perhaps more correctly the phase of life, of a star or nebula, shows us the material of potential suns, other suns in the process of formation, in vigorous youth, and in every stage of slowly protracted decay. It leads us to look on each planet and satellite as having been at one time a tiny sun, a member of some binary or multiple group, and even now (when almost deprived, at least at its surface, of its original energy) presenting an endless variety of subjects for the application of its methods. It leads us forward in thought to the far-distant time when the materials of the present stellar system shall have lost all but their mutual potential energy, but shall in virtue of it form the materials of future larger suns with their attendant planets. Finally, as it alone is able to lead us, by sure steps of deductive reasoning, to the necessary future of the universe --necessary, that is, if physical laws for ever remain unchanged-so it enables us distinctly to say that the present order of things has not been evolved through infinite past time by the agency of laws now at work, but must have had a distinctive beginning, a state beyond which we are totally unable to penetrate, a state, in fact, which must have been produced by other than the now acting causes. Thus also, it is possible that in Physiology it may, ere long, lead to results of a different and much higher order of novelty and interest than those yet obtained, immensely valuable though they certainly are. It was a grand step in science which showed that just as the consump on of fuel is necessary to the working of a steam-engine, or to the steady light of a candle, so the living engine requires food to supply its expenditure in the forms of muscular work and animal heat. Still grander was Rumford's early anticipation that the animal is a more economic engine than any lifeless one we can construct. Even in the explanation of this there is involved a question of very great interest, still unsolved, though Joule and many other philosophers of the highest order have worked at it. Joule has given a suggestion of great value, viz. that the animal resembles an electromagnetic- rather than a heat-engine; but this throws us back again upon our difficulties as to the nature of electricity. Still, even supposing this question fully answered, there remains another-perhaps the highest which the human intellect is capable of directly attacking, for it is simply preposterous to suppose that we shall ever be able to understand scientifically the source of Consciousness and Volition, not to speak of loftier things—there remains the question of Life. Now it may be startling to some of you, especially if you have not particularly considered the matter, to hear it surmised that possibly we may, by the help of physical principles, especially that of the Dissipation of Energy, some time attain to a notion of what constitutes Life-mere Vitality I repeat, nothing higher. If you think for a moment of the vitality of a plant or a zoophyte, the remark perhaps will not appear so strange after all. But do not fancy that the Dissipation of Energy to which I refer is at all that of a watch or such-like piece of mere human mechanism, dissipating the low and common form of energy of a single coiled spring. It must be such that every little part of the living organism has its own store of energy constantly being dissipated, and as constantly replenished from external sources drawn upon by the whole arrangement in their harmonious working together. As an illustration of my meaning, though an extremely inadequate one, suppose Vaucanson's Duck to have been made up of excessively small parts, each microscopically constructed as perfectly as was the comparatively coarse whole, we should have had something barely distinguishable, save by want of instincts, from the living model. But let no one imagine that, should we ever penetrate this mystery, we shall thereby be enabled to produce, except from life, even the lowest form of life. Our President's splendid suggestion of Vortex-atoms, if it be correct, will enable us thoroughly to understand matter, and mathematically to investigate all its properties. Yet its very basis implies the absolute necessity of an intervention of Creative Power to form or to destroy one atom even of dead matter. The question really stands thus :—Is Life physical or no? For if it be in any sense, however slight or restricted, physical , it is to that extent a subject for the Natural Philosopher, and for him alone. It would be entirely out of place for me to discuss such a question as this now and here; I have introduced it merely that I may say a word or two about what has been 80 often and so persistently croaked against the British Association, viz. that it tends to develope what are called Scientific Heresies. No doubt such charges are brought more usually against other Sections than against