180 CHANGE OF THE INDEPENDENT VARIABLE. 



we may use the formulae 



du _du dr du dO 

 dx ~ dr dx dd dx ' 



du _dudr du dd 

 dy ~ dr dy dd dy ' 



204. If the equations which connect x, y, r, 0, are given 

 in the form 



-*>,#, r,0)=0 ..................... (1), 



F t (x,y,r,ff)=0 ..................... (2), 



we may, in order to find the values of T- , -^, ~ , -^, 



required by the formulae of Art. 202, by successively eliminat- 

 ing y and x from (1) and (2), obtain explicitly the values of x 

 and y in terms of r and 6. Or, by Art. 189, we may find 



-^ and ^ from the equations 

 dd dd 



. 



\dd )^\dx) dd^\dy) dd~ 



fdF\ fdF^dx + (<*S}dy 

 \dd)^\dx) dd^\dy) dd 



and use two similar equations for -y- and -~ . 



dr dr 



205. Example. u=f(x,y), 



cc = r cos dy 



dx . .. du 



here -^ = r sm d, ja = r cos 



do at/ 



dx . dy . n 



-j~=cos0, -j- sm d. 



dr dr 



