  The weak gravitational lensing effect, small coherent distortions of galaxy
images by means of a gravitational tidal field, can be used to study the
relation between the matter and galaxy distribution. In this context, weak
lensing has so far only been used for considering a second-order correlation
function that relates the matter density and galaxy number density as a
function of separation. We implement two new, third-order correlation functions
that have recently been suggested in the literature, and apply them to the
Red-Sequence Cluster Survey. We demonstrate that it is possible, even with
already existing data, to make significant measurements of third-order lensing
correlations. We develop an optimised computer code for the correlation
functions. To test its reliability a set of tests are performed. The
correlation functions are transformed to aperture statistics, which allow easy
tests for remaining systematics in the data. In order to further verify the
robustness of our measurement, the signal is shown to vanish when randomising
the source ellipticities. Finally, the lensing signal is compared to crude
predictions based on the halo-model. On angular scales between roughly 1 arcmin
and 11 arcmin a significant third-order correlation between two lens positions
and one source ellipticity is found. We discuss this correlation function as a
novel tool to study the average matter environment of pairs of galaxies.
Correlating two source ellipticities and one lens position yields a less
significant but nevertheless detectable signal on a scale of 4 arcmin. Both
signals lie roughly within the range expected by theory which supports their
cosmological origin.[ABRIDGED]
