PROCEEDINGS 



OF THE 



Cambridge |JI)tl0Sopjwal Stoneig. 



On Lie's Solution of a Partial Differential Equation of the 

 First Order. By 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 Mayer -f". 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, x 1} x 2 ...x n the inde- 

 pendent variables and p 1} p 2 . . . p n the partial differential coefficients 

 of z. If u, v are any two functions of these 2n + 1 quantities, 

 denote the expression 



r = n d(u, v) r = n d (u, v) 



r=l 3 Or, Pr) r=l d (z, p r ) 



by the symbol (u, v). 



* See Forsyth, Theory of Differential Equations, Part I. pp. 238 — 9; or Lie, 

 Math. Annalen, Vol. vm. p. 242. 

 t Math. Ann. Vol. vin. pp. 313—8. 



VOL. IX. PT. VI. 23 