this; but Section A has not been held blameless. It seems to me that the proper answer to all such charges will be very simply and easily given, if we merely show that in our reasonings from observation and experiment we invariably confine our physical conclusions strictly to matter and energy (things which we can weigh and measure) in their multiform combinations. Excepting that which is obviously purely mathematical, whatever is certainly neither matter nor energy, nor dependent upon these, is not a subject to be discussell here, even by implication. All our reasonings in Physics must, so far as we know, be based upon the assumption, founded on experience, that in the universe, whatever be the epoch or the locality, under exactly similar circumstances exactly similar results will be obtained. If this be not granted there is an end of Physical Science, or, rather, there never could have been such a Science®. To use the word “Heresy” with reference to purely physical reasonings about Geological Time, or matters of that kind, is nowadays a piece of folly which even Galileo's judges, were they alive, would shrink from, as calculated to damage none but themselves and the cause which of old they, according to their lights, very naturally maintained. There must always be wide limits of uncertainty (unless we choose to look upon Physics as a necessarily finite Science) concerning the exact boundary between the Attainable and the Unattainable. One herd of ignorant people, with the sole prestige of rapidly increasing numbers, and with the adhesion of a few fanatical deserters from the ranks of Science, refuse to admit that all the phenomena even of ordinary dead matter are strictly and exclusively in the domain of physical science. On the other hand, there is a numerous group, not in the slightest degree entitled to rank as Physicists (though in general they assume the proud title of Philosophers), who assert that not merely Life, but even Volition and Consciousness are mere physical manifestations. These opposite errors, into neither of which is it possible for a genuine scientific man to fall, so long at least as he retains his reason, are easily seen to be very closely allied. They are both to be attributed to that Credulity which is characteristic alike of Ignorance and of Incapacity. Unfortunately there is no cure; the case is hopeless, for great ignorance almost necessarily presumes incapacity, whether it show itself in the comparatively harmless folly of the Spiritualist or in the pernicious nonsense of the Materialist. Alike condemned and contemned, we leave them to their proper fate-oblivion ; but still we have to face the question, where to draw the line between that which is physical and that which is utterly beyond physics. And, again, our answer is— Experience alone can tell us ; for experience is our only possible guide. If we attend earnestly and honestly to its teachings, we shall never go far astray. Man has been left to the resources of his intellect for the discovery not merely of physical laws, but of how far he is capable of comprehending them. And our answer to those who denounce our legitimate studies as heretical is simply this,-A revelation of any thing which we can discover for ourselves, by studying the ordinary course of nature, would be an absurdity. A profound lesson may be learned from one of the earliest little papers of President, published while he was an undergraduate at Cambridge, where he shows that Fourier's magnificent treatment of the Conduction of Heat leads to formula for its distribution which are intelligible (and of course capable of being fully verified by experiment) for all time future, but which, except in particular cases, when extended to time past, remain intelligible for a finite period only, and then * It might be possible, and, if so, perhaps interesting, to speculate on the results of secular changes in physical laws, or in particles of matter which are subject to them, but (so far as experience, which is our only guide, has taught us since the beginning of modern science) there seems no trace of such. Even if there were, as these changes must be of necessity extremely slow (because not yet even suspected), we may reasonably expect, from the analogy of the history of such a question as gravitation, especially in the discovery of Neptune, that our work, far from becoming impossible, will merely become considerably more difficult as well as more laborious, but, on that account, all the more creditable when successfully carried out, indicate a state of things which could not have resulted under known laws from any conceivable previous distribution. So far as heat is concerned, modern investigations have shown that a previous distribution of the matter involved may, by its potential energy, be capable of producing such a state of things at the moment of its aggregation; but the example is now adduced