Resonance cases and small divisors in a third integral of motion. I

In this paper a general discussion of the resonance cases in an axially-symmetric potential field is pre- sented, when the unperturbed frequencies in the radial and z direction have a rational ratio. The general form of the third integral is not valid in these cases because of the appearance of divisors of the form (m2P-n2Q), which become zero in the resonance cases. However, a new isolating integral of the unperturbed case is available, and this can be used to construct a third integral in the form of a power series and eliminate all secular terms. Three cases are distinguished, (a) m+n>4, (p) m+n=4, and ( ) m+n <4. In the first case the orbits are rather similar to those of the general irrational case. In the third case the orbits show a quite peculiar character, which, however, can be explained rather accurately by a first-order theory of the third integral. Numerical integrations were made for the cases P = t6Q, 4P = 9Q, and P = 4Q. The third integral, given in first- or second-order approximation, is rather well conserved. Case p and the cases of small divisors, when m2P-n2Q is near zero but not equal to zero, are discussed in Paper II.