[en] Induced charges on a four-button beam position monitor (BPM) system attached on a beam chamber of narrow rectangular cross sections are calculated as a 2-D electrostatic problem of image charges. The calculation shows that for a narrow chamber of width/height (2w/2h) much-gt 1, over 90% of the induced charges are distributed within a distance of 2h from the charged beam position in the direction of the chamber width. Therefore, a four-button system with a button diameter of (2--2.5)h and no button offset from the beam position is the most efficient configuration. The four-button BPMs used for 8-mm and 5-mm chambers in the APS have relatively low sensitivities because the button locations are outside the range where the induced charge densities are low and the button diameters are less than 2h. Using derived formulae, button sensitivities and beam position coefficients are calculated for the buttons of the most efficient case and of the 8-mm and 5-mm chambers. The formulae may be used to validate the method of computer modeling for BPM buttons on a beam chamber of an arbitrary cross section