OF AN IMPLICIT FUNCTION. 155 



The limit of fr(a^--~.y/-yv.^ when Aa , diminishes, 



/\ /yi 



is the differential coefficient of <j> (x, y) with respect to x, 

 formed on the supposition that x alone varies, and if we put u 



for (j>(x, y), this limit may be denoted by f-r- ] . 



mi v *. f <b(x + Ace, ?/ + A?/) <f> (ce + Ace, y] , , . . 



The limit of -^ -2-* p- -2-i would, if 



Ay 



A# remained constant, be the differential coefficient of 

 <j)(x + Ace, #) with respect to y, formed on the supposition that 

 y alone varies. But as Ace diminishes without limit when 

 Ay does so, the limit is the differential coefficient of u with 

 respect to y, formed on the supposition that y alone varies. 



It may be denoted by f-j- I . 

 \dyJ 



A 11 dy 



The limit of -r 3 - is -?-, Hence finally 

 Aa; cc 



fdu\ dy fdu\ _ 



\dy) dx \dx) 



179. For example, suppose a 2 ?/ 2 + J 2 ce 2 a 2 i 2 = 0. 

 Here u = a 2 y* + Vx* - aV, 



fdu\ = 2 p x 



therefore a*y -^ + b s x = 0, 



?y 6 2 ce / n 



therefore -^ = =- {*) 



dx ay 



Since y = - \/(a 2 - 2 ) f rom ^ e given equation, we obtain 

 a 



directly 



dy_ _ __ bx ,~\ 



dx~ aa 2 - 2 '" 



When in (1) we substitute the value of y in terms of x, 

 the result agrees with (2). 



