350 G. H. KXIBBS. 



A. If p be positive 



In order that a point of inflexion may be between 



and x=U 1 -P must be > o and < 1 

 up 



(a) if n be also positive (curves in heavy cont 



lines), 1 - p must be > and < «p 



or p must be < 1 and > l/(« + l) 



(b) if n be negative (curves in thin 



1-p must be < and > np 

 or p must be > 1 and < l/(« 



B. If p be negative. 



1 (^P) must be > Oand <1 



(the signs of inequality will not change, since 1-P 

 must be a positive quantity). 



(a) if n be positive (curves in broken heavy lines), 

 p must be > (which is impossible as p is positive) 



(b) if n be negative (curves in thin broken lines), 



p must be < and 1-p must be > np 



or p must be < and p(n + l) must be ^ 1 



Since p is negative the first condition is always fulfilled. 



Since p(» + l) must < 1 

 p must be < l/(» + l), if n + 1 is positive, or if 7^ > -1, 

 and p must be > l/(»+l), if n 4-1 is negative, or if < - 1. 

 It may be noted that if only the first mentioned of the 

 conditions above specified be fulfilled in any of the par- 

 ticular cases (a), (b), there may be a positive point of 

 inflexion. Fig. 4 illustrates the form of the curves, 

 Fig. 4a their limiting positions, and Fig. 4b the inflexion- 

 conditions in the respective regions. 



