OF THE HIGHER DEGREES, WITH APPLICATIONS. 125 
§71. Proposition XXVII. Under the conditions that have been 
established, the root 7, takes the form given without deduction in 
Crelle (Vol. V., p. 336) from the papers of Abel. 
For, by Cor. Prop. XIII. (compare also Cor. 2, Prop. XIX..,) 
the expressions 
roe S| OS ee 
Ay 43 M4 49, 4y 4, As 4g, 
Ai, 44 49 4; 43, (73) 
tee, Fs 
As 4 Az 
are the roots of a biquadratic equation. In the corollaries referred 
to, it is merely stated that each of the expressions in (73) is the root 
of a biquadratic ; but the principles of the propositions to which the 
corollaries are attached show that the four exnressions must be the 
roots of the same biquadratic. Let the terms in (73) be denoted 
respectively by 
5A- 1, 5A}, 5Az }, 5AT 
MPS ol 91 3 8 8 
Then 4? 4° A? 4? = A? (4? AP AP 4} ) is an identity. Therefore 
1 Re pe ay 98 
At Ay (4; AP Ae d3-) Smaalarly, 
i es 
4 Ae = A, (4° AP A A?) 
a (a? 48 43 2° 1 
By = Ae (4; 4? 4? dy), anc 
a Le Se 
tla Cae 
Substituting these values in (62), we get 
1 2 1 2 
to 8) f Ne: 
m= A, (4; 43 dg 42) + do (42 At 43 42) 
+ 3 
he NS a bie 
Medes. Ap tdy Ay ) “Ay (de a “Ay AG ): (74) 
This, with immaterial differences in the subscripts, is Abel’s expression; 
only we need to determine A; , Ag, Ag and Aq more exactly. These 
terms are the reciprocals of the terms in (73) severally divided by 5. 
Therefore they are the roots of a biquadratic. Also, no surds can 
appear in A; except those that are present in 4;, 42, dgand dy. 
That is to say, A; is a rational function of ./ s, f/f s,and / 2 But 
it was shown that 4/ 5, ./ s = he fz. Therefore Aj is a rational 
function of ,/ sand ,/ z. We may therefore put 
Ay === K + Ke 4 =|: Ke Ay + Oe A, Ay ; 
