The Rev. R. Murphy on a new Theorem in Analysis, 29 



CO 



Thus far the process is exactly similar to Laplace's, but since 

 <^' in the present case is not a function of y only, the remain- 

 der of the investigation becomes essentially different from his. 

 I put 



d"u „ du f/ /n du\ d^ frt du\ _ 



and now proceed to find the value of the general symbol P„ „, 

 which is a function both of x and y. 



Before differentiating this equation relative to x let it be 

 observed that 



d' P^.n _ p/ d.Vm^n d^ 



dx -'^"'."■^ dy • dx 



_ p/ , ^^»Pm,» ., d y 





and also that 



da 



d du _ d du _^ d / ,du\ 

 dx'da^'da'dx d a\ da) 



by equation (1.); from both of which it follows that 



