260 Mr. A. Cayley on a Theorem of Abel’s relating 
or, what is the same thing, and putting O for F, 
—X0?+ (A?— p)O—3/?=s?. 
Hence in order that the roots of the quartic may be of the 
assumed form, 
a=m+t+/O4Vn+9V0, &e., 
where m, p, g, © are rational, and where also 
/O Jp +qVO /n—qY V@=s, a rational function, 
the necessary and sufficient conditions are that the quartic should 
be such that the reducing cubic 
u>?—)u? + pu—v? =0 
(whoseroots are (2+ 8 —y—86)?, (« +y—B—5S)?, (« +8—B—y)?) 
“may have one rational root @, and moreover that the function 
—r0?+ (A?— pp)O—3y? 
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+/O+Vn+qV0, &e. 
And from what precedes, it is clear that any function of the roots 
of the quartic which remains unaltered by the cyclical substitu- 
tion «Syd, or what is the same thing, any function of the form 
h(a, B, Y 6) + (8, Y> 6, c) +(y; 5, a, B) oh f(S, a, B, Y) 
will be a rational function of m, ©, p, g, 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+/O+/T, 
B=m—/O4VT, 
y=m+/0-VT, 
S=m—/O—1/ 1’, 
then any rational function of a, 8, y, 5 which remains unaltered 
by the cyclical substitution «Sy5 will be a rational function of 
6, T+, TY, /TT(T—-T), /O(T—-T), /O/TT.. In 
fact, suppose such a function contains the term 
(SOS T/T)" 
then it will contain the four terms 
