102 DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 



to x, supposing y constant; the resulting function is to be 

 differentiated with respect to y, supposing x constant ; this 

 last result is to be differentiated with respect to x, supposing 

 y constant. 



132. In considering the equation y=f(x), where we have 

 one independent variable, the student could be referred to 

 analytical geometry of two dimensions for illustrations of the 

 nature of a dependent variable and of a differential coeffi- 

 cient. See Arts. 35... 43. In like manner, if he is acquainted 

 with the elements of analytical geometry of three dimensions, 

 he will be assisted in the present Chapter of the Differential 

 Calculus. For instance, the equation 

 z = ax + by + c 



represents a plane ; x and y are two independent variables, of 

 which a is a function. Here 



dz dz . 



-j- = a, -j- = t>, 

 ax dy 



d*z d s z 

 and all the higher differential coefficients, -^, -j- 3 , ..., 



GLiC tljC 



vanish. 



Again, z = J(r*-x*-y*} .................. (1), 



is the equation to a sphere. If we pass from a point on 

 the sphere, whose co-ordinates are x and y, to a point whose 

 co-ordinates are x -f A# and y, we vary x without varying y. 

 If in this case the value of the third co-ordinate be z + &z, 

 we have 



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



2 



From (1) and (2) we can of course find ; and its limit, 



Cm3& 



dz - /> 



which we denote by -r- , will be -jr^ * _ . . 



The process is the same as if we had given 



* = V( f -A 

 where a is a constant ; from which we deduce 



dz _ x 



dx = J(a*-x*y 

 and finally put r 8 y 2 for a*. 



