Weight functions are derived for the circular channel and the rectangular channel transverse field magnetic flowmeters. A weight function determines how the voltages which are induced by the flow in the channel of a conducting fluid through the transverse magnetic field of the meter, contribute to the output signal at the meter's electrodes. It is a function of the channel geometry and the relative size of the meter's electrodes. The weight functions are derived by means of Green's theorem and conformal mapping. Green's theorem is used to show that a weight function is equivalent to the electric field for a particular electrostatic problem which has a configuration identical to that of the flowmeter channel. Solving for the weight function of a two-dimensional rectangular flowmeter, this paper then uses conformal mapping to determine the weight functions for two practical configurations: the two-dimensional circular flowmeter, and the rectangular flowmeter with finite length electrodes. Weight function contour plots are presented for both of these cases. The circular channel weight functions are given for electrode span angles of 5 degrees , 45 degrees , and 90 degrees . The rectangular channel weight functions are for electrode length to channel width ratios of 0.01, 0.1 and 1.0.