SIMULTANEOUS EQUATIONS. 163 



j J dx \ dt ) dx ' 



from which n equations we can determine the n quantities 



dy dz dt 



dx' dx' " " dx' 



188. Suppose <f> (x, y, z] to be the only equation con- 

 necting three variables, so that z may be considered an im- 

 plicit function of the two independent variables x and y : it 



fi Z CM Z 



is required to find -y- and -y- . 

 dx dy 



dz . 

 By -y- is meant the differential coefficient of z with respect 



dz 

 to x, supposing y constant, and by -7- the differential coefficient 



of z with respect to y supposing x constant. Theoretically 

 we may from the given equation find the value of z in terms 

 of so and y and then effect the differentiation by common 

 rules; (see Art. 131). But to avoid the difficulty of solving 

 the given equation we adopt another method. Suppose y 

 constant, so that we have two variables x and z, and let u 

 stand for (f> (x, y, z), then by Art. 178 



/du\ /du\ dz _ , , 



where ( ] stands for the differential coefficient of u taken 

 \dxj 



on the supposition that x alone varies, and (-y-J for the dif- 

 ferential coefficient of u taken on the supposition that z alone 

 varies. Similarly 



/ du\ fdu\ dz _ , 4 y. 



\dyj \dzj dy 



dz -. dz 

 Equations (1) and (2) determine ^ and -r- . 



We may determine -A and -A by the method of Art. 180, 

 J ax dy 



M2 



