INDEPENDENT DIFFERENTIATIONS. 105 



member becomes <^(x + h, y+k)<j>(x,y + k); by Art. 92 



d 



- becomes -j- + k 7 + 1?B, where 8 remains finite when 

 ax dx dy 



k = ; and v becomes v + kx, where a is a quantity which re- 



mains finite when k = 0, for it tends to - 7 as its limit. Thus 



dy 



jdu 

 =h~ x + hk-~ 



kx ............................ (2). 



Subtract (1) from (2) ; thus 



(f) (x + h, y + k) - (f> (x + h, y} - (]) (x, y + k) + (f> (x, y] 

 ,du 



= kk-^ + 

 dy 



Divide by hJc, and then suppose h and k to diminish inde- 

 finitely; therefore 



c'jc 



, = the limit when h and k vanish of 

 dy 



<j> (x + h, y + k) <f> (x + h, y} < (a;, y + k} + <f> (x, y) 

 h'k 



In a similar way, by first changing y into y + k, and after- 



j au 



cT 

 wards x into x + h, we can prove that ~ is also equal to 



the above limit. 



, du jdu 

 a -7- -f~ 



TT dx ay 



Hence -. = r- . 



dy dx 



135. The object of the preceding Article is to prove that 

 ; this is done by shewing that each of these 



dy dx dx dy 



quantities is equal to the limit of a certain expression. It is 