not for its bearing on heat alone, but as a simple illustration of the fact that all portions of our Science, and especially that beautiful one the Dissipation of Energy, point unanimously to a beginning, to a state of things incapable of being derived by present laws from any conceivable previous arrangement. I conclude by quoting some noble words used by Stokes in his Address at Exeter, words which should be stereotyped for every Meeting of this Association :“When from the phenomena of life we pass on to those of mind, we enter a region “ still more profoundly mysterious. .. . Science can be expected to do but little « to aid us here, since the instrument of research is itself the object of investigation. “It can but enlighten us as to the depth of our ignorance, and lead us to look to a “higher aid for that which most nearly concerns our wellbeing.” MATHEMATICS. a Exhibition and Description of a Model of a Conoidal Cubic Surface called the "Cylindroid,” which is presented in the Theory of the Geometrical Freedom of a Rigid Body. By ROBERT STAWELL BALL, A.M., Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland. We become acquainted with the geometrical freedom which a rigid body enjoys by ascertaining the character of all the displacements which the nature of the restraints will permit the body to accept. If a displacement be infinitely small, it is produced by screwing the body along a certain screw. If a displacement have finite magnitude, it is produced by an infinite series of infinitely small screw displacements. For the analysis of geometrical freedom, we shall only consider infinitely small screw displacements. This includes the initial stages of all displacements. To analyze the geometrical restraints of a rigid body we proceed as follows: Take any line in space. Conceive this line to be the axis about which screws are successively formed of every pitch from - to to. (The pitch of a screw is the distance its nut advances when turned through the angular unit.) We endeavour successively to displace the body about each of these screws, and record the particular screw or screws, if any, about which the restraints have permitted the body to receive a displacement. The same process is to be repeated for every other line in space. If it be found that the restraints have not permitted the body to receive any one of these displacements, then the body is rigidly fixed in space. If, after all the screws have been tried, the body be found capable of displacement about one screw only, the body possesses the lowest degree of freedom. If one screw (A) be discovered, and, the trials being continued, a second screw (B) be found, the remaining trials may be abridged by considering the information which the discovery of two screws affords. It is in connexion with the two screws that the cylindroid is presented. The body may receive any displacement about one or both of the two screws A and B. The composition of these displacements gives a resultant which could have been produced by displacement about a single screw. The locus of this single screw is the conoidal cubic surface which has been called the "cylindroid " (at the suggestion of Professor Cayley). The equation of the cylindroid is -(.22+y)-2axy=0. 8 Any line (8) upon this surface is considered to be a screw, of which the pitch is c+a cos 20, where c is any constant, and 0 is the angle between s and the axis of x. The fundamental property of the cylindroid is thus stated. If any three screws of the surface be taken, and if a body be displaced by being screwed along each of these screws through a small angle proportional to the sine of the angle between the remaining screws, the body after the last displacement will occupy the same position that it did before the first. For the complete determination of the cylindroid and the pitch of all its screws, we must have the quantities a and c. These quantities, as well as the position of the cylindroid in space, are completely determined when two screws of the system are known. In the model of the cylindroid which was exhibited, the parameter a is 2.6 inches. The wires which correspond in the model with the generating lines of the surface represent the axes of the screws. The distribution of pitch upon the generating lines is shown by colouring a length of 2-6x sin 20 inches upon each wire. The distinction between positive and negative pitches is indicated by colouring the former red and the latter black, It is remarkable that the addition of any constant to all the pitches attributed in the model to the screws does not affect the fundamental property of the cylindroid. When a rigid body is found capable of being displaced about a pair of screws, it is necessarily capable of being displaced about every screw on the cylindroid determined by that pair. The theorem of the cylindroid includes, as particular cases, the well-known rules for the composition