  Eighty planetary systems of two or more planets are known to orbit stars
other than the Sun. For most, the data can be sufficiently explained by
non-interacting Keplerian orbits, so the dynamical interactions of these
systems have not been observed. Here we present 4 sets of lightcurves from the
Kepler spacecraft, which each show multiple planets transiting the same star.
Departure of the timing of these transits from strict periodicity indicates the
planets are perturbing each other: the observed timing variations match the
forcing frequency of the other planet. This confirms that these objects are in
the same system. Next we limit their masses to the planetary regime by
requiring the system remain stable for astronomical timescales. Finally, we
report dynamical fits to the transit times, yielding possible values for the
planets' masses and eccentricities. As the timespan of timing data increases,
dynamical fits may allow detailed constraints on the systems' architectures,
even in cases for which high-precision Doppler follow-up is impractical.
