nil -TKRENTIAL CALCULUS. 109 



944. 1 laving given x + y = X, y = XY ; prove thai 



d'u <?n _ d'u d'u du 



~~ 



'J 1 5. If 2x = r (c* -f e-*), 2y = r (c* --); 



d'u d'u _ (Fu _ \_ (Fu 1 d,t 

 dtf~ 7t* = dr* ? dt> a + r dr' 



946. If x = ^ -f -*, y = c 

 /^_rfV\ 



" 



917. If c's 



d'u 9 d'u d'u du. 



y s +r gn 



18. If u be a function of the four independent variables 



^s' ^ *4 ; aud if 



x a r sin ^ in <^, x t = r sin ^ cos <, 

 a5 a = r cos ^ sin \f/ t x 4 = r cos d cos \j/ 



\vill 



ft ,/',/. rf\ dSi 1 <J f t* 1 



~ c/r ^ r' ^ ^in 1 

 1 ^^' 2 



949. If r, y, s be three \ -1 by one equation ; 



prove the formula for changing the di-jirmh-nt variable iron, 

 * to x\ 



dx __ 1 f/r d*x r 



<T _ qr-pt d*x _ p't - 2^7* -I- ffr 

 dy ~f~ 9 ~~jt ' 



