An introduction to the modern theory of equations, by Florian Cajori.

THEORY OF EQUATIONS

Operating upon the subscripts of a in [0, a] by (0 1 2), we get
(X1 -+ a02 + 2(toa
and (a1 + wt2 + w2tAo)2 = A2 + Aoo + A1c2.
We see that Ao, Ai, A2, when operated on by (0 1 2)2, become respectively A2, Ao, Ai.
Ex. 2. Illustrate this theorem by taking v = 3 in Ex. 1, and show that
the function belongs to the cyclic group.
Ex. 3. Show that (0 1 2), applied to the subscripts of ao, Ca1, 2, ill
[C2, a(]2 = (( o w2+ t22i + 2)2 = Ao + Alw + A2&2, produces the same
effect as (0 1 2)4 applied to the subscripts of Ao, A1, A2.
Ex. 4. Show that (0 1 2 3) applied to the subscripts of ao0, (t1, a2, s3,
in [C3, ac]2= (ao co3aj + co6at2 + + 9at)2 = Ao + A1w + A 2w + AA,(os, where
= — i, produces the same effect as (0 1 2 3)6 applied to the subscripts
of Ao, A1, A2, A3.
119. Theorem. If with the cyclic substitution
(O 1 2... (- ))
we operate upon the subscripts of a, the subscrip)t of the coefficient
of each power of w in the product of [(,, a(t]. [1i, a]"i. [(,A2 a] V
~.suffers the substitution (0 1 2 *.. (n - 1))"v+lvl+A2v2+, where
v, v1, V2, ' are positive integers and X1, X2,...positive or negative
integers.
This theorem is a generalization of the preceding and is
proved in the same way. The product yields the equality
[O, X]V. [i, C(]"i. [02, (t.... = Bo + woB 2 + 122 +.. + BW-1B,
where Bo,, B *, *, B,_ are functions of the roots a(, a1,, a, *** l.
Replacing o successively by (, o),,)2, o, o** _, we have all
together n expressions. Multiply them by o-k, %-k, 2-k *..
respectively, then add the resultingp products, and we get

nZ k k1 - - CO (4]11 ~ OX) - l

I