DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 25 



1 2. It will be noticed that in these examples the equation A = (which is of the 

 second degree in p and q) is resoluble, so that it can be replaced by two linear 

 equations, and that the latter have, in turn, been combined with the other equations 

 of the system. Now, these equations are of LAGBANGE'S linear form, and their 

 integral is such that some combination 6 of variables can be an arbitrary function 

 of some other combination ^. Further, it has appeared that the integral of the 

 subsidiary system (other than the original equation) is such as to make some 

 combination i/> of the variable quantities a fxinctional combination of 6 or <f) at the 

 same time that and < are functionally related, so that, as the functional forms are 

 arbitrary, we infer that 



t = v(0, #, 



where V is arbitrary, is an equation that can coexist with the original equation. 

 Hence it is to be inferred that when A = is a resoluble equation, that is, can be 

 resolved into two equations linear in p and q, arbitrary functions of two arguments 

 occur in the most general integral equivalent of the original equation, 



13. The converse also is true, viz., if an integral relation involve at least one 

 arbitrary function of a couple of distinct, arguments and be equivalent to a partial 

 differential equation of the second order, and not to an equation of order lower than 

 the second freed from arbitrary functional forms, then the, characteristic invariant 

 equation can be resolved into two linear equations. (The number of independent 

 variables is, of course, presumed to be three.) 



Let and 77 be two independent functions of x, ?/, z, so that not more than one of 

 the three quantities 



i^y *).,; %,n-, &, &>?.r ~ "?.:>, 



can vanish. As regards the arbitrary function of and rj, let it occur in the 

 integral equation in the form 



where <f> denotes the derivative of the arbitrary function of highest order occurring 

 in <s>. Then we have 



a 



a = || {< n . 3 + 2< lz f,77. c + < 22 7?, 3 } + derivatives of < of lower order, 



VOL. CXCI. A. R 



