We shall be concerned chiefly with the method of Weinstein for the approximate calculation of eigenvalues. The concept of an eigenvalue is of great importance in both pure and applied mathematics. A physical system, such as a pendulum, a vibrating string, or a rotating shaft, has connected with it certain numbers characteristic of the system, namely the period of the pendulum, the frequencies of the various overtones of the string, the critical angular velocities at which the shaft will buckle, and so forth.