R. S. Woodward — Mathematical Theories of the Earth. 341 



stability of the earth's axis of rotation."* In a similar sense we 

 may say all geodesy rests on the direction of the plumb-line. 

 The simple hypothesis of a spheroidal form, assumes that the 

 plumb-line is everywhere coincident with the normal to the 

 spheroid, or that the surface of the spheroid coincides with the 

 level of the sea. But this is not quite correct. The plumb- 

 line is not in general coincident with the normal, and the act- 

 ual sea-level or geoid must.be imagined to be an irregular 

 surface lying partly above and partly below the ideal spheroid- 

 al surface. The deviations, it is true, are relatively small, 

 but they are in general much greater than the unavoidable 

 errors of observation and they are the exact numerical expres- 

 sion of our ignorance in this branch of geodesy. It is well 

 known, of course, that deflections of the plumb-line can some- 

 times be accounted for by visible masses, but on the whole it 

 must be admitted that we possess only the vaguest notions of 

 their cause and a most inadequate knowledge of their distribu- 

 tion and extent. 



What is true of plumb-line deflections is about equally true 

 of the deviations of the intensity of gravity from what may 

 be called the spheroidal type. Given a closely spheroidal 

 form of the sea level and it follows from the law of gravita- 

 tion, as a first approximation, without any knowledge of the 

 distribution of the earth's mass, that the increase of gravity 

 varies as the square of the sine of the latitude in passing from 

 the equator to the poles. This is the remarkable theorem of 

 Stokes, f and it enables us to determine the form, or ellipticity 

 of the earth, by means of pendulum observations alone. It 

 must be admitted, however, that the values for the ellipticity 

 recently obtained in this way by the highest authorities, 

 Clarke;}: and Helmert,§ are far from satisfactory whether we 

 regard them in the light of their discrepany or in the light of 

 the different methods of computing them. In general terms 

 we may say that the difficulty in the way of the use of pendu- 

 lum observations still hinges on the treatment of local anoma- 

 lies and on the question of reduction to sea level. At present, 

 the case is one concerning which the doctors agree neither in 

 their diagnosis nor in their remedies. 



Turning attention now from the surface toward the interior, 

 what can be said of the earth's mass as a whole, of its laws of 



*Toute l'Aslronomie repose sur I'invariabilite de l'axe de rotation de la Terre 

 a la surface du spheroide terrestre et sur runiformite de cette rotation. Me- 

 canique Celeste, Paris. 1882, tome 5, p. 22. 



f Stokes, G-. G-., Mathematical and Physical Papers, Cambridge University 

 Press, 1880, vol. ii. 



% (jeodesy, Chapter xrv. 



§ Helmert, Dr. F. R., Die mathematischen und physikalischen Theorien der 

 hohererj Geodasie, Leipzig, 1880, 1884, II Teil. 



