The proton synchrotron being built at Geneva for the European Council for Nuclear Research employs a new magnetic focusing priciples, and minor irregularities in the magnets will cause perturbations in the free oscillations of the magnets will cause perturbations in the free oscillations of the protons. In order to find out the conditions under which the amplitude of these oscillations will not increase indefinitely, calculations have been made using the Pilot Ace.The Paper deals with the mathematical aspects of two kinds of calculations in which methods for evaluting integrals of the type∫0γ(1−k2sin2t)−1/2dt (∫0γ(1−k2sinh2t)−1/2dt)are developed. These methods also show how to evalute the upper limit γ when the integrals are known.