The projections of a, on axes (1) and (2) are „„, „α, so that its projection on the normal to Σ ka=0 is (cos + sine, a, 'since all the other axes are by construction perpendicular to the said normal), where n cos θ = Ε Σ κ. r=1 But this projection of a, on the normal to Ek,a=0 is clearly also equal to ka. and Hence 1 sin 0 + 1, cos 0 = k„, 1=k, cosec 0-1, cot = k cosec 0-n cot A. Consequently x=21,= cosecka-n cot @ Σa, = 0. Hence, as before, it is readily seen that the mean value of Σa, in this case is equal to §9. As in §5, the results of §§ 6-8 can be readily extended to give the mean of Zam, subject to the conditions Σa, = 0, Σk,a=0, and 2a,' = R2, i.e. over the surface of the (ndimensional sphere so defined. The result is the mean value taken throughout the sphere. n+2m-2 N ( n − 2, times § 10. The extension of the results, by the same method, to the case when further linear relations are required to hold good between the quantities a,, is fairly obvious, and need not be indicated here. Trinity College, Cambridge. CERTAIN CRITERIA FOR THE SUCCESS OF DARBOUX'S METHOD WHEN APPLIED TO THE EQUATION_s=ƒ (x, y, z, P, q). 1. IN By J. R. WILTON, M.A., D.Sc. one of his memoirs, on the partial differential equation 8=ƒ (x, y, z, p, q)..........................................................(1), Goursat very briefly indicates a method of attacking the problem of determining those cases in which equation (1) possesses intermediate integrals of order higher than the second-the memoir is almost entirely concerned with the determination of the cases in which (1) possesses integrals of the second order-and he shows that any equation of the form (1) which possesses integrals of both systems of characteristics must belong to one or other of the four general types,† (1) s = F(x, y, z) p (x, y) + (y, q) In the present communication I propose to consider the same question from a somewhat different point of view. Let = $ (X, Y, Z, P1, P2, P1) 0(2) be an intermediate integral of the nth order; then *Annales de la faculté des Sciences de l'Université de Toulouse, 2 Série, t. 1, pp. 31-78 and 439-463 (1899). † Loc. cit. p. 463. The notation used is that of Goursat, "Leçons sur l'intégration des équations aux dérivées partielles du second ordre, t. 2, p. 106, § 115." VOL. XLVII. D There will be a similar pair of equations derived from the system of characteristics to which the equation dy = 0 belongs. We shall, however, in order to avoid unnecessary complication, always suppose that we are dealing with the system of characteristics of which x=constant is an integral. We shall thus be concerned only with the conditions that (3) and (4) should have a common integral. In order to determine the lowest possible order of an intermediate integral of (1) of the system of characteristics considered, form the successive expressions &c., and let the first of them which vanishes be the nth, so that Then equation (1) cannot possess an intermediate integral® of any order lower than the th. For, since does not contain q, on differentiating (4) n + 1 times in succession with regard to q, and eliminating from the last n equations so *For brevity we omit the phrase "of the system of characteristics of which *=constant is an integral" whenever it can be supplied from the context. derived, we obtain equation (5), which shows that (4) cannot be satisfied for any order less than the nth. The explicit form of the necessary and sufficient conditions that the integral should be actually of the nth order can very easily be obtained. denote the co-factors of, say, the last row of the determinant of equation (5). Then df by substitution of the actual values of A,, ..., A fr then Let A' be defined by the equation n' Hence, on the supposition that the determinant of equation (5) is the first of its type to vanish, the necessary and sufficient conditions for the existence of an intermediate integral of equation (1) of order n are the conditions of coexistence of equations (6), (8) and (9), namely = where pz, P.,=y, and where we have made use of the fact that A does not contain Pn 1a. As a simple illustration of the results obtained, we may remark that the equation 8=4(y, q) + (x, y, z, p)...........................................(11), on account of (5), cannot possess an intermediate integral is a solution, for some value of n, of the equation unless (婦) n-1 יו 12-1 in which Y, Y, ..., Y are arbitrary functions of y; and that the least possible order of the intermediate integral, when it exists, is greater by unity than that of the equation of lowest order satisfied by p. The solution of (12) is, in the most comprehensive form, easily seen to be given by the elimination of μ between the two equations h in which the Y's are arbitrary functions of y; and the least possible order of the intermediate integral is then h 1 + h + Σ k i=1 If (12) is the equation of lowest order satisfied by the co-factors A,, ..., A, are to be calculated from (5), but it must be remembered that no simplification must be made in the original determinant until after the co-factors have been calculated. This renders the determination of A in the general case somewhat laborious. The application of the result of this paragraph to the first of Goursat's four cases is immediate. 2. Still retaining the condition that the determinant of equation (5) is the first of its type to vanish, we proceed to the consideration of the possibility of the existence of an intermediate integral of order m+n, where m > 0, and in particular we shall give the explicit form of the conditions for the existence of an integral of order n + 1. |