INDEPENDENT DIFFERENTIATIONS. 107 



of x only. Thus when y is changed into y + k the value of 

 6 will generally change. This does not affect the preceding 

 proof, because it was not necessary there actually to find the 



value of -j-; but the assumption that does not change 



when y changes has rendered some proofs unsound which 

 have been given of the proposition in Art. 134. 



136. The important principle proved in Art. 134 is 

 enunciated thus : " The order of independent differentiations 

 is indifferent ;" or it is referred to as the principle of the 

 '' convertibility of independent differentiations." It may be 

 extended to any number of differentiations; so that if a 

 function of two independent variables, x and y, is to be dif- 

 ferentiated m times with respect to x, and n times with respect 

 to y, the result will be the same in whatever order the dif- 

 ferentiations be performed. In proof of this we have only 

 to apply the theorem of Art. 134 repeatedly in the manner 

 shewn in the following example. 



, d?u d B u 



To prove that - 



d?u dii d.c , i *. ... 



-y-v -,- = ' , by definition. 

 dy dx cty 



. . 

 ' , by Art. 134, 



r - , by definition, 



dy dx dy 



d*v . f _ du 

 dy dx ' dy ' 



-3^- , by Art. 134, 

 dx dy J 



d*u 

 dx dy* ' 



