight at the image of its own pupil in the mirror, sees the dot in the 28—2 366 centre of the pupil, the normal to the mirror through the dot is the visual axis of the eye—a déterminate line. A right-angled prism is fixed to the board near the eye in such a position that the eye sees the image of its own cornea in profile by reflexion, first at the prism, and then at the mir- ror. > May 17, ,, May 31, ,, Noy. 3, 1873. Mar. 16, 1874. Feb. 22, 1875. ” >? Honorary Members. Col. J. T. WALKER, R.E., F.RS. Prof. A.T. ANGsTRoM. Upsala, (since deceased). M. CHEVREUIL. Paris. M. Otto von StruveE. Pulkova. Prof. W. E. WEBER. GOTTINGEN. Dr T. ANDREW. Belfast. Dr F. C. DonpeErs. Utrecht. W. K. ParKker, F.R.S. Hunterian Professor. Sir W. R. Grove, M.A., F.RS. Prof. J. C. PogGenporF. Berlin. Fellows. Prof. CowELL, M.A., Corpus Christi College. OsBERT SALVIN, M.A., Trinity Hall. G. J. Romans, M.A., Caius College. Rev. A. Rose, M.A., Emmanuel College. F, H. Nrvitze, M.A., Sidney Sussex College. Rev. D. B. BAnHAM, M.A., Caius College. J. G. Ricuarpson, B.A., Trinity College. H. F. Banuam, M.A., St John’s College. D. Burasss, B.A., Corpus Christi College. G. CurystaL, B.A., Corpus Christi College. Associates. Mr A. DEcK. Mr H. BAXTER. © Dr ARMITSTEAD. Mr Bowes. 29 ee 5 6 Ww 1 ge Philosophical Society. ANNUAL GENERAL MEETING, Ocroper 25, 1875. OFFICERS, &c. ELECTED FOR THE ENSUING YEAR. President: Prof. MAXWELL. Vice-Presidents: Prof. MILuER. Mr Munro. Prof. C. C. BABINGTON. Treasurer: Dr CAMPION. Secretaries: Mr J. W. CLARK. Mr C. TROTTER. Mr J. B. PEARSON. New Members of Council. Prof. NEWTON. Mr Bonney. Mr H. M. Taytor. New FELLOWS ELECTED, OCTOBER TERM, 1875. Oct. 18. Prof. DEWAR. Nov. 1. A. M. MarsHAtt, B.A., St John’s College. F. M. Batrour, B.A., Trinity College. H. N. Martin, B.A., Christ’s College. ERRATA (No. XV.) P. 355, 1. 8, for “ones” read “axes.” 5 P. 376, last line, for “classification” read ‘ clarification.” a a ee ee eh ‘ October 18, 1875. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. The following communication was made to the Society: On some Fresh Observations of the Water-holes on the Gorner Glacier. By Mr Trorrer. The attention of the Society was called last year to certain water-holes on the Gorner Glacier associated with hummocks of ice on their southern edges (first observed by the speaker in 1863). (See Proceedings, Oct. 19, 1874.) The holes as they appeared at the latter part of the season were oval, with their longer axes pretty exactly east and west, the larger axis about double of the smaller, the depth on an average nearly double the longer axis, the longer axis varying from about 1‘ to 6‘ or 8 long. The holes as usual had gravel at the bottom, and had usually a hummock of ice at the southern side, the height of which was often nearly equal to the longer axis in the smaller holes, somewhat less in pro- portion in the larger ones. The larger axis was sometimes parallel, or nearly so, to the veined structure, sometimes cut it at a greater or less angle, so that the holes were clearly independent of the veined structure, and seemed to be clearly a meridian phenomenon. The speaker had the opportunity of observing these holes last summer, about the end of June, when they were much less perfectly formed. The surface of the glacier was covered in places with new snow, and in others the winter’s snow was imperfectly melted. Some of the holes, however, were fairly . 30—2 384 developed, though of course not so deep as they were later on in the year. Others were of much less regular form: some had the hummock on the south side well developed ; others had no perceptible hammock, but seemed to have more or less of a raised margin all round; others seemed to show a slight hum- mock on the north as well as the south side. The explanation of the phenomenon which was suggested was as follows :—A collection of gravel gives rise to an irregular or roughly circular shallow water-hole, the water being at first at a considerably lower level than the edge of the hole, formed in part of snow and soft ice. (1) Towards noon the sun’s rays are incident upon the surface at a small angle, a\comparatively small portion are — reflected, and the radiations which enter the water are for the most part absorbed before reaching the wall of the hole, and the resulting heat is carried to the bottom by the descending current of dense water, whose temperature has been raised above the freezing point. This melts ice at the bottom and deepens the hole. On the other hand, towards morning and evening the angle of incidence is larger, a larger proportion of the rays are reflected and strike upon any portions of the east and west boundaries of the hole which are above water. The rays which enter the water have a shorter path to traverse in water before reaching the wall, and therefore will reach and melt it in a greater proportion. The east and west walls will thus be more melted than the south wall, and the hole gradu- ally assumes its oval shape with its longer axis east and west. (2) The ice in the neighbourhood of the holes is for the most part of very rough and irregular surface, very pure, and with small bright crystalline faces inclined in all directions, Consequently a considerable portion of the rays incident on any portion of it, and there reflected, will strike another portion of the surface, so that the wasting of any portion of the surface is due not only to the rays primarily incident on that portion, 385 but also to those incident upon it after reflection at another place. On the other hand, rays falling upon the water will be either absorbed or reflected regularly, so as to pass clear of the ice, except possibly the actual wall of the hole. Hence the ice in the immediate neighbourhood of the hole will receive less radiation, and therefore be less melted than the rest of the surface, and there will be a tendency to the formation of a raised rim surrounding the water-hole. (3) As fast however as this rises, those portions of it which are on the north, east, and west sides of the hole will be melted, as they will at some part of the day receive the sun’s rays not only on their upper surface, but on the vertical face towards the hole. The east and west sides however will be most attacked, for reasons given above in (1). The final result will therefore be an oval hole with its major axis east and west, with a marked hummock of ice on its south side, and sometimes traces of one to the north. (4) The most serious difficulty in the way of this explanation seems to be in the local nature of the phenomenon. Why is it not produced wherever there is a level surface of glacier? The . ice where the phenomenon is conspicuous is of a peculiar soft nature, full of minute air-bubbles, which give it an unusually white appearance. Ablation probably takes place rapidly over ‘the surface, so that phenomena depending upon differential ablation are ‘conspicuous. The peculiar kind of irregularity of the surface would favour the action described in (2). [Communicated Nov. 15, 1875.] Since the notice of my paper read on Oct. 18th, ‘Further remarks on the water-holes of the Gorner Glacier, was pub- lished, my attention has been called to a passage in Agassiz’ ‘Nouvelles études sur les Glaciers,’ &c., Paris, 1847, p. 101—2, in which a similar phenomenon is