solving Equations of partial Differentials. 24 1 



d. 

 effect, and 



c 

 since — does not contain y, the symbol prefixed to it has no 



bx _d_ 



a ' a ' 

 But the integrals of these two expressions with respect to x 

 are not the same. This may be explained by writing instead 



c d cy — — 



°^ ~a 9 ~d~~ ' ~a 9 w ^ cn > Dv prefixing e « dy % j s changed 



t? * c Q+t) _ j <yt# 



Hence the equation 



t . 6 g d b x d 



(e a d y Z ) = f « 

 / 



is the same as 



da? * V'^ + a J- - -j^ b 



(the first side representing the total differential coefficient with 

 respect to z). By integration, 



$ 



r ; bx \ c (y+ir) . *', x 



Finally, by multiplying both sides of the equation by 



— bx d 



s a dy 9 that j Sj changing functions of y into functions of 



*-— ' 



4> (x,y), or2 = -2 +/( fl| y_ft a!: ). 



Hence it may be seen that the essential part of the method 

 consists in making each member of a partial differential equa- 

 tion a total differential coefficient with respect to one of the 

 variables. 



The most general class of equations of the first order, which 

 can (as far as I am at present aware) be solved by this method, 

 are those which fall under the form 



Third Series. Vol. 11. No. 67. Sept. 1837. 2 I 



