_and the Theory of Saturn’s Rings. 267 
disturbances by the primary, the squares and higher powers of 
= and - were retained; and accordingly the very close ap- 
proximation to a true ellipsoid can be exhibited only when the 
size of the satellite is very small compared with that of its orbit. 
If the disproportion between both were not very great, the form 
of the satellite would resemble that of an egg slightly flattened 
by lateral pressure; yet even in such extreme cases the hypo- 
thesis in regard to the ellipsoidal form can lead to no material 
error in estimating the intensity of gravity on its surface, and 
the dimensions of the smallest orbit in which its parts can be 
held together by their mutual attraction. 
From equations (8) and (14), (9) and (15), and (10) and (16), 
the following are readily deduced :— 
P—N 2M aS 
X=a( A, Tora » or =e e - ° (17) 
2—N M bS 
R M cS 
Z=o( G +=), or = G2 ee (19) 
But calling the force of gravity at the given locality F, it is evi- 
dent that F is equal to “X?+ Y?+Z?. On substituting for X, 
Y, and Z their values given by the last equation, there results 
as? 6282 22 Cre RS 
pe — or =* Ta + abt + (20) 
The quantity under the radical in the last expression is the value 
of the normal of the ellipsoid; and hence the force of gravity 
everywhere on the surface is proportional to the length of the 
normal corresponding to the locality. At the extremity of each 
axis this gravitative power, like the normal, is inversely propor- 
tional to the lengths of the axes themselves—a result which might 
be more readily deduced from equations (17), (18), and (19). In 
the first, for instance, if the point be situated at the end of the 
major axis, a becomes equal to A and X, which then expresses 
that the entire gravity at the point is equal to B. ; while the two 
other equations, (18) and (19), treated in a similar manner, would 
give © and C for the values of the intensity of gravity at the 
terminations of the mean and minor axes. 
The cases in which equilibrium is impossible will be indicated 
by the occurrence of imaginary radicals, when we determine the 
relation between the constant quantities in formule (14), (15), 