described as having been observed by Dr Ferdinand Keller. The account of the phe- 386 nomenon observed by Dr Keller on the Aar Glacier so closely resembles that given by me of the water-holes on the Gorner Glacier, that there can be no doubt that they refer to the same phenomenon, and that therefore it was first noticed by Dr Keller, and described by him in 1847. I cannot however think that Dr Keller’s explanation of the phenomenon is satisfactory. He speaks of the holes as semi- circular with the are towards the north, and attributes the greater depth of the northern portion of the hole to the longer time for which the sun will fall upon the gravel on the north portion of the bottom. This does not explain the much more striking phenomenon, the east and west elongation of the hole. Moreover, the holes are so deep in the latter part of the summer that the gravel at the bottom must be in shade all day. No explanation is given of the accompanying hummocks of ice, which are spoken of as if they existed before the water-hole, whereas my observations of last summer make it clear that they are formed subsequently. I therefore still adhere to the explanation given in my paper, which seems to me to explain all the phenomena. October 25, 1875. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. The following communication was made to the Society: On Herwart ab Hohenburg’s Tabule Arithmetice tpocbahaperéws universales, Munich, 1610. By J. W. L. Guaisuer, F, R.S. The title more at length is “‘Tabule arithmetice mpoc- Oapaipecéws universales, quarum subsidio numerus quilibet, ex multiplicatione producendus, per solam additionem : et quotiens 387 quilibet, e divisione eliciendus, per solam subtractionem, sine tediosié & lubricé Multiplicationis, atque Divisionis opera- tione, etiam ab eo, qui Arithmetices non admodum sit gnarus, exacté, celeriter & nullo negotio invenitur. E museo Ioannis Georgii Herwart ab Hohenburg...Monachii Bavariarum...Anno Christi, M.DC.X.”, and the book is a very large and thick folio. It contains a multiplication table up to 1000 x 1000, the thousand multiples of any one number being given on the same page; and there is an introduction of seven pages, in which the use of the table in multiplying numbers containing more than three figures, and in the solution of spherical triangles, is explained. Very little information about the work is to be obtained from the mathematical bibliographers and historians. Heil- bronner (Hist. Math. 1742, p. 801) gives the title not quite correctly, and adds “Docet in his tabulis sine abaco multiplica- tionem atque divisionem perficere.” Kastner (Gesch. der Math. 1796—1800, t. iii. p. 8) quotes the title from Heilbronner and his remark, and adds that the latter could not have known Herwart’s method, or he would have described it. He remem- bers to have read somewhere that the book contained a number of tables of products, arranged by factors, like.a great multipli- cation table. Scheibel (Hinl. zur math. Buch. 1775, t. ii. p. 417) gives the title-page correctly, and explains the method of using the table when the number of figures in the multiplier or multiplicand exceeds three, and concludes with the remark, “So viel von diesem ungeheuren Folianten, den man bloss zur Curiositait und seiner Seltenheit wegen, in einer mathema- tischen Biichersammlung aufbewahret.” Montucla (Hist. des Math. t. ii. p. 13) gives a description of the mode of using the table, remarking that but for the invention of logarithms it might have been of use to calculators, supposing the labour of searching for the products in so large a folio not to be more fatiguing than the direct performance of the work. Murhard 388 (Bibl. Math. 1797—1804, t. ii. p. 199) gives the title correctly, and marks it with an asterisk to show that he has seen the work himself. Rogge (Bibl. Math. 1830, p. 142) merely has, “Hohenburg, Gregor (sic) Herwardt ab, tabulze arithmetic mpoc0ahaipecews universales, 1610.” Neither Weidler (Bzdl. Ast. 1755), Deschales (Cur. seu Mund. Math. 1690), Lalande (Bibl. Ast. 1803), nor Delambre (Hist. de l Ast. mod. 1821), mention the work; but there is a reference to it in Leslie’s Philosophy of Arithmetic (2nd edit. 1820, p. 246). In his article on tables in the English Cyclopedia (1863) De Morgan wrote, “The table goes up to 1000 x 1000, each page taking one multiplier complete. There are then a thousand odd pages, and as the paper is thick, the folio is almost unique in thickness. There is a short preface of seven pages, containing examples of application to spherical triangles. It is truly remarkable that while the difficulties of trigonometrical caleu- lation were stimulating the invention of logarithms, they were also giving rise to this the earliest work of extensively tabulated multiplication. Herwart passes for the author, but nothing indicates more than that the manuscript was found in his collection. The book is excessively rare, a copy sold by auction a few years ago was the only one we ever saw.’ Graesse (Trésor de livres rares, 1859—1867) says that by the book the use of logarithms was first spread in Germany, which is of course erroneous. Herwart was Chancellor of the Palatinate of Bavaria, and published several other works (the most complete list is in the Bodleian Catalogue), among which his “Ludovicus LV. imperator defensus (Munich, 1618—19)” is the best known. In the Biographie Universelle it is described as still useful for the history of Germany; and Scheibel speaks of Herwart as “der bertihmte Staatsmann und Geschichtschreiber.” While I was engaged in preparing the report of the British Association Committee on mathematical tables, I endeavoured ae 389 without success to find something beyond what is quoted above about the table; but the hope is there expressed that, consider- ing the attention so large a work must have received from ~ contemporary mathematicians, some information might still be gained with regard to the calculator of the table, his objects, &e. I recently found a correspondence of six letters between Herwart and Kepler, which took place at the end of 1608, with regard to the table, and which throws light upon these points. The letters are printed in Dr Frisch’s ‘Joannis Kepleri As- tronomi opera omnia’ (t. iv. pt. II. pp. 527—530, 1863). In the first letter, dated September 13, 1608, Herwart writes, “Ich hab bisher in Multiplicatione et Divisione sonder- bare geschriebene praxin gebraucht, dadurch ich den numerum ex quavis multiplicatione productum, per solam additionem, und den Quotienten ex divisione resultantem per solam sub- tractionem (absque tediosa multiplicationum et divisionum operatione) gefunden.” He states that J. Praetorius and others who have seen it recommend him to have it printed, and he adds that if he had not had this method, on account of his continual occupations and because he is not a good calculator, he should long ago have had to give up all mathematics that required calculation. He sends a specimen page of the table, the use of which he explains, and he prays Kepler to give him his opinion on the matter without delay. Kepler replies on October 18, 1608, and remarks that 1000 pages will make a large volume, which the computer will often not have at hand. He suggests that short precepts on the solution of triangles should be added, as