FUNCTIONS OF TWO INDEPENDENT VAEIABLES. 161 



Art. 181, and the equation which may be obtained from (3) 

 will be expressed respectively as follows : 



= 0, 



" + 3 (</>;+ 1^ /+ </>y = o. 



185. Suppose the two equations 

 /(, y, s) = 0, 

 ^(a;,y, ) = 0, 



exist simultaneously, in which cc is the independent variable 

 and y and dependent variables. From the two given equa- 

 tions we may eliminate z, and thus find an equation connect- 



ing y and x, Hence -j- may be determined. Again, from 

 the two given equations we may eliminate y, and thus find 

 an equation connecting z and x, whence -7- may be deter- 



mined. In cases where the elimination is tedious or imprac- 

 ticable we may proceed thus. 



Let u denote f(x, y, z) and v denote F (x, y, z}. Since y 

 and z are functions of x, the differential coefficient of u with 

 respect to x is, by Arts. 172 and 174, 



fdu\ fdu\ dy_ /du\ dz 



and since u always = 0, we have 



d 

 d 



dy fdv\ dz 



- ( du \ 4- ( du \ d V-+ ( du \ dz 

 - ) + 3) dx + &) dx 



. 

 Similarly, = 35 + $ 



from which we find 



/du\ fdv\ _ fdv\ fdu\ 

 dy_ _ \dx) \dz) \dx) \dz) , 



dx (du\ (dv\ _ (dv\ (du\ " "* 



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

 T. D. C. M 



