Let
$\mathbb{Z}$
and
$\mathbb{Z}^{+}$
be the set of integers and the set of positive integers, respectively. For
$a,b,c,d,n\in \mathbb{Z}^{+}$
, let
$t(a,b,c,d;n)$
be the number of representations of
$n$
by
$\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$
with
$x,y,z,w\in \mathbb{Z}$
. Using theta function identities we prove 13 transformation formulas for
$t(a,b,c,d;n)$
and evaluate
$t(2,3,3,8;n)$
,
$t(1,1,6,24;n)$
and
$t(1,1,6,8;n)$
.