Herwart’s table would often be preferable to the ‘ rpoo@adaipeors Vitichiana,’ which is too elaborate to be retained in the memory, confuses sines and their complements, &c. Besides, the reasons for the operations are hidden in work, “At si multiplicemus et dividamus simpli- citer, tunc videmus quid agamus; et possunt varietates trian- 390 gulorum talibus preeceptis comprehendi, que memoria retineri facile possunt.” Kepler then gives a synopsis of the sixteen cases of the solution of spherical right-angled triangles. Herwart writes on November 5, and says he had found that the triangles could be solved better by help of his table than by prosthaphzeresis, so that Kepler was quite right. He men- tions that he has not seen Vitichius, As it is usual to prefix a ‘splendid title’ to books, that they may sell, he suggests the following, “Nova, exacta, certa et omnium facillima ratio Arithmetices, per quam numerus ex multiplicatione productus sine operatione multiplicationis per solam additionem, et quotiens ex divisione resultans absque operosis ambagibus divisionis per solam subtractionem cujuscunque, etiam maxime summe, etiam ab eo qui arithmetices non admodum sit gnarus, citius quam ulla alia ratione invenitur.” He then prays Kepler to send soon his advice as to how the table should be entitled. Not receiving an answer on December 2, he writes a short letter to Kepler again asking for a reply, and suggesting that perhaps it was not prudent ‘so speciosum titulum tantille rei zu pre figiren.’ But in the mean time a letter from Kepler had been received, and, writing on December 5, Herwart ex- plains that he does not expect an answer to his last letter, and that he understands that Kepler has no objection to the title, but thinks it ought to be shortened. He cannot understand the meaning of Kepler’s advice, ‘Greeca compositio imploranda, sed exercito (sic),’ and asks for an explanation. In the last letter of the correspondence, dated December 12, 1608, Kepler explains that as the title seemed long, he had advised that it should be shortened by the composition of two Greek words as épucserredas, puroxivduvos, and as a suitable word did not occur to him he had suggested that some one practised in Greek should be consulted. But perhaps a good idea has occurred to himself. ZecayOera apiOunrixyn. “Nosti enim, XpEwv atroxoTras sic dici. Inest vocabulo et emphasis et pro- a 391 | prietas et similitudinis gratia, quia me Hercule novas tabulas introducis, et uno ictu liberas computatores debitis multiplicandi, et dividendi inextricabilibus.’ But he hopes that this title will - not offend Maginus on account of his ‘Tabula tetragonica’ [Venice, 1592]. At the end he adds the postscript, “Titulus igitur talis: LevcayOea, sive Nove Tabule, quibus Arithmetici debitis inextricabilibus multiplicandi et dividendi liberantur, ingenis, tempori, viribusque ratiocinantis consulitur.” It is thus proved that the table was printed from a manu- seript which Herwart used himself, and which very likely he had had made. The correspondence is of interest, as the table, regarded simply as a multiplication table, has never been surpassed in extent, and has only been equalled by Crelle’s Rechentafeln, first published in 1820, in two volumes, and which, now sold in one volume folio, is one of the best known and most used tables. Scheibel and others who have ridiculed Herwart’s project were only right in so far as the great size of the work renders it unmanageable, as the use of a multiplication table up to 1000 x 1000 has, in spite of loga- rithms, been found to be both practicable and convenient. Herwart’s work is very rare, but there are copies in the British Museum, the Bodleian Library, and the Graves Library at University College, London. It was this last (not quite perfect) copy which, through the kindness of Professor Henrici, I was enabled to exhibit to the meeting. With regard to the word prosthapheresis, it is well known that the prosthapheresis of the orbit was the angle subtended by the eccentricity at the planet, and De Morgan explained the use of the word on the title-page thus: “Prosthapheresis is a word compounded of prosthesis and apheresis, and means addition and subtraction. Astronomical corrections, sometimes additive and sometimes subtractive, were called prosthaphzre- ses. The constant necessity for multiplication in forming pro- portional parts for the corrections, gave rise to this table, which 392 therefore had the name of its application on the title-page.” But the prosthapheresis referred to seems most likely a method of solving spherical triangles, in which the product of two sines, or of asine and cosine, &c., is avoided by the use of formule, such as sina sin b= 4 {cos (a—b)—cos(a+b)}. This explains all Kepler’s allusions to prosthapheeresis, and as Herwart proposed as the chief use of his tables to solve spherical triangles by direct multiplication without previous transformation, as set forth in his introduction, it justifies completely the use of the word on the title-page. Wittich was for a short time an assistant of Tycho Brahe, and his method of prosthapheresis appears to have been a method of solving triangles so as to avoid multiplications by means of formule, such as that just written, but I hope to examine the matter more fully. Laplace (Jour. de [école polyt. Cah. xv. t. viii. 1809), referring to the same formula, sin asin b = $ {cos (a —b) — cos (a + d)}, remarks that “cette maniére ingénieuse de faire servir des tables des sinus 4 la multiplication des nombres, fut imaginée et employée un siécle environ avant l’invention des logarithmes” (see Brit. Ass. Tables Report, p. 23, 1873). November 1, 1875. PROFESSOR BABINGTON, VICE-PRESIDENT, in the Chair. The following communication was made to the Society: On Aristotle's notion of ‘ Right-Handedness’. By Mr Pzarson. After referring to the paper by Dr Hollis on this subject, communicated last year (Nov. 30, 1874), the speaker stated that he had been led by Aristotle’s great reputation to enquire 393 what his views on the subject might have been. Partly from a perusal of much that Aristotle has written on the subject, but mainly from the new Index by Prof. Bonitz, he gave a résumé of the passages bearing on the subject: these passages seemed to shew that Aristotle considered, (1) that the right hand or side was naturally the source or origin of motion, (2) that in nearly all living creatures capable of motion it is the better or stronger side, (3) but that while the heart is always the origin of vitality, it is in the human race only set towards the left side of the body; in all other living creatures it is in the centre of the body or trunk (év peo@ xeitac tod avayKaiov oa@paros). And though it may be a fair question how far Aristotle was misled by the preferential use of the right hand by the human race to attribute an excellence in the right side to the animal world, there can be no mistake about the dis- tinct language in which he does so. The passages referred to by Mr Pearson were as follows: Tlept Zeéwv Mopiwr, Ul. 2; 1. 3, 4, 5; Iv. the whole sec- tion, especially n. 8. Tlepi Zev ‘Ioropias, 1. 17; 1. 1. 17. Tlepi Zwwv Iopetas, XIX. _Nicom. Eth. v.'7 (10). *Magna Moral. 1. 34. Politics 1. (9) 12. * Problems V. 37; VI. 5; XXxXI. 12, 13. De Respir. 