Differential Equations of the Second Order, 37 



If a lt a 2 be the roots of 



= Ra 2 + Sa-fT, 

 we shall have 



« du du _ e^ «fo 



0= 1 ff lT J 0= -= a 9 -7- 



ff # #7/ fife' 2 £&/ 



Hence if the integrals of the equations 



0=zdy + a x dx } = dy + a 2 dx 



be respectively 



co L — const., co 2 — const., 



co x or o> 2 may be substituted at pleasure for u in (2). 

 The equations which succeed (4) may be written 



n dk , dA. n . 



&c. &c. 



The integral by Lagrange's method of any one of these equa- 

 tions, as (5), will be of the form 



A l = L. % (a>)+M, 



where L, M are functions of on and ?/ ; % is arbitrary ; and co = 

 const, is the integral of 



= dy — adx. 



Hence if these values were substituted in (2), and we then put 

 </>(&) = a constant, as we may do, we should have the given 

 equation (1) satisfied by an integral of the form 



where F is a definite, and ^ an arbitrary function ; from which 

 it would inevitably follow that the given equation is soluble by 

 Mongers method, whereas by hypothesis it is not so soluble. 



We must assume, therefore, the arbitrary function in each case 

 to be zero ; so that, in order to find A, A } , &c, we may take the 

 equations 



