In determining the intensity of gravitation from the oscillations of a pendulum, and in estimating the directive power of the earth's magnetism from the frequency of vibration of a needle, we have to make allowance for the extent of the arc through which the motions take place.

The formula usually given for this purpose contains a series converging rapidly enough for very small arcs, but which requires a great amount of labour when the angle of deflection exceeds twenty or thirty degrees.