304 PROFESSOR C4. II. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
The equation to the meridian plane in rectangular co-ordinates is simply y = 0, 
that to the equator is cc = 0. 
In elli|)soidal co-ordinates tlie equation to the equator is simply (j) = hut the 
equation to the meridian is peculiar, for it is in part represented by 6 = Itt and in 
part hy (f) — 0. 
The curve 9 = (f> — 0, which defines the limit between the two regions where 
tlie equation to the plane has different forms, is clearly the hyperbola 
3,2 
o o • 0 
= c- sin-' y. 
K “ 
In the region from 2 = co and x small down to tliis hyperbola the equation is 
0 = \ tt •, and between the origin and the hyj^erbola it is = 0. 
If we follow the arc of the ellipse from the extremity of the c axis we begin with 
6 = ^ = -^TT, and 6 remains constant whilst (f> falls to zero. Then (f) maintains a 
constant zero value whilst 6 falls from \tt to zero. 
On the side of the origin where 2 is negative, 6 is of course negative and undergoes 
parallel changes. 
The hyperbola 6 = (f) — 0 cuts the ellipsoid so near to the extremities of the c 
axis that an adequate idea of the deformation may be derived from the two extreme 
values of cf), namely, and 0. I have also thought it sufficient to compute the 
deformations for ^ = 0, 30°, 60°, 90°. We thus obtain the following scheme of values 
of 9, (f), together with the corresponding rectangular co-ordinates (with c taken as 
unity), at which to compute the deformation :—- 
Meridian (;j = 0). 
9 = 
90°, 
(j) — 
0 
0 
2 = 
1, 
X = 
0 
9 = 
0 
0 
(f, = 
0; 
•961, 
X = 
•096 
9 = 
0 
0 
0 
(j) = 
0; 
2 = 
•832, 
X = 
•191 
9 = 
30°, 
(j) = 
0; 
2 = 
•480, 
X = 
•303 
9 = 
0, 
(f> = 
0; 
2 = 
0, 
X — 
•345 
Equator (x = 0). 
9= 90°, = 90°; 2 = 1 , y = 0 
9=G0°, 4 > = 90°; z= -866, y= •2IG 
9 = 30°, = 90° ; 2 = -5, y - -374 
y = 0°, (/) = 90° ; 2 == 0, y = -432 
It did not seem to be worth wliile to conq')ute the deformations due to the eighth 
zonal harmonic, since it would be quite im2)ossible to show them on a drawing of any 
reasonable scale. 
In order to exhibit the magnitudes of the contributions of the harmonics of the 
several orders, I give tlie normal departures hi at the points 2 = 1, .a' = 0, y = 0. 
