then, from the theory of equations we have a + b + c = -P, ab + ac + bc = Q, abc = R. We square the assumed formula y = [root]a + [root]b + [root]c, and obtain y^2 = a + b + c + 2([root]ab + [root]ac + [root]bc); or, substituting -P for a + b + c, and transposing, y^2 + P = 2([root]ab + [root]ac + [root]bc). Let this equation be also squared, and we have y^4 + 2Py^2 + P^2 = 4(ab + ac + bc) + 8([root]a^2 bc + [root]ab^2 c + [root]abc^2); and since ab + ac + bc = Q, and [root]a^2 bc + [root]ab^2 c + [root]abc^ 2 = [root]abc([root]a + [root]b + [root]c) = [root]R.y, the same equation may be expressed thus: y^4 + 2Py^2 + P^2 = 4Q + 8[root]R.y. Thus we have the biquadratic equation 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.