m 



we have also 



= «M + /3M' + yP', ^*v^AC^^V^■ 



= »M' +fiN +yN' , 

 = «/* + /3N' + yP; 



and therefore /^tuiuo ..^ijii^tj^- -.> - '4-l»:. a,- u« 



Now, if we put ,,,^ ..,3nov.;,„b:- i./ui ; '■■ o i Kmr-e- -; ■ riu'; nibil :: 

 we shall have 





w / >. • , d* «/3 «y 



and therefore by (Q), 



an equation in which ix, hy, iz, are independent, so that it is equi- 

 valent to six separate expressions, for the six partial differential coef- 

 ficients of V, of the second order : and these expressions may easily 

 be shewn to coincide with those marked (0). And on similar prin- 

 ciples we can investigate the relations between the partial differential 

 coefficients of the functions V and W, for the third and higher 

 orders. 



VOL. XVI. O 



