DIFFERENTIAL EQUATIONS OF THE 7i'* OrDER. 15 



Giving in either of the equations (27) successively two values i 

 and ; to the index i and eliminating the m (or m,) occurring in these 

 equations, we obtain 



n: a; = -or ^? (^s) , 



including all the relations, which the two integrals F= and F,. = 

 must satisfy, if independently of each other / and j receive all integral 

 values from i or j = 1 to i ov j = n — 1. 



Taking the last equ. of system (27) together with what it becomes, 

 if s be put for r and / for i, we get by elimination of m the form of the 

 relations, Avhich must be satisfied by three first integrals F=0, F,.~0, 

 F, = of the equation (12), viz.: 



r r Z>"->A" = r C d'" a'-' . . . (29), 



3j,;i— 1 ^n—l.ii i "—V ^/1—1,11 ^O n— 1 J ^— 'i ^ '^ ' 



which for s = r reproduces (28). 



The equations (28) and (29) we, therefore, may regard as complete 

 identities, satisfied by any two or three of the first integrals of (12). 



§. 5. 



Properties of n distinct first integrals of the differential equation (12). 



Firstly let us have the n equations 



i^=0, i^, =0, i^;-, = .... (30), 



involving x , y , : and the partial ditferential coefficients of : with respect 

 to x and y up to the {n — 1)" order, the equations (30) being ?ion-cstricted 

 to be first integrals of a given partial ditferential equation ; we proceed to 

 investigate the general conditions, that must hold, in order that the values 

 of r„_i.o, -„-2.1, ••• , ~».n-i , obtained from (30) may render the equa- 

 tions (15) integrable. 



To the notations, used in the equations (26) and (27), we add the 

 following 



(S=ä' CD— §■)=«"". © = '■" 



where the parentheses are used to denote, that we ditferentiate , for instan- 

 ce, with respect to .r , as far as it is involved in F or F,. both explicitly 

 and implicitly through ~ and its partial differential coefficients of all occur- 

 ring orders except the highest. Lastly we also put 



