OF A FUNCTION OF FUNCTIONS. 153 



Now ( -T-l is itself a function of y and z. If we denote it 



A y . 



by v its differential coefficient with respect to x will be 

 dv\ dy dv\ dz 



which may be written 



\rfyV -dx (dz dy) dx ' 



The differential coefficient of -/ with respect to x is ~ 



dx dx* 



Proceeding in the same way with the term 



A dz 



7-fi 1 



and remembering, (Art. 134), that 



fjt*\ 



\dz dy) \dy dz) ' 

 we have 



d*u fd 3 u\ fdy\* f d?u \ dy dz fd*u\ fdz\' 

 -^- I -= - I I j ) + 2 I -j -y- I -~ -j f- -^ 9 I - 

 \dxj \ay dz/ ax ax \dz~J \dxj 



du\ d?y /du\ d?z 



~7 I ~jf 2 ' I ~7 J ~7 2 



T , , dz d z 



If z = x, we have - 7 = 1, -y-^ = ; thus 

 dx dx 



' z u\dy^(c 



7 I I * I 7 7 I 7 T^ I7VI 17 i7V - 



aaj/ \dydxj ax \dx J \dyj dx 



177. Hitherto in this Chapter we have given methods 

 which, although often convenient, are not absolutely neces- 

 sary, as in all cases, by effecting the required substitutions, 

 we may obtain an explicit function of x, and differentiate it 

 by known rules. But the case we now consider is one in 

 which a new method is frequently indispensable. 



Let < (x, y) = be an equation connecting the variables x 

 and y : it is required to find ~ . If the given equation can 



