Density of rational points on diagonal quartic surfaces

Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P(3) defined by ax(4) + by(4) + cz(4) + dw(4) = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.