16. Plato, Legg. vit. 795. Macrobius vi. 14. Some observations were added about the terms in which Mr Lewes in his work on Aristotle (1864) criticizes some errors into which that writer has fallen in his works on Natural History, while it was admitted that he is probably right in con- sidering that there is no reference in Aristotle’s writings to the anatomical examination of any but animal subjects : though the 394 fact that Macrobius (vu. 14) ascribes to Erasistratus and Herophilus, two celebrated physicians of the succeeding gene- ration, the practice not only of dissection but of vivisection of human bodies, shews, if the story-is true, that public opinion could not have been quite unprepared for it. Mr Pearson also referred to a passage in the Hncyc. Britan., art. Comp. Anatomy (§§ 202—205), ed. 1810 (but not occurring in later editions), in which the preferential use of the right hand is discussed, and ascribed to a natural peculiarity in the form of the sub-clavian and carotid arteries on that side: and in which it is stated that a similar preference for the right side may be traced in some dogs if not in horses. The speaker said, however, that he would not answer for the existence of such a preference himself in those animals, nor in the lion and camel, to which Aristotle (and Pliny after him) especially ascribe it. He concluded by exhibiting a lobster; a kind of shell-fish of which the right claw is distinctly larger and stronger than the left, as is specially mentioned by Aristotle (of xapxwol...t)v SeElav éyovar ynrnv meiSo Kal loyuporépav...). In conclusion an opinion was expressed that the preferential use of the right arm might be due to a natural shrinking from the use of the side nearest the heart, or perhaps a natural wish to pro- tect it as being, to our own sensations at least, the seat of vitality. At any rate such a use must be accepted as a fact, whether the more strictly anatomical reasons given in the former paper were correct or not, and the difficulty of coming to a conclusive view on the subject was suggested as a reason why the point is so little discussed, at any rate in the more popular and simpler treatises on anatomy: nor had the speaker been successful in finding much information on another question: viz., the true position of the heart in animals, Aristotle’s view on this point being very decided, while the practice of dissecting them was evidently common in his time: but still he ventured to think that the situation of the centre of gravity of the human heart 395 to the left of the centre of the body had been a sufficient determining cause in favour of the preferential use of the right arm by our species. Mr NEVILLE GoopMAN said that while he had listened with pleasure to the elucidation of Aristotle’s ideas on the causes of right-handedness he thought these indicated that the inductive method was preferable to the speculative. These ideas shewed how possible it was for an ingenious man with the great reper- tory of nature before him to adduce many facts to support any theory once formed. No doubt there were many facts which would support the idea that the right side had a preferential motor function. In addition to those named, the whole order of Gasteropods might be quoted, in which in the vast majority of instances the opening of the generative organs, the vent and the respiratory chamber, were on the right side. In fact an ordinary snail or whelk exhibited the phenomenon of an ex- cessive development of the right side, which excessive develop- ment had the effect of thrusting that side continuously over the other so as to result in the dextral helix. There were, however, many exceptions to this rule, as the Funis Contrarius of the Red Crag and numbers of existing shells, Clausiliz, &c. Flat- fish (Pleurovectide), the only animals of the vertebrate type markedly asymmetrical in their organs of relation, were by no means constant in having the right side of stronger motor function than the other; the upper, coloured, and more convex and muscular side being in many cases on the left. In the sole the developed side is usually the right, in the turbot usually the left. With regard to the prior motion of the right side from a state of rest he had narrowly observed horses’ paces and thought there was no ground for this supposition. The leading leg in the horse’s canter became so purely from training. Both dogs and young horses constantly change the leading leg when running unrestrained. All the figures in Egyptian Art had 396 their left leg advanced in conformity with Aristotle’s remarks, but he had regarded this attitude as purely conventional. The heart of all mammals was of like asymmetrical position to that of man, and its position on the left side was more apparent than real; the butt end being directed to the right, and its apex or lower end to the left. The beating of the apex on the left side was caused by rhythmical distension of the aortic arch, which thus became periodically straightened, and in re- laxing the apex fell back on the left side of the thorax. It was quite possible and probable that, owing to the depression of the human thorax, (i.e.) its greater lateral than fore and aft diameter as compared with the compression of the same part in the lower animals, the apex of the heart might be thrust more to the left side, but he thought that the centre of gravity of the heart occupied a similar position in both cases. With regard to the cause of right-handedness, as it could not be due to external conditions, it seemed reasonable to attri- bute it to the asymmetry of the internal organs, but he thought that the stomach, whose cardiac end was on the left side and which when distended with food bulged to that side, had a greater claim for consideration. The greater proximity of the right hand to the source of arterial blood through the innom- inate trunk, was also worthy of consideration. The greatest argument against accounting for right-handedness by any of these methods was that while cases of reversal of the whole viscera were known the individuals thus characterized were not left-handed. Right-handedness was so universal throughout the human race, that he thought it could not be accounted for by early education, the customs of the world in all conventional matters being so various. If due neither to asymmetry of the organs of nutrition nor to education, it must be an inherited instinct, accidental in the sense of being due to a cause which has now no bearing on the species, Such a meaningless and persistent habit would go far to prove the unity of the human — — 397. race. It would be very interesting to observe whether the Quadrumana shewed any preference in the use of the right hand. He had observed the smaller monkeys, and concluded that they seized and wrought indifferently with either hand, but the larger anthropoid apes he had not observed. November 15, 1875. PROFESSOR BABINGTON, VICE-PRESIDENT, in the Chair. The following communications were made to the Society: (1) On the behaviour of Nucleus during Segmentation. By F. M. Barrour. The following observations were made upon the eggs of Scyllium and Pristiurus. Ata late stage of the segmentation of these eggs most of the segments contain nuclei, but in some of them there is to be seen in the place of the nucleus a peculiar body. This has