118 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
§56. Cor. 1. The reasoning in the proposition proceeds on the 
assumption that the prime number m is odd. Should m be even, the 
series 4, , 6, etc., is reduced to its first term. The law may be 
considered even then to hold in the following form, The product 
1 1 
mm m 
4, 4, is the root of a rational equation of the (m — 1) degree, 
or is rational. For this product is 4; , which, by Prop. X VIL, is 
the root of an equation of the (m — 1) degree. 
§56. Cor. 2. I merely notice, without farther proof, that the 
generalization in §45 in the case when the equation /’ (a) = 0 is of 
the first (see §30) class holds in the present case likewise. 
ANALYSIS OF SOLVABLE EQUATIONS OF THE FirtTH DEGREE. 
§58. Let the solvable irreducible equation of the m* degree, which 
we have been considering, be of the fifth degree. Then, by Prop. 
1X. and §47, whether the equation belongs to the first or to the 
second of the two classes that have been distinguished, assuming the 
sum of the roots g to be zero, 
1 2 3 4 
n= Liat 5. -Gueimere a Eley ay (62) 
though, when the equation is of the first class, the root, as thus 
presented, is not in a simple state. 
$59. Proposition XX. If the auxiliary biquadratic has a rational 
root 4; not zero, all the roots of the auxiliary biquadratic are rational. 
Because 4; is rational, the auxiliary biquadratic g (x) = 0 is not 
irreducible. Therefore, by Prop. VII. the equation F (~) = 0 is of 
1 
the second (see §30) class. Therefore, by Prop. X1V., Ar is the only 
principal surd in 7, . Consequently, because 4; is rational, 
a, €, and fy are rational. Therefore 4), at Ae a : hi At 4 
which are the roots of the auxiliary biquadratic, are rational. 
$60. Proposition XXI. If the auxiliary biquadratic has a qua- 
dratic sub-auxiliary ¢, (#) = 0 with the roots 4; and 42, then 
A, = hi Ane and 4; = ha 4s; and hy 4; is rational. 
As in §52, ¢ being a certain fifth root of unity, each term in (55) is 
equal to a term in (39). The first term in (55) cannot be equal to 
the first in (39), for this would make 42 = 4,. Suppose if possible 
that the first in (55) is equal to the second in (39). Then, by 
equations (50), applied as in Prop. XVLI., 
