[631 
SYNOPSIS OF THE THEORY OF EQUATIONS. 
9 
631] 
C. X. 
2 
a assignable order) 
f a given equation, 
is possible to find 
a sextic equation ; 
the coefficients of 
d, and therefore 
ic equation. 
of values (obtained 
roots ; for instance, 
e function ab + cd\ 
is 3, viz. ab + cd, 
tain functions of a 
In the case of a quadric equation x 2 +px + q = 0, we can find for x, by the 
assistance of the sign V( ) or ( )l, an expression for x as a two-valued function 
of the coefficients p, q, such that, substituting this value in the equation, the equation 
is thereby identically satisfied, viz. we have 
x=- : kp±V(lp 2 -q), 
giving 
0C 2 = %p 2 -q + p^/(ip 2 - q) 
+px = -^p 2 ± p\/(ip 2 - q) 
+ q = +q 
af+px + q =0, 
the value of any 
letermine rationally 
jans of a numerical 
2, 3, 4; then if it 
2 (viz. a — 1, 6 = 2, 
c = 4, d = 3); and 
md not any other 
,s regards particular 
nines a to be =1 
a 2 b = 16 does not 
, or else 2 and 4, 
i = (a + b) 3 + (c + d) s , 
there are only the 
% + 4 2 2/ 2 +1 3 % will 
Ld therefore rational 
then the equations 
of known functions 
rown. But observe 
. as regards t 2 , t 3 ; 
of tn ~f“ ¿3, t 2 t 3 1 01', 
but these last will 
onally in terms of 
a of t x and of the 
n of equations, or, 
and the equation is on this account said to be algebraically solvable, or, more accurately, 
to be solvable by radicals. Or we may, by writing x = — ^p+ z, reduce the equation 
to z 2 = \p 2 — q, viz. to an equation of the form z 2 = a, and, in virtue of its being thus 
reducible, we may say that the equation is solvable by radicals. And the question for 
an equation of any higher order is, say of the order n, can we by means of radicals, 
that is, by aid of the sign ^/( ) or ( ) m , using as many as we please of such 
signs and with any values of m, find an ^-valued function (or any function) of the 
coefficients, which substituted for x in the equation shall satisfy it identically. 
It will be observed that the coefficients p, q, ... are not explicitly considered as 
numbers, but that even if they do denote numbers, the question whether a numerical 
equation admits of solution by radicals is wholly unconnected with the before-mentioned 
theorem of the existence of the n roots of such an equation. It does not even follow 
that, in the case of a numerical equation solvable by radicals, the algebraical expression 
of x gives the numerical solution; but this requires explanation. Consider, first, a 
numerical quadric equation with imaginary coefficients; in the formula x = — %p + \J(\p 2 — q), 
substituting for p, q their given numerical values we obtain for x an expression of the 
form x = a. + /3i ± V(7 + Si), where a, /3, 7, 8 are real numbers; this value substituted 
in the numerical equation would satisfy it identically and it is thus an algebraical 
solution; but there is no obvious d priori reason why the expression V(y + Si) should 
have a value = c + di, where c and d are real numbers calculable by the extraction 
of a root or roots of real numbers; it appears upon investigation that + 8i) has 
such a value calculable by means of the radical expression \/{V(7 2 + S 2 ) ± 7}; and hence 
that the algebraical solution of a quadric equation does in every case give the 
numerical solution of a numerical quadric. The case of a numerical cubic will be 
considered presently. 
A cubic equation can be solved by radicals, viz. taking for greater simplicity the 
cubic in the reduced form a? - qx — r = 0, and writing x = a + b, this will be a solution 
if only Sab = q, and a 3 + b 3 = r, or say % (a 3 + b 3 ) = \r; whence 
1 (a 3 - V) = ± V(i?’ 2 - 2T? 3 )>