A Boron-lined proportional counter has been developed whose cathode diameter varies continuously, hence varying the gas multiplication along the length of the counter. The sensitive area of the cathode thus becomes a function of the applied voltage, electrical amplification, and pulse-height discrimination. A semi-empirical relationship between counter geometry, gas parameters, and applied voltage is used to develop theoretical expressions for the dependence of count rate on applied voltage and cathode geometry. The behavior of cathodes of hyperbolic and exponential shape is treated. A prototype exponential counter, operated in a pulse counting mode, was constructed and tested. The approximately logarithmic relationship between neutron flux and voltage predicted for such an instrument was confirmed.