( 174 ) 



CHAPTER XII. 



CHANGE OF THE INDEPENDENT VARIABLE. 



196. IN Art. 60 we have shewn that 

 = 



dy 



and in Art. 63 we have shewn that 



dy = dy dz_ ^ ^, 



dx dz dx" 

 and we now proceed to some extensions of these formulae. 



Given x and y, both functions of a third variable z, it 

 is required to express the successive differential coefficients 

 of y with respect to x, in terms of those of y and x with 

 respect to z. 



We have ^ = ^ d ^ bv (ft 



dx dz dx Dy Wl 



z - 



dz 



dy dy 



d'u d dz d 'dz dz . , . 



Hence TT = -y- -r = T T"-T~ ^ ( 2 )> 

 dx dx dx dz dx dx 



dz dz 



d*y dx d*x dy 

 dz 2 ~dz ~ dz* dz dz_ 



/flte\" " * dx 



(dz) 



