162 DIFFERENTIATION OF 



fdv\ /du\ _ fdu\ fdv\ 



dz _ (dx) (dy) (dx) (dy) . . 



dx~ ~ /dv\ (du\ _ fdu\ fdv\" 



(dz) (dy) (dz) (dy) 



186. By differentiating equations (1), (2) of the last Article 

 with respect to x, we obtain 



* d * U d * U 





 dx 2 / \dxdy dx \dxdz dx \dif\dx 



/ d*u \ dy_ dz fd^u\ /dz\* fdu\ d 2 y fdu\ d*z _ 

 (dy~d~z) TxTx* \M) (dx) + \dyl d? + (dz)dx ri ~ 





 dx 2 J \dx dy) dx \dx dz) dx \d^J \dx 



( d*v\ dj^dz_ (d*v\ (dz\ fdv\ d z y (dv\ d*z _ 

 \dy dz) dx dx + (dz 2 ) (dx) + (dy) dx* + (dz) ~dx* ~ ' 



y z 



From these equations we can deduce -~ and -y-^ , which 



may also be found by differentiating equations (3) and (4) of 

 the preceding Article. 



187. Suppose we have n equations connecting n + 1 vari- 

 ables x, y, z, ...... t. Let the equations be 



FI (x, y, z, ...... t) = 0, say u^ = 0, 



^ 2 (x, y, z, ...... *) = 0, say w 2 = 0, 



F n (x, y, z, *) = 0, say w n = 0. 



From these equations all the variables but one may be 

 considered functions of that one. If x be the independent 

 variable, we have by differentiation, as in Art. 185, 



= I * I 4- ( l } -&- 4- 



^dxj \dy ) dx \dz ) dx ^ ^ V dt J dx ' 



w / """"% \ . I "'"'2 1 Wff |^ / WWg > / . . / // \ <Z6 



,G?a;/ (dy)~dx (dz. 



