LAGRANGE'S THEOREM. 113 



Suppose, according to this law, that 

 d n u d"' 1 



n du 



then 



- W 



which shews us that the expression for -7-^1 follows the same 



d n u 

 law as that for -5-^ . Hence, since the law has been proved 



to hold for -y-2 and -7-3, it holds universally. 



CtOO CLOG 



d n u 

 In -y-^ we are to make x = a/fer the differentiation has 



^"i/ 

 been performed; but when we transform -j-^ y by the. above 



formula, into an expression involving only differential co- 

 efficients taken with, respect to z, we may put x = before the 

 differentiation, since x is to be considered as a constant in 

 differentiating with respect to z. When as = 0, 



,, ,. ,,. N 



therefore -j-^ = -=^- =/(), 



dz dz 



T. D. C. 



