1226 
The ratio of the masses of Hyperion and Titan is only a very 
small quantity. The inclinations of the orbital planes of Hyperion 
and Titan to the aequator of Saturn are also small. To simplify the 
problem I shall neglect these inclinations as well as the influence 
of the mass of Hyperion, the sun, the other satellites, the ellipticity 
of Saturn and the rings. I shall, therefore, suppose Hyperion to be 
a particle with the mass zero, moving in the orbital plane of Titan, 
while the latter describes an undisturbed elliptical motion around 
the centre of a sphere with the mass of Saturn. 
Notation : 
a = semi-major axis, e = excentricity ; 
[== mean anomaly, g = longitude of pericentre; 
M = mass of Saturn, m'= mass of Titan; 
The accented letters refer to Titan, those without accents to 
Hyperion. The units are chosen so, that the constant of attraction 
=1; wand y are coordinates in a system of axes in the orbital 
plane of Titan, the origin coinciding with the centre of Saturn. 
From the fundamental equations, which DELAuNay has used in his 
lunar theory’), the following differential equations for the motion 
of Hyperion result: 
dL OR dG OR 
eat ae are 
db OR dg DR 
Eee en: 
eo a ems lla ee m’ 
2L 5 V (a'—2)* + (y'—y)? 
_ The function R is, with regard to the angular elements, a function 
of /-+4—l—q’, 1, l only. The following new quantities are 
introduced: 
l+qg—l—jJ=#4, 
41 — 3l' + 89 — 39 — 180°=8, 
g—g =, 
gt 
One sees at once that R, with regard to the angular elements, 
is a function of ®, 6, 2 only. The reason why these three quan- 
tities are introduced is this: from the observations the mean motion 
of the argument @ appears to be zero and @ to perform a libration 
on each side of the value @ = 0° with an amplitude of about 36°; 
1) Théorie du Mouvement de Ja Lune I, 13. 
