The stability of combined plane Poiseuille and Couette flow of an electricallyconducting fluid under a transverse magnetic field is investigated using linear stability theory.In deriving the equations governing the stability, the so-called magnetic Stokes approximationis made using the fact that the magnetic Prandtl number Pr m for most electrically conductingfluids is extremely small. The Chebyshev collocation method is adopted to obtain theeigenvalue equation, which is then solved numerically. The critical Reynolds number Re c ,the critical wave number α c , and the critical wave speed c c are obtained for wide ranges ofthe Hartmann number Ha and the parameter k = U 0 / (U 0 + nu 0 ), where U 0 is the maximumvelocity of pure Couette flow and nu 0 is the maximum velocity of pure Poiseuille flow. It isfound that a transverse magnetic field has both stabilizing and destabilizing effects on theflow depending on the value of k.PACS Nos. 47.20