the shape of two cones with their bases in apposition. In each cone a series of strie radiate from the apex to the base; and between the two is an irregular row of granules. From the apex of the cone there further diverge into the protoplasm of the cell a series of lines, The author regards these peculiar bodies as metamorphosed nuclei in the act of dividing. He points out that the simple division of the nucleus, as well as its complete disappearance, accompanied by the formation of two fresh nuclei, are well-authenticated modes of behaviour of the nucleus during cell-division. These two processes can only be connected on the supposition that in the second case the two fresh nuclei are formed from the matter of the old nucleus. The author considers that there exist in Selachians modes of behaviour of the nucleus intermediate 31 398° between the two extremes mentioned above, and points out? that in the peculiar striation of the body he described there are indications of the streaming out of its matter into the sur- rounding protoplasm ; while on the other hand it never com- pletely vanishes.. It therefore affords an instance where part of the matter of the nucleus divides and part streams out into the protoplasm of the cell to be again collected to assist in the formation of two fresh nuclei. The author further states that he has found other bodies intermediate between the cone-like bodies mentioned above and true nuclei; and regards these also as nuclei in the act of division, where’a still larger bulk of the protoplasm of the nucleus becomes divided and a smaller part rises with the surrounding protoplasm, | (2) On the effects of Upas Antiar on the Heart. By M. Fostzr, M.A., F.RB.S, | The author recording the movements both of the ventricle: and the auricles of a frog’s heart (Rana temporaria), within the body, by means of two delicate levers, observed in addition to the well-known phenomena of antiar poisoning, a marked slowing of the rythm in the later stages of the action of the poison. The prolongation of each systole was also distinctly marked, especially in the case of the auricles, which, much distended in consequence of the partial occlusion of the con- - tracted ventricle, caused the lever resting on them to make an enormous excursion at each systole. So long as any beat was capable of being recorded by the lever resting on the ventricle; the ventricular systole occurred in its proper sequence. Though the whole rythm often became irregular, the phases of each cardiac cycle remained regular. Repeating the experiment of Schmiedeberg (Beit. zu Anat. eee 399 ‘u. Phys.: also Festgabe, C. Ludwig Gewidmet,. p. 222), the author found that when the ventricle had apparently ceased to beat, forcible distension of its cavity with a normal solution brought back a temporary series of pulsations ; but this restora- tion is possible only within a narrow range of time, and as Schmiedeberg himself seems to admit, cannot be regarded as shewing that the poison’s chief action consists in preventing the normal muscular relaxation following upon each systole. Repeating Neufeld’s observation (Stud. Phys. Inst., Breslau, Ill. p. 97) the author found that strong solutions of potassium cyanide would sometimes restore the beat for a short time— but im this case also the phase at which this could be effected ‘was very transient and very frequently failed, and inasmuch as such solutions are capable of stimulating muscular tissue di- rectly, he was led to the conclusion that the restoration when obtained is not due to any relaxing action of the cyanide but to its chemical stimulation of the cardiac muscles. When the vagus is stimulated in the earlier stages of antiar poisoning, inhibition is obtained as usual, but is followed by ‘a somewhat lengthened period in which the beats are both more rapid and more forcible. When the antiar has produced such an effect on the heart that its beats are exceedingly feeble and hardly capable of being recorded, this secondary action of vagus stimulation becomes exceedingly marked, the pulsations during its con- tinuance being as forcible or even more forcible than normal, and at the same time rapid. Lastly, a stage of poisoning may be witnessed when the ventricle is apparently at rest (i.e. not pulsating at all as far as the eye can judge, though of course in the contracted state so characteristic of antiar), where stimulation of the vagus pro- duces no inhibition (for there is no beat to stop) but is followed by a lengthened series of often very vigorous and rapid pulsa- tions. The author could not satisfy himself that during the 400 stimulation of the vagus any relaxation of the contracted ventricle took place, but on this point he is not sure. These results at first sight seem identical with the phe- nomena observed by Schmiedeberg’ in nicotin poisoning (Lud- wig’s Arbeiten, 1870, p. 41), and explained by him as due to accelerator fibres in the vagus of the frog. The author is unable to accept this explanation: 1. Because both inhibition and secondary action fail when atropin is given with the antiar (the antiar otherwise acting as usual). 2. Because a similar secondary action may be seen in all cases of inhibition not only of the frog’s heart but also of the mammalian heart in which the accelerator fibres are supposed to run not in the vagus. 3. Because a similar secondary action may be seen in the snail’s heart after inhibition by direct application of the interrupted current, and in certain conditions of the snail’s heart may be witnessed when the inhibition cannot be de- tected. The author regards the secondary action as being what for want of more precise knowledge he would call “a reaction” following the direct action of the vagus. And considering that ‘antiar acts essentially, or at least primarily, on the muscular tissue of the heart, the peculiar prominence of this reaction in antiar poisoning may be taken as indicating on the one hand that the effects of antiar are especially favourable to this reac- tion, and on the other that the vagus nerve brings about inhibition by acting directly on the muscular tissue itself—a view which is supported by other facts, . | a é 401 November 29, 1875. Tae Presipent (Proressor CLERK MAXWELL) in the Chair. The following communication was made to the Society: On the temperatures observed in a deep boring at Speremberg near Berlin, as given in a report of a paper by Professor Mohr, of Bonn, in ‘ Nature’ of October 21, 1875. By Mr O. Fisuer. The greatest depth recorded is 3390 feet. The temperatures are given in Reaumur’s scale. The author shewed that the equation 251 10° in which v is the temperature, and x the depth, exactly repre- sents the temperature curve. This curve would give a maxi- mum temperature of 40°°7532 R., or 123°6947 Fah., ~ ata depth of 5171 feet. If there was no cause to disturb the temperature, it ought to conform to a straight line, given by the above equation altered by omitting the term in a*. Conse- quently a cause was sought which would change such a straight - line to the parabolic form. The first cause examined was a change in the conductivity of the strata depending on the depth, and it was found that a law, which would make the conductivity vary inversely as the distance of any point above the level of greatest temperature, would