y4 + 2Py² + P² = 4 (ab + ac + bc) + 8 (√a²bc + √ab²c + √abc²);
and since ab + ac + bc = Q,
and √a²bc + √ab²c + √abc² = √abc (√a + √b + √c) = √R·y,
the same equation may be expressed thus:
y4 + 2Py² + P² = 4Q + 8√R·y.
Thus we have the biquadratic equation
y4 + 2Py² − 8√R·y + P² − 4Q = 0,
one of the roots of which is y = √a + √b + √c, while a, b, c are the roots of the cubic equation z³ + Pz² + Qz − R = 0.
4. In order to apply this resolution to the proposed equation y4 + py² + 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
y4 + py² + qy + r = 0,
y4 + 2Py² − 8√Ry + P² − 4Q = 0,