  We revisit the problem of the Lorentz transformation of time-intervals in
special relativity. We base our discussion on the time-interval transformation
formula $ c\Delta t' = \gamma (c\Delta t - \vec{\beta} \cdot \Delta \vec{r}) $
in which $ \Delta t'$ and $ \Delta t $ are the time-intervals between a given
pair of events, in two inertial frames $ S $ and $ S'$ connected by an general
boost. We observe that the Einstein time-dilation-formula, the Doppler formula
and the relativity of simultaneity, all follow when one the frames in the
time-interval transformation formula is chosen as the canonical frame of the
underlying event-pair. We also discuss the interesting special case $ \Delta t'
= \gamma \Delta t $ of the time-interval transformation formula obtained by
setting $ \vec{\beta} \cdot \Delta \vec{r}=0 $ in it and argue why it is really
\textbf{not} the Einstein time-dilation formula. Finally, we present some
examples which involve material particles instead of light rays, and highlight
the utility of time-interval transformation formula as a calculational tool in
the class room.
