208 ON THE PHYSICAL STRUCTURE OF THE EARTH. 



X = hk |_ J ^,-^ COS a + (^ + ^-J COS ft + {^ + ^^) COS r J, 



^ ='"• LCrfx' + #) "^ « + 2 <,y «o« /< + ( rf. + rf, ) ""^ r} 



r, n zT / '^10 , dn\ , / dw , dv\ ,, , ^ dw ~\ 



'^=''" li,lx + dz)'"""+iji, + di) '°' " + - 7. ™'^ '' J' 



\^lle^e u,v,ic are eomponents of velocity parallel to the coordinate 

 axes, aud where fc is a coefficient depending- on friction and viscidity. 

 If no viscidity and no friction exists we must have fc—0, and hence 

 also 



X=0, r=o, Z=(). 



Now, as A", Y, and Zare the effective components with which the nearly 

 spherical mass of tliiid acts at its surface when each of them is separ- 

 ately equal to zero, it follows that the fluid can do no work at the sur- 

 face, and the motions of the shell would take place quite independently 

 of the contained mass of fluid when the latter is totally devoid of fric- 

 tion and viscidity. 



(3) It has long since been clearly shown that the motion of the axis 

 of the Earth, considered as a solid body, may be determined by the 

 differential equations 



V is the potential of the rotating solid. Cits maximum moment of in- 

 ertia, 6 and if.^ direction angles of the axis of rotation. In the case of 

 the Earth, t) has a particular value when it becomes the obliquity of 

 the ecliptic, and tj:/ the longitude of the first point of Aries. It follows 

 that the determination of //; and (J at any time <lepends upon C and V. 

 By analytical transformations, which are fully given by Poisson in 

 his memoir Siir la Rotation de la Terre autour de son centre de Gravite, 

 and by other writers, it finally appears that the variations of aud //' 

 depend on equations in which abactor enters of the form 



2 C-A-B 



where A, B, (7, are the three principal moments of inertia of the Earth. 

 In a spheroid of revolution A = B, and the factor becomes — —p . 



