The direct and exchange scattering amplitudes for helium ${(1s)}^{2}$ with antiparallel spins in a three-body approximation, and for hydrogen $1s$, are treated. The fundamental differential and integral equations are arrived at by using a simple atomic model given by Allis and Morse. Analytical formulas for differential elastic cross sections are derived for the collisions of an unpolarized electron wave with helium and hydrogen atoms in the above-mentioned quantum states. The second Born approximation is given, and distortion and exchange effects are taken into consideration. The exchange effect is calculated in the second Born approximation and the perturbed exchange integral is calculated analytically to a good degree of approximation.