OF THE HIGHER DEGREES, WITH APPLICATIONS. 101 
values of the generic expression 4 are obtained by changing C4 suc- 
cessively into C; , C2, etc., namely, 
Aq Cy +d,Co+ .... + dnCn, 
ee ss ve ~~ is dm —il Can ’ 
dh C, Nai Ce +. .+ dy Cane 
To show that these expressions are all unequal, take the first two. 
If these were equal, we should have 
(dm — dy) Cy a (dy — dz) Ce air etc. = 0. 
Therefore, by §13, d, — d; = 0, dy — dz = 0, and so on; which, 
because d, , d2, etc., are not all equal to one anenee is impossible. 
Since then 4 has at east m unequal particular cognate forms, 4; is, 
by Prop. IIIL., the root of a rational irreducible equation of a degree 
not lower than the m*; which, by Prop. VIL, is impossible. 
Therefore k, , k2, etc., are rational. Hence each of the expressions 
in (27) is a rational function of f, . 
§37. Cor. Any expression of the type k; + hz t; + kg ty + ete, 
which is such that all the unequal particular cognate forms of the 
generic expression under which it falls are obtained by substituting 
for ¢; successively the different primitive m* roots of unity, while 
ky , ko , etc., remain unaltered, is a rational function of ¢,. For, in 
the Proposition, 4; or ky + kz t, + etc. was shown to be a rational 
function of ¢; , the conclusion being based on the circumstance that 
4; satisfies the condition specified. 
38. Proposition IX. If g be the sum of the roots of the equation 
g q 
eae) —' 0", 
1 2 3 
1 m 1 m ‘| m 
m=—(g9t+4 +4, +64, 4+. 
m — 2 m—1 il 
m Wie 
iy Rae, (29) 
For, z being one of the whole ‘vente Lp is... 971, put 
= (71 + ti ro + fe v3 + ete.) (1 + & re + t 3 + etc.)—?. (30) 
Multiply the first of its factors by ¢ * and the second by tj. Then 
= (r2 + fore ti; rg + ete.) (r2 + t rg + ‘i r4 + ete.) —%. (31) 
ome pz does not alter its value when we change 7; into 72, 72 into 
rz, and soon. In like manner it does not alter its value when we 
