56 ON A THEOREM OF ABEL’S RELATING [311 
It is to be noticed that in the expressions for a, a x , a 2 , a 3 , the radicals are such 
that 
Jl +e 2 Jh(l + e 2 + Vl + e 2 ) Jh (1 + e 2 - Vf + e 2 ) = he (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 
x = 0 + Aa. r; /3 : ’ r ÿ r, 8 r ‘ 4- B¡3*7* 8* a 5 + Cq s 8*a. i (3 : ' + D8~ s a. s /3 5, y r ' > 
a = ???,+ V© + Jp + q V0, 
/3 = m — n V© + Jp — q V©, 
7 = m + n V© — Jp + q V©, 
.1 2 4 3 
where 
8> = m — n'J®—Jp — q V© ; 
the radicals being connected by 
and where 
\/© Jp + q V© Jp — q d© = .s, 
4 = HT + Xa + ÜÎ7 + Na.q, B = K + L/3 + M8 + N¡3 8, 
G = K + Lq + M a + Nay, D = K + L8 + M/3 + N/38, 
in which equations 6, m, n, p, q, ©, 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 x 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, x will be the root of a rational quintic ; and 
conversely. 
Now attending to the equation 
V© Jp + q V© Jp — q V© = s, 
the different admissible values of the radicals are 
\/©, 
v p + q V©, 
Jp — q V©, 
— V©, 
Jp — q V©, 
— Jp + q V©, 
V©, 
— Jp + q V©, 
— Jp — q V©, 
— V©, 
1$ 
> 
C5-* 
1 
S' 
1 
Jp + q V©,