1897.] Differential Equation of the First Order. 



291 



and eliminating a x . . . a n by means of the equations 



^ + b r ^-=0 (r = 1, 2... n-1). 



da r oa n 



We still have a 1 = u 1 , a 2 = u 2 ... a n = u n and the values of 

 bi>b 2 ...b n are solutions of the auxiliary equation. Thus the 

 In — 1 necessary solutions of the auxiliary equation are u l> u 2 ... u n 

 and the ratios of 



dU dU dU 

 du x ' du 2 ' ' ' du n ' 

 where U denotes 



F(z, X\, 0B 3 ... X n , Uj, u 2 ... u n ). 



It is easy to shew that these are independent, and thus we are in 

 a position to write down the general solution of the auxiliary 

 equation. 



§ 16. It would appear at first sight that the determination of 

 the forms of the arbitrary functions to satisfy limiting conditions 

 would involve the solution of differential equations, since the 

 derivatives of the arbitrary functions occur in the solution ; this is 

 however not the case. 



Suppose for instance that the value of z is given when x x = 0, 

 say z = t(w a , x 3 ...x n ). 



The values of p 2 , p 3 ...p n are therefore given when #i = 0, for 



then 



*-£(•■-*«...•> 



These values may be substituted for x 1} z, p 2 , p 3 ... p n in the 

 equations 



f—0, Wj = «! , u 2 = a 2 ... u n = a n , 



and the n quantities^, x 2 ... x n may be eliminated. 



Thus one or more relations will be found among a u a 2 ... a n 

 and these are what were required. 



More generally if the solution is to include all systems of values 

 that satisfy the equations 



%i {?, x l} x 2 ... x n ) - 0, x* ( z > x l} x 2 ... x n ) = 0, 



= 0. 



