258 Mr. A. Cayley on a Theorem of Abel’s relating 
radicals are such that 
SI FeV 1 t+e+ Vite) VA +e—V1+e)she(1 +e), 
a rational number. 
The theorem is given as belonging to numerical equations ; 
but considering it as belonging to literal equations, it will be 
convenient to change the notation; and in this point of view, 
and to avoid suffixes and accents, I write 
e=O+ Aa Bi y8 85 + BBS 258 as + Cys d5 a5 Be 4 Ddtak Biys, 
where 
a=m+mn/O+Vp+qV0, 
B=m—n/O+V p—qV@, 
y=m+n/O—Vn+qV@, 
S=m—m/O—-V p—qvO; 
the radicals being connected by 
SOV p+yVOV p—qvO=s, 
and where 
A=K+lLa+My+Ney, B=K+L6+M6-+ N68, 
C=K+Iy+Ma+Nay, D=K+L6+M6+ N68, 
in which equations 0, m, n, p, g, ©, s, K, L, M, N are rational 
functions of the elements of the given quintic equation. 
The basis of the theorem is, that the expression for 2 has only 
the five values which it acquires by giving to the quintic radicals 
contained in it their five several values, and does not acquire any 
new value by substituting for the quadratic radicals their several 
values. For, this being so, # will be the root of a rational quintic ; 
and conversely. 
Now attending to the equation 
JOVp+qVOV p—IVO=s, 
the different admissible values of the radicals are 
JO, Vpt+qv®, Vp—gv®, 
—/6, Vp—qv0, —Vp+qVv®, 
JO, —Vp+qv@, —Vp—qv®, 
-J®, —Vp=4q¥6,  VptqV® 
corresponding to the systems 
