A simple proof is given of the modular identity for the theta series
$\sum exp(2\pi\sqrt{-1}\tau r^2/4m + 2\pi\sqrt{-1}z r)$,
where the sum is over all integers $r$ which are congruent to
a fixed integer $a$ (mod $2m$).
The formula describes the transformation of this series under
the change of variables
$\tau \to -1/\tau$ and
$z \to z/\tau$.