It is shewn that, if N(P) be the number of solutions of the
indeterminate equation
ax^3+by^3+cz^3+dw^3 =0 \qquad (a,b,c,d \neq0)
for which \vert x \vert,\vert y\vert, \vert z\vert, \vert w\vert \leq P, then
N(P) = KP^2 + o(P^2),
where, to within a term O(P),KP^2 is the contribution to N(P) corresponding to the
rational lines in the projective surface defined by the equation. This proves a conjecture made by
Heath-Brown, who has studied N(P) under the assumption of the Riemann Hypothesis
for certain Hasse-Weil L-functions. The remainder term o(P^2)
in the formula represents O(P^{ {4 \over 3}+ϵ}),O(P^ {{5 \over 3}+ϵ}),
or O(P^{2}/^3 \sqrt{ \log P}) according as the surface contains three, one, or no
rational lines.