19] VARIABLES AND FUNCTIONS. 33 



has as roots X, /*, />, and being reduced to the common denominator 



has a numerator of the third degree in p. As this vanishes for 



p=\, p=p, p = v 

 it can only be 



-(p-\)(p 



Hence we have the identity 



_ - (p - V) (p ~ /*) (p - " 



Multiplying this by p + a 2 and then putting p = a 2 we get 



- (a' + X 



(a 2 

 and in like manner 



(6 2 - c 2 ) (6 2 - a 2 ) 



(c 2 + X)(c 2 + /*)(c 2 +v) 

 (c 2 -a 2 )(c 2 -6 2 ) 



If X, fj,, v are contained in the intervals above specified, these will 

 all be positive, so that the point will be real. 



If we insert these values in S x , we shall have Ti^ expressed in 

 terms of X, ^ v. 



This is more easily accomplished as follows. 

 Differentiating the above identity (14) according to p, 



-X p + a 2 p-p p+tf p-v p + c 1 } 



If we put p = X, all the terms on the right except the first, being 

 multiplied by p X, vanish, and we have 



(a 2 + \y T (6 2 + X) 2 """ (c 2 + X) 2 ~ (a 2 + X) (6 2 + X) (c 2 + X) ' 

 w. E. 3 



