284 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



{/>'d + g' S + r') {d + a d^ + b) x = {p d + q di + r) (d + a' d^ + If) z 



or (P-P') r+{P''' + 9-P'"-9')--r-;+{p''' + 9°' + ^ 



d t- d t'- 



-p' b-q'a-r')j-^+{qb' + ra'-<;/b-r'a)-^^ 

 + rb'-'/b)x = 



which coincides with the general form Cor. 2, Ex. 3, Class 1. 



Section III. Partial Differential Equations. 



21. In order to effect the solution of partial differential equations in which 

 the operation with respect to x is totally independent of the operation with 

 respect to i/, we must distinguish the operation of differentiation in the two cases 



by different symbols. Let d stand for ^, d for ^ : then in solving the equation 



with respect to d, we may treat 5 as a constant, and uice versa. 



F 1 i^_fti^ = 



dj:- dy^- 



Write this equation (rf'-6 2'')r = 0: In this form it coincides with Ex. 1, 

 Class 1, and its solution is 5=Ae'''^^. 



Now A is an arliitrary function of y ; call it f {y) : then s=e''^-''/{y), where 

 gb^xi represents an operation on /(»/). 



But since /{>/-t-/c)=/y + -jj k + 



={l+k8 + &c)/{y) 



it is evident that z=f{y + lPx). 



diz dz 



This equation may be written (d^-aS) z=0. 



Now /'°°rfw«"'"~'''^=\/7r (Gregory's ^.ramjofe«, p. 499.) 



z's/'K— j _^do^e .e /(!/; 



7-* 



