370 Mr. B. Williamson on the Solution of 



that the operation x-j- + y -j- is transformed into t -r-, and ac- 

 cordingly the solution of all partial differential equations of the 

 form (}){xDi. +_?/D^)z = V is immediately reducible to that of the 



differential equation ^ U -i- W = V. More generally^ if we make 

 w = f, 



tJ = Vt , 



then the operation axD,i + bijDy is transformed into tJ)t. Ac- 

 cordingly we can solve all partial differential equations of the 

 form (f> . {axDj. +6?/D,^)3 = V whenever the solution of the corre- 

 sponding equation ^lt^)2 = Y' is known. In order to exem- 

 plify the advantage of this method of solving such classes of 

 partial equations, I will apply it to a few examples. 

 Ex. 1. rx^ + 2sxi/ + ty^ + a%2Z = 0, 



where Ucf=x^f—, 



^ •' X 



it is immediately seen that 



o ^ o { d d\{ d d .\ 



r^' + ^sxy + tf=(x^^+y^^)(x^^+y^-l)z. 



Accordingly the proposed equation is transformed into 



the solution of which is _ _ 



z=ylr^v . cos at \^fv + '^^v.smat \/fv, 

 or 



^• = '(|rji^COSfl! ^M2 + 'x|r2-sin« -/Wg' • ' * C^^) 



Ex.2. rx^-^2sxy-^ty'^ + a%-.^z=0, 



The solution of this equation evidently depends on that of 



xVdi) "^ ~^r~^' ^^^ accordingly is by (6), 



z=x^''-cosa Vu-<i-^x-^ci — s\aat/u-2' • • (17) 



Ex. 3. rx"^ + 2sxy + ty'^—2n{px + qy) + ahic^^O. 

 The solution of this equation is immediately seen from (1) to be 



z=L\--a-^\ U^i^cosa -v/^^ + 1/^2^ sin ^VwaJ. (18) 



