Without entering into the consideration of the mode in which Poisson obtained the particular integral it may easily be shown, by actual differentiation and substitution, that the integral does satisfy our equations. The function ƒ being arbitrary, we may assign to it any form we please, as representing a particular possible motion, and may employ the result, so long as no step tacitly assumed in the course of our reasoning fails. The interpretation of the integral (1) will be rendered more easy by the consideration of a curve. In Fig. 1 let oz be the axis of z, and let the ordinate of the curve represent the values of w for t=0. The equation (1) merely asserts that whatever value the Fig. 1. Fig. 2. velocity w may have at any particular point when t= 0, the same value will it have at the time t at a point in advance of the former by the space (a+w) t. Take any point P in the curve of Fig. 1, and from it draw, in the positive direction, the right line PP' parallel to the axis of z, and equal to (a+w) t. The locus of all the points P' will be the velocity-curve for the time t. This curve is represented in Fig. 2, except that the displacement at common to all points of the original curve is omitted, in order that the modification in the form of the curve may be more easily perceived. This comes to the same thing as drawing PP' equal to wt instead of (a+w) t. Of course in this way P' will lie on the positive or negative side of P, according as Plies above or below the axis of z. It is evident that in the neighbourhood of the points a, c the curve becomes more and more steep as t increases, while in the neigh Here it is observable that no relation exists between the points of no velocity and the points of maximum velocity. As m, t1, and λ are arbitrary constants, we may even have in which case the points of no velocity are also points of maximum velocity."] bourhood of the points o, b, z its inclination becomes more and more gentle. The same result may easily be obtained analytically. In Fig. 1, take two points, infinitely close to each other, whose abscissæ are z and z+dz; the ordinates will be w and dw After the time t these same ordinates will belong to points whose abscissæ will have become (in Fig. 2) z+wt and Hence the horizontal distance between the points, which was dz, will have become and therefore the tangent of the inclination, which was dw/dz, will have become At those points of the original curve at which the tangent is horizontal, dw/dz = 0, and therefore the tangent will constantly remain horizontal at the corresponding points of the altered curve. For the points for which dw/dz is positive, the denominator of the expression (A) increases with t, and therefore the inclination of the curve continually decreases. But when dw/dz is negative, the denominator of (A) decreases as t increases, so that the curve becomes steeper and steeper. At last, for a sufficiently large value of t, the denominator of (A) becomes infinite for some value of %. Now the very formation of the differential equations of motion with which we start, tacitly supposes that we have to deal with finite and continuous functions; and therefore in the case under consideration we must not, without limitation, push our results beyond the least value of t which renders (A) infinite. This value is evidently the reciprocal, taken positively, of the greatest negative value of dw/dz; w here, as in the whole of this paragraph, denoting the velocity when t=0. By the term continuous function, I here understand a function whose value does not alter per saltum, and not (as the term is sometimes used) a function which preserves the same algebraical expression. Indeed, it seems to me to be of the utmost importance, in considering the application of partial differential equations to physical, and even to geometrical problems, to contemplate functions apart from all idea of algebraical expression. In the example considered by Professor Challis, where m may be supposed positive; and we get by differentiating and putting t = 0, the greatest negative value of which is 2m/; so that the greatest value of t for which we are at liberty to use our results without limitation is λ/2mm, whereas the contradiction arrived at by Professor Challis is obtained by extending the result to a larger value of t, namely λ/4m. Of course, after the instant at which the expression (A) becomes infinite, some motion or other will go on, and we might wish to know what the nature of that motion was. Perhaps the most natural supposition to make for trial is, that a surface of discontinuity is formed, in passing across which there is an abrupt change of density and velocity. The existence of such a surface will presently be shown to be possible*, on the two suppositions that the pressure is equal in all directions about the same point, and that it varies as the density. I have however convinced myself, by a train of reasoning which I do not think it worth while to give, inasmuch as the result is merely negative, that even on the supposition of the existence of a surface of discontinuity, it is not possible to satisfy all the conditions of the problem by means of a single function of the form ƒ{z-(a+w) t}. Apparently, something like reflexion must take place. Be that as it may, it is evident that the change which now takes place in the nature of the motion, beginning with the particle (or rather plane of particles) for which (A) first becomes infinite, cannot influence a * [Not so: see the substituted paragraph at the end.] particle at a finite distance from the former until after the expiration of a finite time. Consequently even after the change in the nature of the motion, our original expressions are applicable, at least for a certain time, to a certain portion of the fluid. It was for this reason that I inserted the words "without limitation," in saying that we are not at liberty to use our original results without limitation beyond a certain value of t. The full discussion of the motion which would take place after the change above alluded to, if possible at all, would probably require more pains than the result would be worth. [So long as the motion is continuous, and none of the differential coefficients involved become infinite, the two principles of the conservation of mass and what may be called the conservation of momentum, applied to each infinitesimal slice of the fluid, are not only necessary but also sufficient for the complete determination of the motion, the functional relation existing between the pressure and density being of course supposed known. Hence any other principle known to be true, such for example as that of the conservation of energy, must be virtually contained in the former. It was accordingly a not unnatural mistake to make to suppose that in the limit, when we imagine the motion to become discontinuous, the same two principles of conservation of mass and of momentum applied to each infinitesimal slice of the fluid should still be sufficient, even though one such slice might contain a surface of discontinuity. It was however pointed out to me by Sir William Thomson, and afterwards independently by Lord Rayleigh, that the discontinuous motion supposed above involves a violation of the principle of the conservation of energy. In fact, the equation of energy, applied to the fluid in the immediate neighbourhood of the surface of discontinuity, and combined with the two equations deduced from the two principles first mentioned, leads in the case of pop to where p, p' are the densities at the two sides of the supposed surface of discontinuity; but this equation has no real root except p=p'.] [From the Transactions of the Cambridge Philosophical Society, ON THE FORMATION OF THE CENTRAL SPOT OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. [Read December 11, 1848.] WHEN Newton's Rings are formed between the under surface of a prism and the upper surface of a lens, or of another prism with a slightly convex face, there is no difficulty in increasing the angle of incidence on the under surface of the first prism till it exceeds the critical angle. On viewing the rings formed in this manner, it is found that they disappear on passing the critical angle, but that the central black spot remains. The most obvious way of accounting for the formation of the spot under these circumstances is, perhaps, to suppose that the forces which the material particles exert on the ether extend to a small, but sensible distance from the surface of a refracting medium; so that in the case under consideration the two pieces of glass are, in the immediate neighbourhood of the point of contact, as good as a single uninterrupted medium, and therefore no reflection takes place at the surfaces. This mode of explanation is however liable to one serious objection. So long as the angle of incidence falls short of the critical angle, the central spot is perfectly explained, along with the rest of the system of which it forms a part, by ordinary reflection and refraction. As the angle of incidence gradually increases, passing through the critical angle, the appearance of the central spot changes gradually, and but slightly. To account then for the existence of this spot by ordinary reflection and refraction so long as the angle of incidence falls short |