DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 79 



also 



and, as 

 we have 



that is, A behaves as a constant with regard to the operator -p ' --- \- <i - The 



L il.i: J <l>j 



required equation will therefore be satisfied it' 



But it is necessary that the other three equations 



1_ ill 1 din _ I tin fa 



<J dlf, ' ' 1' du ' r. ' du du ' 



should also be satisfied ; that they lead to no new condition for c/> can bo seen a.s 

 follows. We have 





so that 



On account of the equation satisfied by <j>, this is equal to 



