The resistive tearing instability of an incompressible plasma is investigated for the plane sheet pinch in which the equilibrium magnetic field, x̂Bx0+ẑBz0, depends only on y. The usual assumption is to take v0=0, but here the effect of a nonzero v0 is studied. A linear, time-dependent model is used in which perturbations take the form f1(y,t)exp [i (kxx+kzz)]. A new initial-value code has been developed to solve the resulting higher-order system of equations. For a symmetric magnetic equilibrium and modes α<1, where α=a (kx2+ky2)1/2, an exponential growth develops. The growth rate, p=ωτr, is computed as a function of α and S=τr/τh, for several values of v0. The effect is to reduce p for all α, and to reduce the marginal α for instability for values of v0 of the order of the resistive diffusion velocity. Results for larger values of v0 are briefly discussed. For asymmetric tearing, the effect of the diffusion velocity depends on its sign. The velocity may have either a stabilizing or destabilizing influence on both the growth rates and the critical α for instability.