1299 
Wwe cannot but take this latter as the normal surface. In that case 
the normal surface is very nearly an equipotential surface. The 
deviations of the geoid from the ellipsoid, or, which is the same 
thing, of the normal surface from the equipotential surface, are 
caused by the irregularities in the crust. They would be very much 
larger — in fact of the order of 1000 meters *) — if there were 
no isostatic compensation. If this point of view is adopted, then the 
normal surface can differ only very little from the “ideal” surface 
S, as detined above. This will be assumed in what follows and no 
further reference will be made to the surfaces S, and S,. They were 
only discussed here to point out the necessity of precision in the 
‚definition of the relation between the isostatie and the normal surfaces. 
3. Let A< B<C be the moments of inertia of a body about 
the axes of a, y, 2. If the body rotates about the axis of z with the 
velocity w, then the outer surface, if it is an equipotential surface, 
is very nearly *) an ellipsoid whose principal axes are 
b, b(1—v), &6(1—4pr) (1—e). 
If C—A and C—B are of the first order of smallness, and B—A 
of the second order, and if 
26 A-~ B BA 
fa ee a 
el 2 Mb? Mb? 
then to the second order inclusive we have 
sd koke hen, AB, 2. ed) 
aen ne kend a (EN 
The radius of the equator in longitude 2 is b[1—v sin? (A—2,)], 
if A, be the longitude of the axis of z. The compression of the 
meridian in longitude 2 is thus &, = € + 5 r cos 2 (à —2,). Consequently 
e is the average compression of the meridians. 
The value of o, in (4), viz. 
N 1 
* = 0.0034496, 
g 1 
‘an be assumed to be exactly known. Further 
, = 0.0000029. 
The equation (1) can thus be written 
Bh OOP TOS EEN fP) 
1) Hetmert, Höhere Geodäsie, Il, p. 356. 
*) The deviation from the ellipsoid is — xb sin? 2c, where 
%=eo-- fe? + 8 B, — 0.0000051, 
or bs = 3.26 meters. DARWIN, Scientific Papers, Vol. Ill, p. 102. 
86 
Proceedings Royal Acad. Amsterdam. Vol. XVII. 