account for the observed facts. But it was argued that such a law was entirely improb- able. The next cause examined was the effect of the descent of water through the strata, and the author believes that this cir- cumstance will account for the observed temperatures. a? + 0:012982 2 + 71817, "Aegan 402 : It was remarked that the results of this investigation make it appear, that the true law of underground temperature would be better obtained from borings of moderate than of very great -depth, because the disturbance of the temperature curve from the rectilinear form is greater the further we descend. April 19, 18757. THE PRESIDENT (PROFESSOR BABINGTON) in the Chair. un the Physvological Action of Jaborandi. By Mr J. N. Lanatey. [Abstract.] The preparations used are _ (1) The alcoholic extract of the crushed Jaborandi leaves. (2) The glycerine solution of this extract evaporated to dryness. , The results of experiments point to there being more than one active principle. Injected subcutaneously there is one striking difference in the action of Jaborandi on the Frog and the Rat. In the former it causes convulsive movements with occasional tetanic spasms, in the latter it acts as a narcotic. Death in one case is probably more immediately caused by the stopping of the heart’s beat; in the other by paralysis of the respiratory centre. In the Frog convulsive movements are noticeable if any part of the spinal cord be left intact, but not otherwise. Reflex action is greatly depressed. Jaborandi has little or no effect on nerves, their endings in striated muscle, or on striated muscle itself. If an arterial blood-pressure tracing be taken of a Mammal, and Jaborandi injected into a vein, it causes | (1) : L, c 2 2L*)m whence at={(5-1) = - ate. In the experiments 5007 w = 2500 rey. in 30” = 7 ad, a=7'5 in. =*1905 metres nearly, J =424 gramme-metres. 426 Substituting these values we find "ee \(¢- 1) 101077 _ = mm C be ae r The units being gramme, metre, second, centigrade. C ae The experimental value of a for aluminium has not been found, but Prof. Maxwell has pointed out to me that Edlund has determined it for some other metals. So, though we are unable to calculate the amount of the heating for an aluminium disk, we may get an idea of its magnitude by taking some other metal. If we take silver we shall have 3 ~~ © =1-0203, Cc c= ‘057, w = "000057, » = 8481 x 10° grammes per square metre, m=10'4 x mass of cubic metre of water, = 10°4 x 10° grammes. Substituting these we find for a silver disk under the con- ditions of the experiment, Fall of temperature on expanding = *4° C. As the conductivity of silver is very high, the heat absorbed during rotation would be rapidly conducted inwards, and there- fore after 40” the disk will almost have risen to the surround- ing temperature, and consequently-on stopping the disk we should get the whole effect of ‘4°C. showing itself. If we take into consideration the effect of the conduction at the end of 40”, the surface would be ‘04° C. below the surrounding space, and therefore on stopping the disk the temperature observed ought to be 36°C. The order of magnitude of the effect is thus the same as ~ Saree 427 that of the experiments, and the explanation proposed seems sufficient to account for all the results. If the heating is due to friction, the amount was shown to be about ‘0006 Ibs. per square feet, and that this would produce an alteration in the length of the day of not less than 006” in a century. (2) On the Equilibrium of Heterogeneous Substances, By Prof. Cuerx Maxwett. The thermodynamical problem of the equilibrium of hetero- geneous substances was first attacked by Kirchhoff in 1855, who studied the properties of mixtures of sulphuric acid with water, and the density of the vapour in equilibrium with the mixture. His method has recently been adopted by C. Neumann in his Vorlesungen tiber die mechanische Theorie der Warme (Leipzig, 1875). Neither of these writers, however, make use of two of the most valuable concepts in Thermodynamics, namely, the intrinsic energy and the eutropy of the substance. It is probably for this reason that their methods do not readily give an explanation of those states of equilibrium which are stable in themselves, but which the contact of certain sub- stances may render unstable. I therefore wish to point out to the Society the methods adopted by Professor J. Willard Gibbs of Yale College, published in the Transactions of the Academy of Sciences of Connecticut, which seem to me to throw a new light on Thermodynamics. He considers the intrinsic energy (e) of a homogeneous mass consisting of n kinds of component matter to be a function of n+ z variables, namely, the volume of the mass v, its eutropy », and the n masses, m,, m,...m,, of its component substances. Each of these variables represents a physical quantity, the 428 value of which, for a material system, is the sum of its values for the parts of the system. By differentiating the energy with respect to each of Ges variables (considered as independent), we obtain a set of n +z differential coefficients which represent the intensity of various properties of the substance. Thus, = =-— p, where p is the pressure of the substance ; “s = 6, where @ is the temperature’ on the thermodynamic scale 5 in =, where pw, is the potential of the component (m,) with respect to the compound mass. Each of the component substances has therefore a potential with respect to the whole mass. The idea of the potential of a substance is, I believe, due to Prof. Gibbs. His definition is as follows :— If to any homogeneous mass we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous, and its eutropy and volume remaining unchanged, the increase of the energy of the mass, divided by the mass of the substance added, is the potential of that substance in the mass considered. The condition of the stable equilibrium of the mass is ex- pressed by Prof. Gibbs in either of the two following ways: I. For the equilibrium of any isolated system tt 1s necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variation of tts eutropy shall either vanish or be negative. Il. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its eutropy, the variation of the energy shall either vanish or be positive. 