WITH MORE THAN ONE VARIABLE. 139 



Hence, putting h and Jc each zero, we have 



This result may also be found by examining the limit 

 towards which z tends as x approaches y, and then the limit 

 towards which this result tends as y approaches a. 



The next Article must be omitted until the student has 

 read Chapter XI. 



I \ Of* 77 I 



166. Generally, if z = V,, ' ^4 , and both numerator and 

 F(x,y}' 



denominator of z vanish for certain values of x and y, the 

 value of z is really indeterminate, and in fact depends upon 

 the arbitrary relation we choose to establish between x and y. 

 Suppose that x=a, yb, are the values which make z assume 



the form - ; and assume that y=ty (x}, where ty (x) is any 

 function the value of which is l> when x = a. 



Thus the numerator and denominator of z become func- 

 tions of x only ; and by previous rules for ascertaining the 



value of a fraction which takes the form - , we have 



x being put = a and y = b after the differentiations are per- 

 formed. This value is indeterminate, since ^'(x) is a function 

 which is quite arbitrary. 



But if MM and f -?- ) both vanish, 



\dxj \dxj 



or if (-\ and f-r-J both vanish, 



then the value of z becomes determinate. 



