  Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive
integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let
$N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by
$a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let $t(a_1,a_2,\ldots,a_k;n)$ be the
number of representations of $n$ by
$a_1\frac{x_1(x_1-1)}2+a_2\frac{x_2(x_2-1)}2+\cdots+a_k\frac{x_k(x_k-1)}2$
$(x_1,\ldots,x_k\in\Bbb Z$). In this paper, by using Ramanujan's theta
functions $\varphi(q)$ and $\psi(q)$ we reveal many relations between
$t(a_1,a_2,\ldots,a_k;n)$ and $N(a_1,a_2,\ldots,a_k;8n+a_1+\cdots+a_k)$ for
$k=3,4$.