22 | ne et “ae aia he a ns 429 _ The variations here spoken of must not involve the trans- - portation of any matter through any finite distance. Tt follows from this that the quantities 0, p, u,...u, must have the same values in all parts of the mass. For if not, heat will flow from places of higher to places of lower temperature, the mass as a whole will move from places of higher to places of lower pressure, and each of the several component substances will pass from places where its potential is higher to = where it is lower, if it can do so continuously. Hence Prof. Gibbs shows that if ©, P, V...M, are the values of 0, p, w,...4, for a given phase of the compound, and if the quantity Bee th Pe ee — &e. — Mm,, is zero for the given fluid, and is positive for every other phase of the same components, the condition of the given fluid will be stable. If this condition holds for all variations of the variables the fluid will be absolutely stable, but if it holds only for small variations but not for certain finite variations, then the fluid will be stable when not in contact with matter in any of those phases for which K is positive, but if matter in any one of these phases is in contact with it, its equilibrium will be destroyed, and a portion will pass into the phase of the substance with which it is in contact. Thus in Professor F. Guthrie’s experiments, a solution of chloride of calcium of 37 per cent. was cooled to a temperature somewhat below — 37° C. without solidification. ‘In this state, however, the contact of three different solids determines three different kinds of solidification. A piece of ice causes ice to separate from the fluid. A piece of the cryo- hydrate of chloride of calcium determines the formation of eryohydrate from the fluid, and the anhydrous salt causes a precipitation of anhydrous salt. 430 The phase of the fluid is such that K is positive for all phases differing slightly from its own phase, and its equilibrium is therefore stable, but for certain widely different phases, namely, ice, cryohydrate and anhydrous salt, K is negative. If none of these substances are in contact with the fluid, the fluid cannot alter in phase without a transport of matter through a finite distance, and is therefore stable; but if any one of them is in contact with the fluid, part of the fluid is enabled to pass into a phase in which XK is negative. The conditions of consistent phases are that the values of 0, p, w,...4,, and K are equal for all phases which can coexist in equilibrium, the sur- face of contact being plane. This was illustrated by Mr Main’s experiments on co-exist- ent phases of mixtures of chloroform, alcohol and water. Monpay, May 22, 1876. THE PRESIDENT (PROFESSOR CLERK MAXWELL) in the Chair. The following communication was made to the Society : On Curvilinear and Normal Co-ordinates. By the Rev. J. W. Warrey, M.A. (Communicated by Prof, Cayuzy.) THE Memoir refers partly to the general theory of curvilinear _ co-ordinates, partly to the special case of normal co-ordinates. Taking (u, v, w) each of them a given function of the rect- angular co-ordinates (#, y, 2), so that a point is determined either by its rectangular co-ordinates (a, y, z) or by its curvi- linear co-ordinates (u, v, w), and writing dx’ + dy’ + dz’ = (a, b, ¢, f, g, h) (du, dv, dw)’, 431 and then, 0 being an arbitrary function of (#, y, z), or of (u,v, w), : (as) #(G) + (Ze) -42 oe om (TPG) then (a,...), (A,...) are given functions of the differential ; d. d coefficients a I 7 &c., that is of (a, y, z), or, what is the same thing, of (wu, v, w), such that eee t. Cot. KF £hhGas =be—f* : ca—g® : ab—h’®: gh—af : hf—bg : fg—ch, and es eee ae ee | soe eee | =BC-F”’ : CA—G: AB-H’ : GH-AF: HF—~BG : FG—CH, and the theory of curvilinear co-ordinates is in fact a theory of the mutual relations of these coefficients (a, ... ) and (A, ...). In Lamé’s system of curvilinear co-ordinates where the surfaces u=0, »=0, w=0 are orthotomic, f=g=h=0, and therefore also F= G= H=0: and the remaining coefficients correspond to Lamé’s h, h,, h,, H, H,, H,; viz. we have 1? “2? HafaBava, Ha=p=T=VB Beap= 77 V% VA VB ar. and Lamé gives six differential equations of the second .order satisfied by h, h,, h,, or H, H,, H,, considered as functions of the variables which correspond to (wu, v, w). In the author’s system of normal co-ordinates, w, v, w denote the normal distances of the point (#, y, z) from three given surfaces u=0, v=0, w=O0 respectively: and the coefficients are then such that d= B= C=1. He obtains on this assump- 432 tion six differential equations of the second order satisfied by a, b, c, f, g, h considered as functions of (u, v, w); viz. the forms are sh! + Ae ue = given function of first derived functions, d’g ‘ ah _ da dt dudv' dudw dvdw dw = given function of do. and as a consequence of these he obtains a seventh differential equation of the second order, symmetrical as regards the coeffi- cients (a, b, c), (f, g, h), and the variables (wu, v, w) ; which seven equations are the chief analytical results arrived at in the memoir. The memoir:contains various developments in rela- tion to the curvature of the surfaces, &c, HONORARY MEMBERS ELECTED. Feb. 28, 1876. Prof Luigi Cremona, Rome. March 13, Joseph Prestwich, F.R.S., Prof. of Geology at Oxford. . NEW FELLOWS. Feb. 14, 1876. Edward Tanner, M.A., Christ’s College. — J. N. Langley, B.A., St John’s College, March 13. W. M. Hicks, B.A., St John’s College. G. T. Bettany, B.A., Caius College. INDEX OF NAMES. VOU... 11, NB. a larger figures indicate separate papers ; the smaller, incidental references. Adams, Prof, 60, 120, 213, 269 — Adams, W. G. 211 Airy, Sir G. B. 47, 119, 123, 125, 250 oo B- Boe 4 19 Ansted, D. T Babington, Prof. C. C. 8, 128, 181 Bacon, Dr, 263 Balfour, F. M. 373, 377, 397 Baxter, H. F. 419 Bonney, Rey. T. G. 15, 57, 108, 150, 152, 182, 196, 238, 266, 268, 284, 301, 305, 372 Boole, Prof. 114 Bourbouse, M. 117 Bow, Mr, 407 Bradbury, J. B. 271 Candy, F. J. 323 Caron, Capt. 117 Cayley, Prof. 106, 120, 123, 153, 159, 219, 240, 243, 260, 314, 332, 430 Challis, Prof. 38, 60, 120, 143, 184, 200 Clark, J. W. 26, 299, 357 Clark, W. G. 15, 42, 109 Clausen, Prof. 269 Clifford, W. K. 120, 155, 156, 211 Cope, Rev. E. M. 27 Cunynghame, H. H. 319 De Morgan, 24 Denison, E, B. 283 Dew-Smith, A. G. 407 Drosier, W. H. 1 Earnshaw, S. 360 Ellis, Rev. J. C. W. 155, 197, 244, 245, 248, 256, 276, 317, 347 Faye, M. 266 Fennell, C. A. M. 123, 144, 297 Fisher, Rev. O. 51, 115, 152, 195, 222, 239, 285, 324, 367, 401 Fizeau, M. 268 Foster, Dr M. 256, 289, 309, 313, 340, 344, 398, 407 Gibbs, Prof. J. W. 427 Glaisher, J. W. L. 224, 241, 260, 270, 317, 386 Goodman, N. 267, 285 Goodwin, Rev. H., D.D. 64 Govi, Prof. 302 Grassmann, Prof. 75 Green, W. C. 144 Groux, M. 283 Gunn, Rey. J. 53 Hammond, Mr, 144 Herschel, Sir J. 71 Herwart ab Hohenburg, 386 Hicks, W. M. 422 Hiern, W. P. 215, 227, 261 Hill, Rev. E. 372 Hollis, W. A., M.D. 361 Hudson, W. H. ~ 204 Hughes, Prof. 3 Humphry, Prof 49, 116, 129, 185, 258; 321, 349, 363, 364 Huxley, T. H., F.R.S. 348 Jebb, R. C. 296, 375 Keller, Dr F. 385 Kennedy, Rey. Prof. 143 Kingsley, Rev. W. T. 283, 369, 370 Langley, J. N. 402 Latham, P. W. 254, 270 Lewis, Rey. 8. S. 177, 305 Liveing, Prof. 108, 117, 177, 194, 239, 265, 282, 298, 302 Lumby, Rev. J. 156 Marshall, A. 318 Maxwell, Prof. 242, 289, 292, 294, 302, 318, 338, 365, 372, 407, 427 Miller, Prof. W. H. 34, 75, 107, 117, 182, 186, 219, 239, 251, 266, 268, 276, 302 Monro, C. J. 113 Moon, R. 217 Moulton, J. H. 211 Munro, H. A. J. 143 434 Newman, F. W. 314 Paget, Prof. G. E. 290, 362 Paley, F. A. 13, 40, 107, 143, 148, 155, 264, 295, 305, 331 Palmer, E. H. 115, 153, 191, 303 Pearson, Rev. J. B. 298, 332, 351, 357, 392, 414 Peaucellier, M. 334, 407 Phear, Rey. 8. G., D.D. 240 Pirie, Rev. J. 374 Potter, R. 21, 38, 64, 114, 180, 14s, 160, 201, 240, 241, 306 Pritchard, Rev. W. E. 109 Robinson, Rev. 0. K., D.D. 369 Rohrs, J. H. 149, 152, 168, 220, 354 Salter, J. W. 125, 128 Savory, Mr, F.R.S. 345 Seeley, H. 11, 39, 40, 99, 118, 129, 130, 152, 155, 186, 187 Selwyn, Rev. Prof. 123, 143, 144, 149, 201, 296, 324 ~ Sollas, W. J. 299 Stanley, W. H. 276, 291 Stokes, Prof. G. G. 115, 222 Stuart, Jas. 199 Taylor, S. 183, 323, 344 Todhunter, I. 16, 218, 238 Tresca, M. 276 Trotter, Rev. C. 145, 184, 224, 311, 355, 370, 383 Tyndall, Prof. 136 Walker, Col. J. T. 201 Warren, J. W. 430 Wiener, Dr, 219 Wiesnegg, M. 117 Williams, Rev. G. 194 Wilson, Dr H. 8. 