  In this paper we study the cones corresponding to sums of squares of $n$-ary
$d$-ic forms with at most $k$ terms. We show that these are strictly nested as
$k$ increases, leading to the usual sum of squares cone. We also discuss the
duals of these cones. For $n \ge 3$, we construct indefinite irreducible
$n$-ary $d$-ic forms with exactly $k$ terms for $2 \le k \le
\binom{n+d-1}{n-1}$.
