1301 
mean value of a function / of 4 which differs very little from 
unity for values of 4 between O and 4,. 
If the formula (3) is extended to the second order, it becomes 
very complicated. The range of #, becomes wider, and therefore 
also of g and «. The formula has been elaborated by Darwin!) and 
Vironnet*). The formulae given by these two authors are very 
different. Darwin starts from a definite assumption regarding the 
constitution of the earth, and thus finds a definite value of «. 
VÉRONNET introduces no assumptions, and consequently only gives 
limits for e. Introducing the above value of MH we find: 
Darwin . …. . &-! = 296.03. 
VÉRONNET . . . 295.84 < e—! < 296.68. 
The lower limit of et corresponds to the case of homogeneity, 
the upper limit to concentration of the whole mass in the centre. 
There can be no doubt, but that the actual distribution is nearer 
the first limit. The agreement of the results of Darwin and VÉRONNET 
is thus complete, and we can adopt the value derived from Darwin’s 
formula. The m. e. of e-! due to the uncertainty of H is + 0.16. 
From the agreement of the results of Darwin and VÉRONNET we may 
conclude that any probable hypothesis regarding the constitution of 
the earth differing from that of Darwin would not cause in st a 
difference exceeding say + 0.10. le thus estimate the total uncer- 
tainty of es! at + 0.19. 
4. However, the value of H used above is the ratio of the true 
moments of inertia. The equation (3) on the other hand is only 
applicable to the 7deal: surface. We must thus try to derive the . 
values of /, and MH, for the ideal surface from the true values J 
and H, and at the same time determine the difference e—e, of the 
compressions of the normal and the ideal surfaces. This will be 
done on the basis of the hypothesis of isostasy. 
Tbe normal surface is the ellipsoid best fitting the geoid. The 
potential on the geoid depends on the true momeuts of inertia. The 
compressions pv and ¢ of the normal surface are therefore derived 
by the equations (1) or (1) and (2) by using the true values of ./ 
and A. The equation (1) or (1') also applies to the ideal surface. 
Consequently 
1) The theory of the figure of the earth to the second order of small quantities. 
Scientific Papers, Vol. Ill, p. 78-118. 
2) Rotation de lellipsoide hétérogène et figure exacte de la Terre. Journal des 
Math. 1912, 4me fascicule. 
86* 
