  When the discriminants $\Delta$ and $\Delta p^2$ are idoneal, Patane proved a
theorem which connects the theta series associated to binary quadratic forms of
each discriminant. This paper generalizes the main theorem of Patane by no
longer requiring $\Delta$ and $\Delta p^2$ to be idoneal. In particular, we
state and prove an identity which connects the theta series associated to a
single binary quadratic form of discriminant $\Delta$ to a theta series
associated to a subset of binary quadratic forms of discriminant $\Delta p^2$.
Here and everywhere $p$ is a prime.
