404 
PROFESSOR G. H. DARWIN ON FIGURES OF 
We next have to consider the values of r, the radius-vector of the ellipsoid, due to 
rotation. 
We might compute from the spherical harmonic formula 
The results so computed will be compared with the others computed as shown below. 
The following Table of the angular velocity and corresponding eccentricity e of the 
equilibrium ellipsoid of revolution is extracted from Thomson and Tait’s ‘ Natural 
Philosophy,’ §772 :— 
o 
or 
€ 
27r 
•3 
•0243 
•4 
•043G 
•5 
•0G90 
•6 
•1007 
•7 
T387 
•8 
TS1G 
From this we find by interpolation that, when 3&r/47r = '09175, e— '472; and, 
when 3 w 2 /47t = T481, e = '594. 
These, then, are the eccentricities of the ellipsoids whose radius-vector is r in the 
two cases /3 = y, /3 = 
The equations to the generating ellipses are 
1 - -0806 
1 - -2228 cos 2 9 
for /3 = y ? 
and 
r 
a 
l - T353 
1 — ‘3535 cos 2 0 
for {3 
1 
5 
The following are the computed values of r/a for each 15° of 6, the latitude, the 
small figures written below appertaining to the case of /3 = T. 
0 = 
0° 
15° 
o 
O 
CO 
45° 
o 
O 
to 
75° 
90° 
II 
r 
a 
1-0429, 
1-0330, 
1-0074, 
•9753, 
•9461, 
•92G4, 
•9194 
1-075, 
1-056, 
1-009, 
•953* 
•906, 
"875, 
•865 
Computing from the spherical harmonic formula, I find 
/3 = }: -= 1-0382, 1'0305, 1'009G, '9809, '9522, '9312, '9235 
CL 
= U0G16, 1-0490, 1-0154, -9692 -9230, -SS92, -8768 
