which when n=1 agrees with (59). Multiplying into de and integrating, we find B22 + C§ + nA2 n- - 1 {f($}"-=N.......(60), where N, like B and C, is an arbitrary constant. The formula fails when n=1, but this case has already been fully discussed. In virtue of (xxiv) equation (60), becomes 108. In proceeding to the integration of (43), and (44)„, we may adopt at once the equations which, in Art. 6, are marked (45) to (53), both numbers inclusive. But the value of t therein contained must be altered in conformity with the article next following (i.e. Art. 109). 109. Multiplying either side of (43), into 2 on integration, Bv1"+m (1-n) 1- - n (57)', and in order to complete the solution, we must take steps analogous to those taken in Art. 9. When C vanishes, we have a case corresponding to that discussed in Art. 103. The value of twill be the same as that in Art. 108. In the rest of the solution for this case we must be guided by the analogies afforded by Art. 103. 110. When B vanishes but C does not vanish, we have a case corresponding to that discussed in Art. 13. In the present case equations (92) and (93) of Art. 13 will hold, but in place of equation (94) we must put and then follow out the solution accordingly. 111. When n=2, we have a case corresponding to Laplace's law. In such case (96) 27 T= Cw2 log E (t + m) + D (t + m) which last is derived from (94),. 112. If in the problem of Art. 1 we suppose the cylinder to be not of small but of any diameter, our formulæ will still apply, the motion being supposed to be all parallel to the axis, and all the elementary particles of a disc to have the same velocity, density and direction of motion. For this reason the case may be termed, and I have termed it, one of plane disturbance. In this paper I have not entered upon second integrations or initial conditions, but much of the necessary work is already done in Arts. 10, 11, 12, 15 and 16. There are cases, I allude more especially to that in which E and E vanish simultaneously, wherein it may be desirable to complete the details of the discussion. When E and E are both zero, we have the case of a spherical mass of elastic fluid expanding in virtue of its own elasticity only. "Oakwal," near Brisbane, Queensland, Australia, A DEMONSTRATION OF A PROPERTY OF By WILLIAM WALTON, M.A., Fellow of Trinity Hall. IF and are connected by the equation where μ F (c, 0) + F(c, 4) = F (c, μ) ................ ........ (1), is a constant, then will cos cos-sin sin p (1-c2 sin3μ) = cosμ. The differential equation on the integration of which is based the demonstration of this well known property, will be seen by the following method of proof to belong intrinsically to Clairaut's form of differential equations. The differential equation, cleared of radicals, is d0* – d�* = c (sin*pd0* — sin* 0d,*). Multiplying by cos' - cos' p, we have (de-do") (cos10-cos p)=c(cose-cos3p) (sin pd0"-sin30dp3). Now the first member of the equation is equal to sin (4+0).(d0 + dø).sin (p − 0). (d0 – dø) =-d cos(+0).d cos(4-0) = dy2 - dx3, where a represents cose cosp and y represents sine sind. Again, the second member of the equation is equal to c2 multiplied by the product of the two expressions and (cos + cos p) (sin pd0 + sin @dø), (cos - cos) (sin pd0 – sin Odp). The former of these expressions is equal to and the latter, as is evident by writing T instead of 4, is evidently equal to dy- xdy+ydx. Thus the differential equation in 0 and 4 is transformed into dx" - dy = c* {(xdy — ydx)2 — dy"}, whence 1 y = xp = {(c* − 1) p2 + 1}*, a differential equation of Clairaut's form. or 1 sin@ sin4 = y cose cos¿ − = {(c2 − 1) y2 + 1} *. Now μ when =0: hence whence = с { (c2 − 1 ) y2 + 1}3, y2 (1-c2 sin3μ) = 1. Hence the integral becomes cos cos-(1-c sinμ) sine sin & = cosμ. September 16, 1870. A DEMONSTRATION THAT EVERY EQUATION By WILLIAM WALTON, M.A., Fellow of Trinity Hall. LET ET fx represent the polynomial of the theory of equations, and suppose the coefficients of the various powers of x to be possible quantities. Write u+v √√(−1) for x, u and v being possible quantities, and let ƒ {u + v √(− 1)} = P+ Q √(− 1), P and Q being possible quantities. Differentiate this equation + times with regard to u: then, f (x) denoting the (+)th derivative of f(x), λ+μ |