OF THE HIGHER DEGREES, WITH APPLICATIONS. 81 
11. Solution of the Gaussian. §43. 
12, Analysis of solvable irreducible equations of the fifth degree. 
The auxiliary biquadratic either is irreducible, or has an irreducible 
sub-auxiliary of the second degree, or has all its roots rational. The 
three cases considered separately. Deduction of Abel’s expression 
for the roots of a solvable quintic. §58-74. 
PRINCIPLES. 
$1. It will be understood that the surds appearing in the present 
paper have prime numbers for the denominators of their indices, 
1 
unless where the contrary is expressly stated. Thus, 215 may be 
1 1 
regarded as hi, a surd with the index }, % being 23. It will be 
understood also that no surd appears in the denominator of a fraction. 
: : 2 , wh WSs 
For instance, instead of —————— we should write ——_———.. 
1+ ¥-3 2 
When a snrd is spoken of as occurring in an algebraical expression, 
it may be present in more than one of its powers, and need not be 
present in the first. 
§2. In such an expressionas ¥ 2 + (1 + ¥ 2)* , Vv 21s subordi- 
nate to the principal surd (1 + VY 2)° , the latter ay the only prin- 
cipal surd in the expression. 
§3. A surd that has no other surd subordinate to it may be said to 
1 
be of the first rank ; and the surd h¢ , where h involves a surd of the 
(a — 1) rank, but none of a higher rank, may be said to be of the 
a‘ rank. In estimating the rank of a surd, the denominators of the 
indices of the surds concerned are always supposed to be prime 
numbers. Thus, 34 is a surd of the second rank. 
1 
§4. An algebraical expression in which 4 is a principal (see §2) 
1 
surd may be arranged according to the powers of 4” lower than the 
1 
m*h, thus, 
1 on 120) ay ™m — m—-1 
eae ce Ae +. fh ered” + d™ } 
a 1 1 ay 1 1 1 Ay! by 1 (1) 
where gj, k,, ai, etc., are clear of J," , 
