2 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



adopted, viz., the construction of subsidiary equations in which the number of 

 expressed independent variables is less by unity than the number in the original 

 equation; consequently the number is two. The subsidiary equations thus are a 

 set of simultaneous partial differential equations in two independent variables and 

 a number of dependent variables. 



It then appears that such differential equations of the second order having no 

 intermediary integral divide themselves into two classes, discriminated by- the 

 distinction that, for the one class, what I have called the characteristic invariant, viz. : 



_3F , 3F , 3F 3F 3F 3F 

 w 2 r- -f pn -5- + q~ -57 p 3 -- 2 &2 + a" = > 



1 3 L L 3/t 1 cb ay * of tic 



can, qua function of p and q, be resolved into two linear equations, and. for the other 

 class, the characteristic invariant is irreducible. 



The first section of this paper is devoted to the general theory of equations of the 

 second order, so as to construct systems of equations subsidiary to the integration ; 

 occasional paragraphs in the other sections develop the general theory in connection 

 with particular types of equations. The second section is devoted to the integration 

 of equations whose characteristic invariant is reducible : a method is devised whereby 

 the integration can be effected in those cases where the integral can be expressed in a 

 finite form without partial quadratures ; and various examples are given in elucidation 

 of detailed processes. The third section is devoted to the integration of equations 

 whose characteristic invariant is irreducible ; and the method is applied with 

 considerable detail to some of the equations that are important in mathematical 

 physics. 



It should be added that the case of three independent variables has been selected 

 for detailed treatment, as being that of complexity next greater than the case of two 

 independent variables, the general theory of which is fairly complete. An inspection 

 of the results, as well as of the processes, will make it manifest that, for many of 

 them, generalisation to the case of n independent variables is immediate.* 



SECTION I. 

 General Theory, - 



1. Let the number of independent variables be three, and denote them by x, y, z. 

 Denote the dependent variable by ?;, and write 



dv 



- = n, 



oz 



--'' 



* See a note at the end of the paper. 



