DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 63 



the term involving x explicitly vanishes, as it should, and therefore we have 



JU.*- -S J&4..-JL_d, 



(1 + ) V (1 + tf p f- (1 + ,)3 q ' * 



2 ?" 2 



Hence 



the explicit value of IT, where A and B are arbitrary functions ; and thus the value 

 of iv is known. But, as already remarked, the value of 3 2 H/3^ 2 is sufficient for the 

 identification of the solution. 



37. After the preceding investigation, it is natural to inquire whether a solution 

 more general than that which is expressible in terms of u and tj, and the form of 

 which has been suggested by the theory, can be obtained directly from the original 

 differential equation. 



When the variables are taken to be x, y, u, where 





z = xp + yq + 



_ ,3 I 3 I j f" 



|? and 2 being functions of u, the equation + b -f c = becomes 



- _j_ Ip \- q - ) = 0, 



where 



7 = xp' -\- yq. 



Let be defined by the equation 



= /' + yq", 



so that M, >?, C are three variables functionally equivalent to x, y, z; and let us 

 inquire what solutions of the form 



are possessed by the foregoing equation. We have 



p' = qd, q' = pQ, 



p" = qff - p&\ q" = -p0' - 



so that 



