96 CERTAIN IMPORTANT FUNCTIONS 



etc., when V q is a continuous function of e q commencing with 

 the value V q = 0, or when we choose to attribute to V q a 

 fictitious continuity commencing with the value zero, as de- 

 scribed on page 90. 



If we substitute hi these equations the values of V p and e^p 

 which we have found, we get 



^= r/il /^ < - <) <* ^ ' (304) 



(305) 



where e^ c?e g may be substituted for d V q in the cases above 

 described. If, therefore, n is known, and V q as function of 

 p V and e^ may be found by quadratures. 



It appears from these equations that F"is always a continu- 

 ous increasing function of e, commencing with the value V= 

 0, even when this is not the case with respect to V q and e q . 

 The same is true of e^, when n > 2, or when n = 2 if V q in- 

 creases continuously with e q from the value V q = 0. 



The last equation may be derived from the preceding by 

 differentiation with respect to e. Successive differentiations 

 give, if h < } n + 1, 



d h Vjd<? is therefore positive if A < J n + 1. It is an in- 

 creasing function of e, if h < Jw. If e is not capable of 

 being diminished without limit, d h Vjd^ vanishes for the 

 least possible value of e, if h < \n. 

 If n is even, 



n 



(307) 



