CHANGE OF THE INDEPENDENT VARIABLE. 175 



d' l y dx d*x dy 



dz* dz dz* dz , ,,. 

 V (1). 



d*y dx d*x dy 



d 3 y d dz* dz dz* dz dz 



Again, V* = -; 5 g . -r 



dx 3 dz /dx\* dx 



dz 



fd*y dx__d*x,dy\ /dx\ a _ /dx\* d*x fd*y dx d z x dy 

 _(dz 3 dz dz 3 ~dz)\dz)~ (dz) d \f* fa ~ dz* ~dz 



dx\* ~" dx 





dz) 



fd s y dx d 3 x dy\ dx d 2 x fd^y dx d 2 x dy\ 

 \dz 3 dz dz 3 dz) dz dz* \dz* dz dz* dz) 



. ., , . , , y 



Similarly we might express -rj , 



This process is called " changing the independent variable 



72 



from x to z-" since in -T-| the independent variable is x, 



doc 



d*y dx d*x dy 



, , . ,, dz 3 dz dz* dz ,, . , , 



but in the expression , 8 the independent va- 



/ doc \ 



(dz) 

 riable is z. 



197. Suppose in the preceding Article we put z = y. 



We have $-1. ^? = 0, &-,.. 



dz dz dz 6 



dx dx d*x d 2 x 



