Duality in Siegel's theorem on representation by a genus of quadratic forms, and the averaging operator

Let $S$ and $T$ be two integral positive definite quadratic forms in the same number of variables, and let $S_1,\dots,S_H$ and $T_1,\dots,T_h$ be complete systems of representatives of the different classes in the genus of the form $S$ and $~T$, respectively. The author proves, in particular, that
$$ \bigg(\sum_{i=1}^He(S_i)^{-1}\bigg)^{-1}\sum_{i=1}^He(S_i)^{-1}r(S_i,T)=\bigg(\sum_{j=1}^he(T_j)^{-1}\bigg)^{-1}\sum_{j=1}^he(T_j)^{-1}r(S,T_j), $$
where $r(S',T')$ denotes the number of integral representations of the form $T'$ by the form $S'$, and $e(S') = r(S',S')$.
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