and it immediately appears that
2P = p, −8√R = q, P² − 4Q = r;
and from these equations we find
P = ½ p, Q = 1⁄16 (p² − 4r), R = 1⁄64 q².
Hence it follows that the roots of the proposed equation are generally expressed by the formula
y = √a + √b + √c;
where a, b, c denote the roots of this cubic equation,
| z³ + | p | z² + | p² − 4r | z − | q² | = 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 √a, √b, √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 √a, √b, √c must be equal to √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 = ¼ v. Thus the equation