10 
SYNOPSIS OF THE THEORY OF EQUATIONS. 
[631 
and therefore 
a = \/{ïr ± 
a six-valued function of q, r. But then writing b = ~~, we have, as may be shown, 
a + b a three-valued function of the coefficients; it would have been wrong to com 
plete the solution by writing b = \/{\r + \/(|r 2 — ^rq^)), since here (a + b) would be 
given as a 9-valued function, having only 3 of its values roots, and the other 6 values 
being irrelevant. An interesting variation of the solution is to write x — ab (a + b), 
giving a 3 b 3 (a 3 + b 3 ) = r and 3a?b 3 — q, or say ^ (a 3 + b 3 ) = f ^, a?b s = %q; whence 
{l (a 3 - b 3 )} 3 = ^ (ir 2 - i Y q% 
and therefore 
-■Mi 
± - V(i^ 2 - 2V2 3 ) 
and here although a, b are each of them a 6-valued function, yet, as may be shown, 
ab (a + b) is only a 3-valued function. 
In the case of a numerical cubic, even when the coefficients are real, substituting 
their values in the expression 
X = \/{&' ± V(i*’ 2 - 2T? 3 )} + -5- \/{\r ± \/(ir 2 - M 3 )}1 
this may depend on an expression of the form ^/(7 + hi), where 7 and 8 are real 
numbers (viz. it will do so if \r 3 — ^ T q 3 is a negative number), and here we cannot 
by the extraction of any root or roots of real numbers reduce v'Xy+Sf) f° rm 
c + di, c and d real numbers; hence, here the algebraical solution does not give the 
numerical solution. It is to be added that the case in question, called the “irreducible 
case,” is that wherein the three roots of the cubic equation are all real; if the roots 
are one real and two imaginary, then, contrariwise, the quantity under the cube root is 
real, and the algebraical solution gives the numerical one. 
The irreducible case is solvable by a trigonometrical formula, but this is not a 
solution by radicals; it consists, in effect, in reducing the given numerical cubic (not 
to a cubic of the form z 3 = a, solvable by the extraction of a cube root, but) to a 
cubic of the form 4oc 3 — 3x= a, corresponding to the equation 4 cos 3 0 — 3 cos 0 — cos 3d 
which serves to determine cos 0 when cos 3d is known. 
A quartic equation is solvable by radicals; and it may be remarked, that the 
existence of such a solution depends on the existence of 3-valued functions such as 
ab + cd, of the four roots (a, b, c, d); by what precedes, ab + cd is the root of a cubic 
equation, which equation is solvable by radicals; hence ab + cd can be found by radicals; 
and since abed is a given value, ab and cd can each be found by radicals. But by 
what precedes, if ab be known, then any similar function, say a + b, is obtainable 
rationally; and, consequently, from the values of a + b and ab we may by radicals