1897.] Differential Equation of the First Order. 



283 



since m may be any of the integers 1, 2, ... n — 1, and the com- 

 plete and singular integrals are extreme types for which m has the 

 values n and respectively. 



All solutions of the equation /= are included among those 

 thus derived from any one system of functions 



Ui , u 2 . . . u n , 



but the system of functions is by no means unique, as is clear from 

 their mode of formation. 



§ 7. We may verify these solutions in the following way. 

 Considering z, p u p., ...p n as functions of x lt x 2 ... x n we have 

 at once, since f=0, 



$t + * .% + ?£. $t._ (r=1> 2...„), 



ox r ox r dz s=x ox r op s 



and also 



dui dz din *= n dp s dui _ dm (r = 1, 2 . . . n\ 



dx r dx r ' dz 5=1 dx r dp s dx r \i= 1, 2 ... n) ' 

 Thus 



Y d«H Jf = ( f y (dz_ _ \ d( Ui ,f) 



r=i dx r dp r iJ r ~i\dx r r 'Jd(z,p r ) 



and 



«=lr=l far d(Ps>Pr) 



'=» fdu i du l _du l ^ duA = + r ^ n /dz _ \ 8 (u ( , itj 



r =i \dx r dp r dx r ' dp r ) ' r =\ \dx r ^ r ) d(z,p r ) 



dp s d (m, Uj) 



s=n r=n 



+ 2 2 



(4 a). 



s =i r =i dx r ' d(p s ,p r ) 

 (In the double summations r, s must be unequal.) 



The first term on the right-hand side in each of the equations 

 (5 a) and (4 a) vanishes, since f= 0. 



Now let these equations be multiplied respectively by the 

 determinants of the matrix 



df du x du 2 du n 

 dp 2 ' dp 2 ' dp 2 '" dp 2 



df dui du n 



dp 3 ' dp 3 ' dp 3 



dp n ' 



