106 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
§43. Proposition XII. To solve the Gaussian. 
The path we have been following leads directly, assuming the pri- 
mitive m*? root of unity ¢; to be known, to the solution of the reducing 
Gaussian equation of the m* degree to the equation a”— 1 = 0. 
For, as in §41, the roots of the Gaussian are Cy, Cz, etc. Therefore 
g, the sum of the roots, is — 1. Therefore 
a wd 1 
q=—(—14 4 4 fe (35) 
By Prop. VIII., 41, 42 , ete., are rational functions of #;. Therefore 
2 Lag 
Ay = ky + ke tp ig] +... 1 ele 
2 4 2(m—1 
Ay = ty “+ ke katy, +...» =e L (36) 
a =e 
Am—1 = hi + kot + ks th + .... thm ; J 
where hk, , ko, etc., are rational. From the first of equations (26), 
putting Cj for 7, , C2 for rz, and so on, 
= (Ci + t Co + etc.)™. 
By actual involution this gives us ky , kz , etc., as determinate functions 
of C; , C2, etc., and therefore as known rational quantities. For 
instance take k,. Being a determinate function of Cy, C2, etc., 
we have 
ky = 1 + 92 Ci + 93 Co + tee + dm Cm —13 
where qi , gz, etc,, are known rational quantities. But, by §13, the 
rational coefficients gq; — k,, go, etc., are all equal to one another. 
Therefore k; = gq; — q2. In like manner kz, 3, etc., are known. 
Therefore, from (36), 41, 42, etc., are known. Therefore, from 
(35), 71 is known. 
§44, Proposition XIII. The law established in Prop. X falls 
under the following more general law. The m — 1 expressions in 
each of the groups 
pie u 1 ae 
m m m m m m 
(4, iN ae 1? A, 4,2) y eee 4, ») 
2 1 2 1 2 1 
m m m m m m 
sere | 37 
(4, ane 4, ord) > m—A1 A, ») ( ) 
Reo a so 
m m m m m m 
i herr 
(4, aes ’ 4, 4n—6? ia 4, :) J 
and so on, are the roots of a rational equation of the (m — 1) degree. 
