58 
ON A THEOREM OF ABELS RELATING 
[311 
Hence in order that the roots of the quartic may be of the assumed form, 
a = m + f® + Jp + q V®, &c., 
where m, p, q, ® are rational, and where also 
V® Jp + q V® Jp — q V®= s, a rational function, 
the necessary and sufficient conditions are that the quartic should be such that the 
reducing cubic 
v? — Xu 2 + — v 2 = 0 
(whose roots are (a + /3 — 7 — 8) 2 , (a + 7 — /3 — 8) 2 , (a + 8 — ¡3 — y) 2 ) may have one rational 
root ®, and moreover that the function 
— A® 2 + (A 2 — f) ® — Sv 
shall be the square of a rational function s. This being so, the roots of the quartic 
will be of the assumed form 
a = m + V® + Jp + q V®, &c.; 
and from what precedes, it is clear that any function of the roots of the quartic 
which remains unaltered by the cyclical substitution a/3yS, or what is the same thing, 
any function of the form 
<P («, /3, 7, 8) + $ (/3, 7, 8, a) + (f) (7, 8, a, /3) + cf> (8, a, /3, 7) 
will be a rational function of m, ®, p, q, s, and consequently of the coefficients of the 
quartic. The above are the conditions in order that a quartic equation may be of 
the Abelian form. 
It may be as well to remark that, assuming only the system of equations 
a = m + V® + Vt , 
¡3 = m — V ® 4- VT, 
7 = m + V® — Vt , 
8 = m — V® — VT', 
then any rational function of a, /3, 7, 8 which remains unaltered by the cyclical sub 
stitution a/378 will be a rational function of ®, T + T', TT', VTT'(T — T'), V® (T — T'), 
V© vTT. In fact, suppose such a function contains the term 
(V©)“ (ffy (fry ; 
then it will contain the four terms 
( V©)«( v"r>( fry, 
(_ v©)“ ( fry (- ff )y, 
( V©)“ (- fry (- fry, 
(_ V©)* (- fry ( fr )y,