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PHILOSOPHICAL SOCIETY OF WASHIXGTOX. 
S4th Meeting. November 16, 1887. 
The Chairman presided. 
Present, ten members. 
Mr. Ormond Stone read a paper on 
THE ORBIT OF HYPERION. 
[Abstract.] 
The principal difficulty in the integration of the equations of motion 
in the case of the problem of three bodies arises in the integration of 
terms involving the inverse powers of the distances between the dis¬ 
turbed and disturbing bodies. When the ratio between the radius 
vectors is not too near unity, the inverse powers referred to can be 
developed in rapidly converging series in terms of cosines of multiples 
of the elongation. When, however, these ratios do not differ greatly 
from unity the convergence of the series mentioned is very slow. If, 
in addition, the mean motions of the two bodies are nearly com¬ 
mensurate, the ordinary methods of solving the problem become in¬ 
applicable. 
Such a case presents itself in the determination of the perturba¬ 
tions of Hyperion produced by Titan. On the other hand the mutual 
inclination of the orbits of these satellites is so small as to have eluded 
detection; the eccentricity of the orbit of Titan is less than 0.03, and 
the position of the aposaturnium of Hyperion so nearly coincides, 
at least at present, with the point of conjunction of the two satellites 
as to give rise to a suspicion that the eccentricity of its orbit is, in 
reality, small, and that the apparent eccentricity is principally due 
to the perturbations produced by Titan. 
In view of these circumstances, that part of the disturbance has 
been investigated, which may be determined by neglecting the mutual 
inclination and the eccentricities of the orbits of both bodies, reserv¬ 
ing a discussion of the remaining portion for another paper. It was 
accordingly assumed that 
r = a (1 -b cos 0 cos 2^ 
^ (1 q- cos d -f- cos -b . . .) ; 
where r and w are the radius vector and longitude in orbit of Hype¬ 
rion, 0 is the mean angular distance between the radius vectors of 
