A general relation is derived for the number of counts registered by a Geiger-M\"uller tube counter or similar electrical counting device exposed to a radioactive source whose strength varies arbitrarily with the time when the counter has a finite, constant resolving time. This is applied specifically to the case of an exponentially decaying source superposed on a uniform background, the solution of the resulting formula being represented nomographically. The number of spurious coincidences observed in a set of $P$ counters used coincidentally is calculated when the resolving times of the individual counters are neglected in comparison with the resolving time of the combining electrical circuit for coincidences. This general expression is applied to the cosmic-ray "telescope" and the double coincidence magnetic spectrometer such as that of Henderson and Alichanow. The constancy of the resolving time of a single counter and the justification for neglecting individual resolving times in comparison with the coincidence resolving time are discussed.