Abstract The change in the adiabatic invariant (action integral) of a one-dimensional system is investigated in the case that an external parameter is initially a constant, then a slowly varying function of time, and finally a constant again. The method consists of making a canonical transformation from the conjugate position and momentum to a pair of variables which reduce to the action-angle variables when the external parameter does not vary. The canonical equations governing the new variables are solved by a method of iteration, and the solution is used to determine the change in the action integral when the external parameter varies at a finite rate. The adiabatic invariance of the action integral is proved to the approximation in which the canonical equations are solved. The case of a harmonic oscillator whose frequency is changing is discussed.