THE MATHEMATICAL THEORIES OF THE EARTH. IS? 



auy knowledge of the distributiou 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,* and it enables ns to determine the form or ellipticity of the 

 Earth by means of pendulum observations alone. It must be admitted, 

 however, that the values of the ellipticity recently obtained in this way 

 by the highest authorities, Clarke t and Helmert.f are far from satis- 

 factory, whether we regard them in the light of their discrepancy or 

 in the light of the diftereut methods of computing them. In general 

 terms we may say that the difficulty in the way of the use of pendulum 

 observations still hinges on the treatment of local anomalies and on the 

 question of reduction to sea level. At present, the case is one concern- 

 ing which the doctors agree neither in their diagnosis nor in their 

 remedies. 



Turuiug attention now from the surface towards the interior, what 

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

 and of the pressures that exist at great depths? Two facts, namely, 

 the mean density and the surface density, are roughly known ; a third 

 fact, namely, the precession constant, or the ratio of the difference of 

 the two principal moments of inertia to the greater of them, is known 

 with something like precision. These facts lie within the domain of 

 observation and require only the law of gravitation for their verification. 

 Certain inferences, also, from these facts and others, have long been and 

 still are held to be hardly less cogent and trustworthy, but before stat- 

 ing them it will be well to recall briefly the progress of opinion con- 

 cerning this general subject during the past century and a half. 



The conception of the earth as having been primitively fluid was the 

 prevailing one among mathematicians before Clairaut published his 

 Theorie de la Figure de la Terre in 1743. By the aid of this conception 

 Clairaut proved the celebrated theorem which bears his name, and 

 probably no idea in the mechanics of the earth has been more suggest- 

 ive and fruitful. It was the central idea in the elaborate investigations 

 of Laplace and received at his hands a development which his succes- 

 sors have found it about equally difficult to displace or to improve 

 From the idea of fluidity spring naturally the hydrostatical notions of 

 l)ressure and level surfaces, or the arrangement of fluid masses in strata 

 of uniform density. Hence follows, also, the notion of continuity of in- 

 crease in density from the surfjice toward the center of the Earth. All 

 of the principal mechanical properties and effects of the earth's mass, 

 viz, the ellipticity, the surface density, the mean density, the preces- 

 sion constant, and the lunar inequalities, were correlated by Laplace § 



* Stokes, G. G., Math einatical and Physical Papers, Cambridge University Press, 18e0, 

 vol. II. 



t Geodesy, Chap. xiv. 



\ Helaiert, Dr. F. R., Die Matliematischen und PhyaikaUschen Theorieen der HiJheren 

 Geodiisie, Leipzig, IHdO, 1H84, ii Teil. 



^ Mecanique Celtate, Tome v, Livre xi. 



