174 G. F. BECKER ISOSTASY AND RADIOACTIVITY 



In 1849 stokes^" showed that to the first order of small quantities the 

 relation between gravit}^ and latitude discovered by Clairaut can be de- 

 duced from the Newtonian law of gravitation without any explicit as- 

 sumption as to the form of shells of uniform density. The only express 

 assumption made is that equipotential surfaces, external or internal, are 

 approximately spherical. It follows that the mean figure of the earth, 

 though not its dimensions, can he determined from pendulum observa- 

 tions alone, a task actually performed some time later by the famous 

 geodesist, F. R. Helmert." 



differing but little from that which it would asi5ume in virtue of the laws of equilibrium 

 if, the sea ceasing to cover it, the spheroid were to become fluid. 



"4. The depth of the sea is a small fraction of the difference between the two axes of 

 the earth. 



"5. The irregularities of the earth and the causes which disturb its surface extend to 

 but a small depth. 



"6. Finally, the whole earth was originally fluid. 



"These results of analysis, observation, and experiment ought, it seems to me, to be 

 set down among the few truths which geology' has to offer." 



Laplafp in many passages refers to the earth as solid. The liquid shells, he says, 

 "would changp their shape only very slightly during solidification." Nearly a century 

 has elap.'jed since these conclusions of Laplace were made known, yet it is dotibtful 

 whether they could be modified to advantage. 



In this memoir Laplace is by many supposed to have extended Clairaut's theorem on 

 the variation of gravity with latitude ; but Todhunter's review in his histoiT of the 

 theories of attraction and the figure of the earth, vol. 1, p. 229, denies this, splendid as 

 he considers Laplace"s analysis. Clairaut did not assume, according to Todhunter, that 

 the component shells were fluid or of the configuration corresponding to fluidity, but 

 only that the bounding outer surface has the same form as if it were fluid, and that it is 

 in relative equilibrium when rotating with uniform angular velocity. In the 11th edi- 

 tion of the Encyc. Brit, the article on the figure of the earth is by Clarke, revised by 

 Helmert, and in it Todhunter's conclusion is accepted. 



10 Stokes published two papers on this subject : Camb. and Dublin Math. .Tour., vol. 4. 

 1849, p. 194. and Trans. Camb. Thil. Soc, vol. 8, 1849. p. 672. Both are included in his 

 Collected Papers. 



" Stokes's investigation has been extended by Helmert, who has added to the expres- 

 sion for the variation of gravity with latitude a small negative term of the second order, 



which is maximum in latitude 45°, and there amounts to 7 X lO"*" times gravity at the 

 equator. All modern geodesists accept Helmert's emendation, which is in accord with 

 investigations by George H. Darwin and E. AViechert, indicating a depression of sealevel 

 in latitude 45° of some 3 meters. Besides being an essential part of the history of the 

 subject, this correction serves to show how very nearly the geoid coincides with the 

 ellipsoid excepting for local attractions. 



For a rotating homogeneous mass without rigidity the only figure of equilibrium is an 

 oblate ellipsoid. Strictly speaking, this figure is that of equilibrium only for the case of 

 homogeneity. The external equipotential surface of a heterogeneous globe, in which the 

 masses are either liquid or disposed as if they were liquid, is represented by an algebraic 

 equation of the tenth degree, differing but little, however, from an ellipsoid. The equa- 

 tion of the potential, V, may be written as in Pratt's Figure of the Earth, article 122, 



Here /a. = z'r and of course r^ = x'^ -\- ■tp -'r i*. 

 Making V = C, a constant, 



