THE ROOTS OF EQUATIONS. 27 



By putting S = k^A cos {n<l> + 0) + ^c, and T—k"A sin (n(f>+6)-\-^c, 

 this becomes 



y (x,) =(P, cos Ai + -S) + y^ (Pi sin X, + T); 

 which again if P|= (P, cos A, + S")* + (P, sin A, + T)*, may be written 

 /(a;,) =P2(cos /8 + y^ sin iS). 



Since J^ is a particular value of P^, and since the least value of P* is 

 P'i Pg — Pi cannot be negative. But 



Pi-P?=2 P,(;8cos A, + TsinAi) + S'+T' 



=2 k^AP, cos (n(l> + $-Xi)+^c (4) 



We give only the first term in the expansion of P|— Pf according to 

 the ascending powers of k. The other terms contain powers of k 

 higher than the n"'. Now suppose if possible that P, is not zero. 

 From the manner in which F (x) was taken in e(Juation (3), F(Xi) is 

 not zero ; for if it were, P(«) would be divisible by x-xi, and there- 

 fore there would be more than n roots of equation (2) equal to a;, : 

 which we supposed not to be the case. Hence A also, which is a 

 factor of F (x^), is distinct from zero. Take then n(f) such [^and Aj 

 being determined, the former from F(x{), and the latter from/(ir,)] 

 that cos (n<f>-\-0—X{) may be distinct from zero, and have its sign 

 opposite to that of ^P,. Then cause k, always remaining positive, 

 to approach indefinitely near to zero ; till the sign of the whole 

 expression forP|— Pf in (4) is the same with that of its first term. 

 The sign of that first term is necessarily negative. Therefore the 

 sign of P|— Pf is ultimately negative : which, however, we have seen 

 to be impossible. Therefore P, cannot but be zero. Hence/(a;i) is 

 zero ; and xi is a root of equation (1). 



