Demonstrations of certain Theorems c/Lagrange $ Laplace. 25 5 



And, since 



d n .(xy) = d n x.y + n d~ l x.dy + n.?~- d^x. 

 we obtain the following results : 



6 P (p l )*q l 



+ 4. q lPa p" 



+ 



&c. &c. 



Hence the theorem is obtained. 



2ndly. Demonstration of the theorems of Laplace and La- 

 grange. 



Lemma. Let t be a function of y and = z + xf(y\ where 

 z and x are independent variables ; also let , be any other 

 function of y : 



then 



Case 1st, when n 1. 



dt dt dy 

 For, -=-- = -r- . ~/~ and =/(#) + x 



dy 

 hence 



. --; 

 dx 



dy dy 





 hence, -7 - 



" i* . 



But, 



