y^4 + 2Py^2 - 8[root]R.y + P^2 - 4Q = 0,
one of the roots of which is y = [root]a + [root]b + [root]c, while a,
b, c are the roots of the cubic equation z^3 + Pz^2 + Qz - R = 0.
4. In order to apply this resolution to the proposed equation y^4 +
py^2 + qy + r = 0, we must express the assumed coefficients P, Q, R by
means of p, q, r, the coefficients of that equation. For this purpose
let us compare the equations
y^4 + py^2 + qy + r = 0,
y^4 + 2Py^2 - 8[root]Ry + P^2 - 4Q = 0,
and it immediately appears that
2P = p, -8[root]R = q, P^2 - 4Q = r;
and from these equations we find
P = 1/2 p, Q = (1/16)(p^2 - 4r), R = (1/64)q^2.
Hence it follows that the roots of the proposed equation are generally
expressed by the formula
y = [root]a + [root]b + [root]c;
where a, b, c denote the roots of this cubic equation,
p p^2 - 4r q^2
z^3 + -- z^2 + -------- z - --- = 0.
2 16 64
But to find each particular root, we must consider, that as the square
root of a number may be either positive or negative, so each of the
quantities [root]a, [root]b, [root]c may have either the sign + or -
prefixed to it; and hence our formula will give eight different
expressions for the root. It is, however, to be observed, that as the
product of the three quantities [root]a, [root]b, [root]c must be
equal to [root]R or to -(1/8)q; when q is positive, their product must
be a negative quantity, and this can only be effected by making either
one or three of them negative; again, when q is negative, their
product must be a positive quantity; so that in this case they must
either be all positive, or two of them must be negative. These
considerations enable us to determine that four of the eight
expressions for the root belong to the case in which q is positive,
and the other four to that in which it is negative.
5. We shall now give the result of the preceding investigation in the
form of a practical rule; and as the coefficients of the cubic
equation which has been found involve fractions, we shall transform it
into another, in which the coefficients are integers, by supposing z =
1/4 v. Thus the equation
p p^2 - 4r q^2
z^3 + -- z^2 + -------- z - --- = 0
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becomes, after reduction,
v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0;
it also follows, that if the roots of the latter equation are a, b, c,
the roots of the former are 1/4 a, 1/4 b, 1/4 c, so that our rule may
now be expressed thus:
Let y^4 + py^2 + qy + r = 0 be any biquadratic equation wanting its
second term. Form this cubic equation
v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0,
and find its roots, which let us denote by a, b, c.
Then the roots of the proposed biquadratic equation are,
when q is negative, when q is positive,
y = 1/2([root]a + [root]b + [root]c), y = 1/2(-[root]a - [root]b - [root]c),
y = 1/2([root]a - [root]b - [root]c), y = 1/2(-[root]a + [root]b + [root]c),
y = 1/2(-[root]a + [root]b - [root]c), y = 1/2([root]a - [root]b + [root]c),
y = 1/2(-[root]a - [root]b + [root]c), y = 1/2([root]a + [root]b - [root]c).