- 
OF THE HIGHER DEGREES, WITH APPLICATIONS. 107 
fie 
The m — 1 terms in the first of the groups (37) are the 
terms in (32) each taken twice. Therefore, by Prop. X., the law 
enunciated in the present Proposition is established so far as this 
groupe is concerned. The general proof is as follows. By (30) in 
m— Zz 1 
. . . mm m 
§38, taken in connection with (26), pm—z,4, = 4)... There 
; HART cs 
fore ae ti , = Pm—z 4,. But, by §38, yp», is a rational 
function of #; ; and, by Prop. VIII., 4; isa rational function of t . 
z 1 
Therefore oa shi 18 a rational function of 4. Also from the 
manner in which p,,—,z is formed, when ¢, in p,—2 4) is changed 
z 1 
d : | ; ™ mm ‘ 
sucessively into 4 ti ye acume. H. » the expression ae nas aeuis 
changed successively into the m — 1 terms of that one of the groups 
z 1 
— 
m 
(37) whose first term is 4 A Se . Therefore the terms in that 
group are the roots of a rational equation. 
§45. Cor. The law established in the Proposition may be brought 
under a yet wider generalization. The expression 
ra, (38) 
is the root of a rational equation of the (m — 1)* degree, if 
a+ 26+ 3c + .... + (m—1)8= Wm, 
W being a whole number. For, by (30) in connection with (26), 
: ss 
ww = ps3 4," , and so on. Therefore (38) has 
the value 
ae a+ 2b+ 38e+ ....+(m—1)s 
(pa pa ses) 4y m 
This is a rational function of ¢; , and therefore the root of a rational 
equation of the (m — 1)th degree. 
b oc Ww 
, or (p2 ps ....-) 41 . 
