FIGURE AND STABILITY OF A LIQUID SATELLITE. 
177 
The three roots of the cubic 
* +Ji-+^U = i 
were 
a 2 + u b 2 +u e 2 +u 
7 2 2 . 7 2 2 7 2 1 — ft COS 2(f) 
Ul = k V , u 2 /•>“, u 3 = k - 1 Q 
Lastly v ranges from oo to 0, /x between + 1, <ft from 0 to 2n. 
In the two later papers, I jJut 
.2 _ 1 — ft >2 i 2 1 
K = 
1 +/3 
'2 1 2 
, K = 1 — K , V = 
/c sin y 
ix 
= sin 6 ; 
and for convenience I introduced an auxiliary constant ft (easily distinguishable from 
the ft of the previous notation) defined by sin ft = k sin y. 
The squares of the semi-axes of the ellipsoid were then 
a = 
7 2 2 
k cos y 
_ Z' 2 cos 2 /3 _ Z; 2 
sin 2 ft ’ sin 2 /3 ’ sin 2 ft 
The rectangular co-ordinates became 
X COS y ,, 2*2 2 / 1 / COS 2 ft o /] * 2 I 2 2 1 * 2 /W t /2 2/\ 
F = sm^ (1_K " Sm ^ 00S F = *F^ C0S#Sm ^ P“3n i /5* m *< 1 -* «**>• 
The roots of the cubic were 
w, = 
Z: 2 
% = F sin 2 9, u 3 = — ^ ( 1 — k:' 2 cos 2 
1 sin 2 /3 ’ 
The notation employed for the harmonic functions is that defined in “ Harmonics.” 
§ 5. The Determination of Gravity on Roche’s Ellipsoid. 
In Roche’s problem a mass of liquid, which assumes approximately the form of an 
ellipsoid, revolves in a circular orbit about a distant centre of force without any 
relative motion. In the present section it is proposed to evaluate gravity on the 
surface of this ellipsoid. I intend to solve the problems of the present paper by 
means of the principles of energy, and for that purpose it is necessary to determine 
the law of gravity. 
Suppose that the ellipsoid of reference, defined by z' 0 , is deformed by a normal 
displacement defined by the function pf ( y, <£), where p is the perpendicular from the 
centre on to the tangent plane at y, <f>. This deformation must be expressible by a 
series of ellipsoidal harmonic functions, and therefore we may assume 
/(/X, (-/») = 2e/i/(/x) (£/(<£). 
2 A 
VOL. CCV1.—A. 
