80 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
1 au 2 3 m-2 m—1 
™ 
Rye het m ™. m m ‘ 
Oo g+ 0," +a,0,° +A,” +.. #64) +h dO, : 
where g is rational, and a,, b,, ete., involve only surds subordinate to 
1 
By, iso 888, 47, 
7. The equation (x) = 0 Las an auxiliary equation of the 
(m — 1) degree. §35, 52. 
8. If the roots of the auxiliary be 4,, 0, 9,,.., Om —1, the m — 1 
expressions in each of the groups 
1 ig aL 1 i 1 
m mM , mm , m m 
ZL ¢ (3) 0 2) 
A; et 2 m—2? ’ m—1 4, ? 
a 4 a oe 2 1 
mm mm m m 
( 
1 Fae J, Oe —4? ? Cn —1 D5 ’ 
3 1 3 1 1 
mm mm m m 
EM eee 
1 m—3 2 m— 6 m—1 3 
and so on, are the roots of a rational equation of the (m— 1)™ degree. 
- 1 
The — terms 
I 
at 1. 2 1 1 
m mv m0 m™m ™m m 
0) pags O 
Oi 0s Oy. Ome » 9, 9 (a 
2 2 
F : m— | 
are the roots of a rational equation of the CS degree. 
$39, 44, 55. 
9. Wider generalization. §40, 57. 
10. When the equation /(x) = 0 is of the first class, the auxiliary 
equation of the (m — 1)» degree is irreducible. §35, Also the roots 
of the auxiliary are rational functions of the primitive mt root of 
unity. §36. And, in the particular case when the equation F(x) =0 
is the reducing Gaussian equation of the m*® degree to the equation 
m — : 
x” — 1 = 0, each of the 5 expressions, 
nat cos! 
m m m m 
O 0 ZC, 
ite 11 9. m—2? ao 
has the rational value m. $41. Numerical verification. §42. 
