Mathematical Proceedings Volume 9 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 Excerpt: ... A. C. Dixon, Sc.d., Trinity College. 1. The present paper gives an account and a proof of Lie's method of solving a differential equation in one dependent and any number (including unity) of independent variables. The arrangement of the proof is such as to facilitate the examination of certain cases of exception. Such are afforded by the tac-locus and cusp-locus of the ordinary theory with one independent variable, and their analogues, and also by an extensive class of equations in which the linear partial form is included, and the integration of which has been discussed by Mayerf. Notes are added on the nature of a complete primitive, the complete solution of the auxiliary linear equation, and the satisfaction of limiting conditions. Method of Solution. 2. Let z be the dependent variable, xu X2...xn the independent variables and px, p2... pn the partial differential coefficients of z. If u, v are any two functions of these 2n + 1 quantities, denote the expression 'n d(u, v) -2- 8 (u, V) r-l 3 (r, Pr) r-3 (, Pr) by the symbol (u, v). See Forsyth, Theory of Differential Equation , Part I. pp. 238--9; or Lie, Math. Annalen, Vol. vm. p. 242. t Math. Ann. Vol. vm. pp. 318--8. VOL. IX. PT. vi. 23 Then we know that the integration of any partial differential equation f(xux, ...xn, z, p1, p.2, ...pn) = 0 (1) depends on the solution of the linear equation (/) = 0 (2) for in terms of a, ... xn, z, p, ... pn. The general solution of this linear equation is derivable when that of the equation /= 0 is known. 3. With regard to the expression (/, ) two things are noticeable. In the first place it is indifferent whether we do, or do not, suppose that wherever pn, say, occurs in it is to be regarded as a known function of the other 2n...