OF SOLVABLE EQUATIONS OF THE FIFTH DEGREE, 95 
IJ].— REDUCTION OF THE VALUE OF x IN (5) TO ABEL’s Form. 
§ 4. Taking Q,, @., etc., as above, it will he found that the value of x in (5) admits 
of being written 
Q, (6) 4, 8, a3) 4 Q, (6, 0, 4, 4°) ry Q, (4, 4, 4, POS Q, (6, 8, 6, ae), (9) 
For, as was proved in “Principles of the Solution of Equations of the Higher Degrees” 
(American Journal of Mathematics, Vol. VII), the separate members of the value of 
x in (5), namely, as, a A, À & a”, À a pô, are fifth roots of the four values of 4 That is 
to say, if a, which, by ( 
4), is a rational fraction of A, becomes a, when À has the value J,, 
1 4 
2 L 
A; = À a,” A; ef = À a CA : (10) 
4 
1 
5 
mn 
Os Open lve US 
1 1 
5 5 
( 6; A, A, A, ) NT (A C4 ais ) A, = 
1 1 
Therefore, by (8), 4° = Q, (4; 6, 4, 4, )°. (11) 
As A, runs through the four values 1,, 1,, À,, A,, the expressions #, and Q, run through 
their corresponding values. Hence, from (11), 
RON (RON, a) | 
= Q, (65076, 0°)* | 
ee 
1 
Q, ( A, A. 6, 6, ) i 
D 
I 
But (11), (12) and (10) give us for the value of x the expression in (9). This is Abel’s 
form; only we require to prove that #, 6, 4, 6,, have the forms in (6), which are those 
assigned to them by Abel. Since 4, is a rational function of À,, its form is 
A=ptk/YAt+s)+ ijmtnrnyvl¢ iv 2(1+s) +24 (1+s)}; (13) 
where p, k, m and » are rational. Put 
D=m+n14+s) +2mn(1+ 8°) 
and G= m+ nr? (1+ 8s?) +2mn. 


2 _m (1 2 
Then ‘pa fae et Ro (14) 
D 
2 2 
and À — CURE (15) 
G 
e and h are rational. From (14), keeping in view the values of D and G, we get 
D? (1 + e)=G (1 + s’); therefore 
VA+H=TVA+e). (16) 
Also, (15) and (16) give us 
im+nvy (A+) f2a4+s) +2 YVd+*fay f~adteytavatey}. (7) 
