A theoretical analysis is given of the correlation to be expected between the fluctuations in the outputs of two photoelectric detectors when these detectors are illuminated with partially coherent light. It is shown how this correlation depends upon the parameters of the equipment and upon the geometry of the experiment. The correlation may be detected either by linear multiplication of the fluctuations in the two outputs or by a coincidence counter which counts the simultaneous arrival of photons at the detectors. The theory is given for both these techniques and it is shown that they are closely equivalent. A laboratory test is described in which two photomultipliers were illuminated with partially coherent light and the correlation between the fluctuations in their outputs measured as a function of the degree of coherence. The results of this experiment are compared with the theory and it is shown that they agree within the limits of accuracy of the test; it is concluded that if there is any systematic error in the theory it is unlikely to exceed a few parts per cent.