  Let ${\mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being
a prime and $k$ be a positive integer. For any $y, z\in\mathbb{F}_q$, let
$N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+\cdots+x_s^3=z$
and $x_1^3+\cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p,
p\equiv1\pmod3$ and $y$ is non-cubic, then
$T_3(y)=p^2+\frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined
by $4p=c^2+27d^2,~c\equiv 1 \pmod 3$ except for the sign of $d$. In 1978,
Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being
a non-cubic element of ${\mathbb F}_p$. But the sign problem is kept open for
the remaining case of $2$ being cubic in ${\mathbb F}_p$. In this paper, we
solve this sign problem by determining the sign of $d$ when $2$ is cubic in
${\mathbb F}_p$. Furthermore, we show that the generating functions
$\sum_{s=1}^{\infty} N_{s}(z) x^{s}$ and $\sum_{s=1}^{\infty} T_{s}(y)x^{s}$
are rational functions for any $z, y\in\mathbb F_q^*:=\mathbb F_q\setminus
\{0\}$ with $y$ being non-cubic over ${\mathbb F}_q$ and also give their
explicit expressions. This extends the theorem of Myerson and that of Chowla,
Cowles and Cowles.
