164 DIFFERENTIATION OF IMPLICIT 



or by that of Art. 181. If we adopt the latter method, the 

 two equations we obtain are 



dxdz dx \cUf \dx \dz I dx* 



/<Fu\ / <Fu'\ dz_ /o[V\ /dz\* fdu\ <Fz _ 

 (dy*J 4 \dydz) dy + (dz 2 ) \dy) + \dz) df ~ 



d*z 

 We can obtain an equation for finding . , by differen- 



tiating (1) with respect to y, or by differentiating (2) with 

 respect to x. We thus deduce 



/ d*u \ t d*u\ dz_ ( d*u \ dz_ /d*u\ dz_ dz 

 \dxdy) \dzdx) dy \dzdy) dx \d&) dy dx 



\dzj dydx 



189. Suppose we have two equations connecting four 

 variables; for example, 



f(v, x,y,z}=Q, say M, = 0, 

 F(v, x, y, z) = 0, say u 2 = ; 



from these equations v and z may be considered functions 

 of the independent variables x and y. If we eliminate v we 



obtain an equation connecting 2, x, and y, so that -y and -j- 



L*)G (JuU 



may be obtained by the preceding Article; and similarly 

 if we eliminate z we may find -y- and -j- . Or we may pro- 

 ceed thus: from the equation u^ = we deduce, by Art. 174, 

 du\ (du\ dv fdu\ dz_ 

 )^\dv) dx^\dz) dx~ 



and from the equation w 2 = we deduce 



/du 3 \ (du\ dv rdu\ dz _ 

 \dx) + \dv) dx *\ds) dx ' 



from which 3- and -y- can be found. 

 dx dx 



