[631 
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 11 
as may be shown, 
obtain the value of a or b, that is, an expression for a root of the given quartic 
expression; the expression finally obtained is 4-valued, corresponding to the different 
values of the several radicals which enter therein, and we have therefore the expression 
by radicals of each of the four roots of the quartic equation. But when the quartic 
ien wrong to com- 
(a + b) would be 
the other 6 values 
write x = ab (a + b), 
is numerical, the same thing arises as in the cubic: the algebraical expression does 
not in every case give the numerical one. 
It will be understood from the foregoing explanation as to the quartic, how in 
the next following case, that of a quintic equation, the question of the solvability by 
whence 
radicals depends on the existence or non-existence of ^-valued functions of the five 
roots (a, b, c, d, e); a fundamental theorem on the subject is that a rational function 
of 5 letters, if it has less than 5, cannot have more than 2 values; viz. that there 
^)}> 
are no 3-valued, or 4-valued, functions of 5 letters; and by reasoning, depending in 
part upon this theorem, Abel showed that a general quintic equation is not solvable 
by radicals: and d fortiori the general equation of any order higher than 5 is not 
solvable by radicals. 
, as may be shown, 
The general theory of the solvability of an equation by radicals depends very 
much on Vandermonde’s remark, that supposing an equation is solvable (by radicals) 
and that we have therefore an algebraical expression of x in terms of the coefficients, 
,’e real, substituting 
then substituting for the coefficients their values in terms of the roots, the resulting 
value of the expression must reduce itself to any one at pleasure of the roots a, b, c,...; 
}], 
thus in the case of the quadric equation where the solution is x = + ^p + — f)> 
writing for p, q their values a + b, ab, this is x = | [(a+ 6) ± V{( a _ &) 2 }]> = a or b 
7 and 8 are real 
ad here we cannot 
y + 8i) to the form 
does not give the 
ed the “irreducible 
real; if the roots 
3r the cube root is 
according to the value of the radical. But it is not considered necessary in the 
present sketch to go further into the theory of the solvability of an equation by 
radicals. It may oe proper to remark that, for quintic equations, there are solutions 
analogous to the trigonometrical solution of a cubic equation, viz. the quintic equation 
is here in effect reduced to some special form of quintic equation; for instance, to 
Jerrard’s form x? + ax + b = 0 or to some form presenting itself in the theory of elliptic 
functions; but the solutions in question are not solutions by radicals. And there are 
various other interesting parts of the theory which have been excluded from consideration. 
but this is not a 
imerical cubic (not 
be root, but) to a 
3 6 — 3 cos 6 = cos 36 
emarked, that the 
functions such as 
the root of a cubic 
found by radicals; 
j radicals. But by 
+ b, is obtainable 
e may by radicals 
2—2