282 



Mr Dixon, On Lie's Solution of a Partial 



§ 6. The n equations (5) offer three alternatives : 

 (a) u„ u 2 ... u n are all constants ; 

 (/3) u„ u 2 ... u n are variable, but 

 d (u„ u 2 . .. u n ) _ 



(X^ , X 2 . . • X n ) 



so that they are connected by one or more relations ; 



<*> #T 



(r = l, 2...w). 



If we take (a) we may have an integral, the complete primitive, 

 which is the result of eliminating p„ p 2 ... p n from 



f= 0, u 1 = a 1 , u 2 = a 2 ... u n = a n . 



If we take (7) we may have an integral, the singular solution, 

 which is the result of eliminating p,,p 2 ...p n from 



f=0 # = o ^ = -^ = 0. 



J ' dp, ' dp 2 dp n 



For (/3), suppose the relations to be m in number, namely, 



<f>i (u 1} u 2 . . . u n ) = (i = 1, 2 . . . m) (6). 



Then only n — m of the equations (5) are independent — from 

 (4) we may deduce that 



£ du r d(f>i _ 5 dfa du r _ _ /r = 1, 2 . . . n \ 

 s =i 9# s ' 3p s s =i 9# s ' 8p t \v= 1, 2 ... m/ * 



Each value of i gives a set of n equations of which only n — m 

 are independent and by comparison with (5) we deduce the n — m 

 relations 



= 0. 



.(7). 



If (/3) leads to an integral it will be the result of eliminating 

 P,,Pv>-Pn from/=0 by means of the n equations (6) and (7). 

 The equation has thus n — 1 distinct types of general integral 



