PHILOSOPHICAL TRANSACTIONS. 



I. Memoir on the Integration of Partial Differential Equations of the Second Order in 

 Three Independent Variables ivhen an Intermediary Integral does not exist in general. 



By A. R, FOKSYTH, F.R.S., Sadler ion Professor in the. University of Cambridge. 



Received November 23, Read December 16, 1897. 



THE general feature of most methods for the integration of partial differential 

 equations in two independent variables is, in some form or other, the construction 

 of a set of subsidiary equations in only a single independent variable ; and this 

 applies to all orders. In particular, for the first order in any number of variables 

 (not merely in two), the subsidiary system is a set of ordinary equations in a single 

 independent variable, containing as many equations as dependent variables to be 

 determined by that subsidiary system. For equations of the second order which 

 possess an intermediary integral, the best methods (that is, the most effective as giving 

 tests of existence) are those of BOOLE, modified and developed by IMSCHENETSKY, and 

 that of GOURSAT, initially based upon the theory of characteristics, but subsequently 

 brought into the form of Jacobian systems of simultaneous partial equations of the 

 first order. These methods are exceptions to the foregoing general statement. But 

 for equations of the second order or of higher orders, which involve two independent 

 variables and in no case possess an intermediary integral, the most general methods 

 are that of AMPERE and that of DARBOUX, with such modifications and reconstruction as 

 have been introduced by other writers ; and though in these developments partial dif- 

 ferential equations of the first order are introduced, stijl initially the subsidiary system 

 is in effect a system with one independent variable expressed and the other, suppressed 

 during the integration, playing a parametric part. In other words, the subsidiary 

 system practically has one independent variable fewer than the original equation. 



In another paper* I have given a method for dealing with partial differential 

 equations of the second order in three variables when they possess an intermediary 

 integral ; and references will there be found to other writers upon the subject. My 

 aim in the present paper has been to obtain a method for partial differential equations 

 of the second order in three variables when, in general, they possess no intermediary 

 integral. The natural generalisation of the idea in DARBOUX'S method has been 



* " Partial Differential Equations of the Second Order, involving Three Independent Variables and 

 possessing an Intermediary Integral," Camb. Phil. Trans., vol. xvi., 1898, pp. 191-218. 



VOL. CXCI. A. B 12.4.98 