312, 364 Yule, C. 321 SUBJECTS. Agricultural terms, antiquity of, 155 Air-cells in birds, 1 Alphabet, physiological, 323 Alum Bay, pipe in the Chalk at, 194 Are of meridian in taplens, 218 Aristophanes, 109 Aristotle, Meteorologica, 144 a on music, ‘375. Rn on ‘‘right-handedness,” 392 Asymmetry, case of, 185 Attraction of a thin shell, 213 Aurora Borealis, 199 Bedawin of Sinai, 153 Bernouilli numbers, logs. of, 224, 264, 269 Birds, on their respiration, &c. 1 Blindness, Half-, form of, 254 Bos Primigenius, skeleton of, 357 Brachiopoda, passage to bivalve and univalve, 128 Calculus of Variations, problem in, 294 Camera Lucida prism, 107 a invention of, 186 on an improved, 302 Capillary attraction, 21 Carboniferous Limestone, burrows in, 150, 182, 266 Carmine, and cochineal, 168 Cauchy, theorems by, 332 Caves at Beit-Jibrin, so-called ‘‘Ho- rite,” 303 Celestial globe, a new, 115 Cetacea, 26 Chimpanzee, 49 Clepsydra, on a, 347 Clock, Remontoir, 283 Clouds, generation of, 136 Coal-seam, on a boulder in, 301 Constants, discontinuity of arbitrary, 115 Coordinates, curvilinear and normal, 30 Crotalocrinus Rugosus, 128 Crystals, and Grassmann’s crystallo- graphic method, 75 Curves, degeneration of, 155 »» machine for tracing, 256 » of the fourth degree, 314 Darwinian theory of species, 116 Degeneration, human, influence of, 263 Denudation, phenomena of, 195 Digit, extra, appearance of, 285 435 Double-sixer, construction of, 219 Dynamics, method of demonstrating some propositions in, 16 Earth, figure of, 238 » friction on, ascribed toether, 422 » inequalities of its surface, 325 » temperature of, 354, 401- Ebenacexw, monograph on, 261 Eclipse, solar, of Dec. 22, 1870, 204 Electrical problems, solution of, 242 Electro-motive machine, 197 Ellipses, method of describing, 197 Ellipsoid, centro-surface of, 159 Endothelium, use of term, 340 Equations, algebraical, 317 ° differential, 360 % machine for solving, 155 A of motion, proof of, 292 Equations, rational, 156 Eseapement, Denison’s gravity, 283 Ether, friction attributed to, 422 Euripides, Pheenisse, passage from, 332 Eustachian Tube, mechanism of, 321 Eye, affected by malformation, 47, 250 3, centre of motion of, 365 Factorial resolution, a, 119, 125 Figure of the earth, theory of, 238 Flint implement exhibited, 152 Fluid, motion of an imperfect, 220 Function, on the rvot of any, 24 Galileo, experiment by, 345 »» trial of, 823 Galvanometer circuit, 374 Geodesy, 201 Glaciers of North Wales, 283 Gravels, Farringdon, 99 Greek authors, when committed to writing, 123 Green Sand, lower, 130 ” upper, 299 Halo of 22°, 34 Heart, effect of constant current on, 407 Herschel’s Astronomy, 144 Heterogeneous substances, equilibrium of, 427 re Hi tamus, notes on, Ba picic texts, late date of, 40 », tumuli, 13 Hyperbolas, rectangular, 319 Ice Hummocks, on some, 355, 383 » prismatic structure in, 57 India, Trig. survey of, 201 Insanity, on, 263 Instruments for sounding, 870 Jaborandi, action of, 402 Lanx, on a Roman, 177 Latitude, observation for, 351 Laws of life and rocks, 39 Leaves, floating, theory on, 215 Levelling, method of, 276 Life, succession of plant, 125 Ligamentum Teres, use of, 345 Light, effects of, on Portland stone, 264 Lobster, model of a, 347 Longitude, observation for, 357 te of the society’s clock, 60 Loudness, test of, 217 Lunar distances, on a set of, 414 Machine for solving equations, 155 Mammals, spinal cord and tail of, 419 Matter, space-theory of, 157 Measures “a bouts,” 182 3 4 compared with **& traits,” 251, 266 Mechanical solution of equations, 317 Mercury, transit of, 123 Meteoric shower of Nov. 1866, 60 Métres Internationaux, form of, 276 Molluse borings in limestone, 182 Monopolies, representation of, 318 Moon, period of rotation, &c., 180 Mortar, composition of, 117 Mountains, their elevation by lateral pressure, 115 Mountains, formation of, 367 Moving bodies, resistance experienced by, 160 Music, Aristotle on, 375 Musical consonance, Helmholtz and Tyndall on, 183 Musical scales, modern, 64 Narwhal, Rete mirabile of, 312 Neolithic Age, 15 Neutral series, 24 Nuseirfyeh, the, 191 Nutrition, problems of, 344 Odysseus, Odyssey; meaning of, 295 Optics, geometrical, 338 Organic acids, 38 Ornithosauria, 186 Paganism, vitality of, 191 Pangenesis, theory of, 285 Papyrus of Lake Gennesaret, 8 Peaucellier, parallel motion of, 334 436 Pendulum, on maintaining the oscil- lations of, 75 Pentagraph, on a. 240 Peritoneum, on the, 364 Perspective, method of drawing in, 197, 276 Pholas burrows, 150 Photographs of Mercury, 123 Planet, as a habitable world, 149 Planetary configurations, 149 Platea, city walls of, 331 Platine, Chalumeau a, 117 Plato, Thestetus, 27 Pleiad, the lost, 144 Plesiosaurus, new species of, 187 Porismatic problems, 120 Prism-circle, observations with, 351, 35 Prismatic structure in ice, 57 as subjective and objective, Propelling vessels, on, 245 Propositions numerically definite, 114 Pterodactyles, 113, 130, 155, 186 ” brain and metatarsus in, 1 Purbeck, ile of, 130 Reflex action, on, 309 Remontoir clock, 283 Reptiles of the Kimmeridge Clay, 155 Resistance to moving bodies, 160, 240, 241 Rete mirabile, the, 312 Right-handedn 861, 392 Rocks, weather of 114 Root of any function, 24 Roslyn, or Roswell Hill Clay-pit, 51, 268 Rotatory motion, on a method of, 248 a model for transfer- ring, 197 Sands, the Potton, 40, 99 Saving life at sea, design for, 291 Sea terraces, elevated, 107 Segmentation, on, 397 Sextic torse, 153 Shell, attraction of a thin, 213 Ship, Hindu and Greek compared, 298 Skull, morphology of, 348 3» parietal bones of, 321 », and Skeleton, theory on, 11 Solar dise, 149 Spinal cord of mammals, “419 Statics, diagrams in graphical, 407 Steamship for conveying trains, 197 Stonehenge, 1 Surface, cubic, model of, 219 os developable, 123 Surfaces, equilibrium of flexible, 145 Syenite hills in Skye, Cirque in, 238 Teichopsia, 254 Thucydides, rv. 30, 143 Tidal phenomena, on, 152, 220 Transmutation of species, &c., 116 Tumuli, the Homeric, 13 Tympanum, condition of, with refer- ence to sound, 217 Upas Antiar, effects of, 398 Uremic poisoning, on, 270 Vases, group of figures on, 148 Venus, transit of, 324 Vertebrates, alimentary canal in, 373 5 fossil remains of, 11 segmental organs of, 377 Violoncello, ‘“* Wolf” in, 369 Vision, empirical theory of, 224 Vital force, 419 Volition, 149 Vowel-letters and vowel-sounds, 306 Wealden strata, 130 Zirconia light, 117 END OF VOL. II. _ CAMBRIDGE: PRINTED BY ©. J. CLAY, M.A. AT THE UNIVERSITY PRESS, . © BINDING SECT. NOV 81987 Q Cambridge Philosophical 4l society, Cambridge, Ing. C17 Proceedings Vol=-2 Physical & Appiied Sci. Serialg ~~. PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY he Wine | hse, Tiers ¥ 4) as eA i et peer eeri dl bits vigie tiriseths ar seria bt! enenetiae y BANGS ATE ihe f i r at iti} { IRN ESE be ¢ MIREEOULaneh Bewbod ve tots hea i 2 earl iteen bias onepy Ppezen aE sighetes mateeeshrin 3: And = ch Fei ty Ee a ASA 4 wg i at Asse Og oyaery . ld id nae. of 8 Sea ey =o ats ~ i se * fh elite et Wen at ’ Dek Siesiestsr. ae : ay ae + bass. S94 pa mete tik = he pent ee) ( ey } th Fhdlie : a ie oe Bika Ley ta BED hie i x ee sc @ kee oes ebieee ath ervoil hats pro ee Geet \ RE a eh Ay eA heteendinrtge 7 % eu re *, i