WITH MORE THAN ONE VARIABLE. 141 



Also then m = 0and ^ = 0. 

 \dxj \dxj 



Hence z has a determinate value, namely, - . 



a 



Example 3. Suppose z = , *- 



or + y 



Here, when x = and y = Q, we have 



D=' (I)-- -* -* 



_ 



Here the value of z is indeterminate ; but it will be found 

 that it is confined between the limits and 2, as may be 



2u 

 shewn by writing the fraction just given in the form 1 + : - *, 



A ~r U 



2u 

 remembering that - - ^ ^ s never greater than unity. 



167. In solving examples on this Chapter there are 

 various considerations which will abbreviate the labour of the 

 operations, as will be seen in the following case. 



, log(l+a; + # 2 ) + log(l-a:4-a; 8 ) 

 Find the value of 



sec x cos x 



when x = 0. 



The proposed expression takes the form - when x = 0. If 



we proceed in the ordinary way, we shall find after reduction 

 that the differential coefficient of the numerator is 



2x + 4a; 3 



l + x'+x" 

 and that the differential coefficient of the denominator is 



sin a; 



cos x 



+ sin x. 



