DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 103 



On the other hand, if we pass from the point (x, y) to 

 a point having x and y + A?/ for its co-ordinates, we have, 

 as before, 



(3). 



Now, in (2) and (3) we have used A^; but we do not 

 mean that the value attached to the symbol is the same in both 

 cases. If there were any risk of error by confounding them, 

 we could use biz in (3), or something similar. But in fact 



we only use (3) to assist us in forming a conception of -y- ; 



and since we look on -y- and -y- as whole symbols not admit - 

 dx dy 



ting of decomposition, the question can never occur, " Is the 



dz in -r- the same as the dz in -^ ?" 

 ax dy 



133. When u is a function of two independent variables. 



^ j-jy j.- i 01 du du d' 2 u d*u 

 the differential coefficients -y- . -y-, -^ a . -? y- , ... are 



ax dy dx dxdy 



often called "partial differential coefficients." Each of these 

 differential coefficients is obtained by one or more operations, 

 every operation being conducted on the supposition that only 

 one of the possible variables x and y is actually variable. 



T* 



Let us suppose for example that u = tan" 1 - ; then 



7 



du x 



d*u _ 2xy 

 dy* (x* + y r f ' 

 and so on. 



By differentiating -y- with respect to y we obtain 



..du 



dx x*-y* 4 

 dy 