of two displacements parallel to given lines, or of two small rotations about intersecting axes. If the parameter a be zero, the cylindroid reduces to a plane, and the pitches of all the screws become equal. If the arbitrary constant which expresses the pitch be infinite, we have the theorem for displacements, and if the pitch be zero, we have the theorem for rotations. As far as the composition of two displacements is concerned, the plane can only be regarded as a degraded form of the cylindroid from which the most essential features have disappeared. On the Number of Covariants of a Binary Quantic. By Professor CAYLEY, D.C.L., F.R.S. The author remarked that it had been shown by Prof. Gordan that the number of the covariants of a binary quantic of any order was finite, and, in particular, that the numbers for the quintic and the sextic were 23 and 26 respectively. But the demonstration is a very complicated one, and it can scarcely be doubted that a more simple demonstration will be found. The question in its most simple form is as follows: viz. instead of the covariants we substitute their leading coefficients, each of which is a “seminvariant” satisfying a certain partial differential equation ; say, the quantic is (a, b, c....kDx,y)", then the differential equation is (ado+26Dc.... +njdk)u=0, which quà equation with n+1 variables admits of n independent solutions : for instance, if n=3, the equation is (ado +260.+3cda)u=0, and the solutions are a, ac— b, a’d-3abc +263; the general value of u is u= any function whatever of the last-mentioned three functions. We have to find the rational non-integral functions of these functions which are rational and integral functions of the coefficients; such a function is 1 {(a’d– 3abc+26)*+4(ac —b)"}, =a’d? + 4ac3 +463d - 36c* — 6abcd, and the original three solutions, together with the last-mentioned function a’d+&c., constitute the complete system of the seminvariants of the cubic function; viz. every other seminvariant is a rational and integral function of these. And so in the general case the problem is to complete the series of the n solutions a, ac – 6, C a’d-3abc +26*, a'e-4aRbd +6abc3b*, &c. by adding thereto the solutions which, being rational but non-integral functions of these, are rational and integral functions of the coefficients; and thus to arrive at a series of solutions such that every other solution is a rational and integral function of these. n is even, + ... On a Canonical Form of Spherical Harmonics. By W. K. CLIFFORD, B.A. The canonical form in question is an expression of the general harmonic of order n as the sum of a certain number of sectorial harmonics, this number being, when 5n-10 5n-9 2 d? Laplace's operator, + d.ca be obtained from the tangential equation may d d d of the imaginary circle & +n +$=0, by substituting for $, n, Š. LE dx dy az therefore, a form U=(x, y, z)" is reduced to zero by this operator, it follows from Prof. Sylvester's theory of contra variants that the curve U=0 is connected by certain invariant relations with the imaginary circle. I find that U can be expressed in the form U=A"+B" +C"+ .. where A=0, B=0,.... are great circles touching the imaginary circle, the number of terms being as above. Now if L=0, M=0 be two such great circles meeting in a real point a, and if ø be a longitude and 8 latitude referred to a as pole, it is easy to see that L" +M"=l sin" & sin ng+msin" cos no, a sum of two sectorial harmonics, which is the proposed reduction. When n is less than five, exceptions of interest occur. For n=3, if we take a, b, corresponding points on the hessian of the nodal curve U=0 (Thomson and Tait, Treatise on Natural Philosophy, $ 780), and if we call ov $, the longitudes, 0, 0, the latitudes referred to these poles, we have U=lsin, sin 39,+ msin', cos 3°, un sin 0, sin 302 +8 sin 0, cos 3° . For n=4, the nodal curve is of the species first noticed by Clebsch, of which many most beautiful properties have been pointed out by Dr. Lüroth. The form U is expressible as the sum of five fourth powers ; so that if we take b real points of intersection of two pairs of them, c a real point on the fifth, calling ou da u 0,, 09, 0, longitudes and latitudes referred to them, we have U=lsin. O, sin 40. msin, cos 40, +p sin, sin 40, +9 sin* , cos 402 a, 4 On certain Definite Integrals. By J. W. L. GLAISHER, B.A., F.R.A.S. The integrals *sin (zn)dr, Socos (27)dx have been evaluated in several different ways, and the investigations all present points of interest. The integrals have usually been written in the forms 8,2P-1 sin ædr, So** XP-1 cos x dx, deducible by an obvious transformation ; and so universally have the latter forms been adopted, that the former are not to be found in Prof. De Haan's Tables. The most natural way of obtaining So* sin andx and scos andx is by writing so p=i(=V-1) in the well-known form of the Gamma Function, 1 e-po" dx = PM |