FUNCTIONS OF TWO INDEPENDENT VARIABLES. 165 



Similarly, from w x = and u z = we deduce 



fdu\ fdu\ dv^ + fdu\ d^ = Q 

 \dy) \dojdy \dzjdy 



/du\ 



~ , which = 5-r , assumes the indeterminate form - . 

 dx fdu\ 



from which -j- and -y- can be found. 

 dy dy 



In such equations as those in the present Article it is 



., df df dF du 



very common to wnte -f- , -y- . -7- , . . . , to denote -r 1 , 



dy' dv' dy' dy ' 



dv dy' 



190. If values of x and y which satisfy an equation u = 

 involving x and y, also make (-T-) and (-T-J vanish, then 

 /e??A 



w 



/Jw\ 



w 



If we apply the method of Art. 145, we have 



/du\ fd*u\ f d?u \ dy 



,, -,. ., /s \dx) , .. . \rfa; 2 / \dxdy) dx 

 the limit of . - = the limit of >*, * . 7 , 



fdu\ / du \ fdu\ dy 



\dy) \dx dy) \dy z j dx 



the numerator and denominator of the second fraction being 

 respectively the differential coefficient of f -5- J and of f y J 

 with respect to x. 

 We have then 



| ___ ] -4- ( -^ I ii 



dy_ _ _ \dx*J \dx dy dx 



tx ^ ~ 7H?u \ (d*u\ ~dsy 



\dx dy) \dy 2 ) dx 



