DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 19 



With this valuo, the first set gives 



P 

 that is, 



(a\ - &',) (a r x + Ifl + c, 2 ) = (a t - 6j) (a\x + b\y + c\~). 



Since x, y, z, are independent variables, this can be satisfied only if 



a\ l\ c\ 



"i^^i" >Say ' 

 so that 



a- = kp, 



Similarly, the second set gives 



a 7 a' 7' 



, , 



X ^ 



leading to 



' /^ 7' 



= > = v = K > sa >'' 



so that 



^ = K\. 



The first set of equations can now be replaced by 



f + r) + C = 0, 



! 6j /D^ == 0, 



f + 17 + L + & t i? + c^ = 0, 



^ + * K + 6,17 + c T = o. 



Hence k = 1, and so cr p ; also 



so that 



g <L_ _r_ 



Similarly, by the other set, we find = '], and so /x, = X ; and 



As the values of ^, >;, , must be the same in the two determinations, and as the 

 variables x, y, z, are independent, we have 



D2 



