FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
303 
Po 
eS^ + 
9 cos^ P COS" 7 
r,> 
f 5 - 2G ) 
In order to construct a figure it will be convenient to adopt as unit of length c, 
the greatest axis of the ellipsoid which is deformed. We know that 
c = ^ ^ h = k cot /3, a = -p , so that h = c cos /3, a ~ c cos y, and the mass oi 
sin /S’ sin /3 
the ellipsoid is ^npc^ cos (3 cos y. But since the mass of the pear is ^TTph^ 
where kQ = k^ (1 it follows that it is 
cos /3 cos 7 
sin'^ /3 
fTrpc^cos/Scosy (1 + -01368666^). 
Hence the mass of the pear is a little greater than that of the ellipsoid whose 
deformations we shall draw, and the protuberances above the surface slightly exceed 
in volume the depressions below it. 
We have 
Bo = 
c cos /3 cos 7 
A.Fi 
c cos (3 cos 7 
(1 — sin^/3 siiF 9f (cos ^7 + sin- 7 cos^ (f)}- ’ 
and the expression for the orthogonal arc, measured from the ellipsoid to the pear, is 
therefore 
- - -+- - - 
2(1 — sin" /3 sin^ 0) 2 (cos" 7 + sin- 9 cos" (p) 
— 1(1 + sec'2 /3 + sec^ y) | + . 
It appears to me that it will afford a sufficient idea of the corrected form of surface 
if I draw two principal sections, namely, first, a section through the axis of rotation 
and the longest axis of the ellipsoid, and, secondly, a section at right angles to the 
axis of rotation. It is not worth while to consider the third section drawn throuorh 
o 
the axis of rotation and the mean axis of the ellipsoid, since it will hardly differ 
sensibly from the uppermost figure shown in the “ Pear-shaped Figure.” 
For the sake of brevity I will call the first and second sections the meridian and 
the equator. 
The three ellipsoidal co-ordinates v, 6, (p of any point are connected with x, y, z by 
the relationships 
X ■= c sin y. (kV" — 1)* (1 — k" sin^ 6)^ cos (p, 
y = c sin y. k[v^ — l)^ cos 6 sin (p, 
z = c sin y. Kv sin 6 {l — cos^ <p)^. 
The equation to the surface of the ellipsoid is ^ • 
^ K sm 7 siu a 
Bo 
eSo + 
