The resonant extraction method previously proposed for the normally constant gradient synchrotron has been extended to alternate gradient accelerators. It is found that perturbation field gradients which contain circular functions of arguments 2&ngr;x, 2&ngr;x+1, 2&ngr;x+2 are particularly effective in causing the radial betatron oscillations to grow exponentially at a particular azimuthal position. The analytical procedure for predicting which perturbations give optimum results, and digital computer calculations verifying these predictions, are presented.