104 CONVERTIBILITY OF 



and by differentiating -y- with respect to x we obtain 



Thus we see that in this example 



,du jdu 

 dj- <*j- 

 J- 32 

 7 -~ 



or, as we may write it, 



= f ^ (<\ 



dydx dxdy " 



"We shall prove in the next Article that this result is 

 universally true. Of the two modes of writing the result 

 given in (1) and (2) the second is the more commodious, but 

 it has the disadvantage of making the theorem which we 

 have to prove appear obvious to the student, because it sug- 

 gests to him that he is merely comparing two fractions. But 

 as we have already remarked, a symbol for a differential 

 coefficient is denned as a whole, and is not to be decomposed 

 into a numerator and a denominator. See Arts. 26 and 77. 



134. If u be any function of the independent variables x 

 and y, 



, du jdu 

 dx dy 

 dy dx 



Let u= <j> (x, y) ', change x into x + h, then by Art. 92, 



. / v / \.Tf Cf-lv . fv t It t 



we may therefore write 



where v is a certain function of x and y, which remains finite 

 when h = 0. In (1) write y+k for y; then the left-hand 



