Physical Structure of the Earth. 241 



the same, the precession would be unaffected by the fluid, and 

 any small inequality of nutation would be totally inappreciable 

 to observation, p. 423, Phil. Trans. 1839. This may be ren- 

 dered more manifest by recalling the general equations for the 

 surface of a fluid obtained by Poisson, Navier, Meyer, and 

 other mathematicians, when the internal friction of the fluid 

 is taken into account. If a, /3, 7 be the angles made by the 

 normal to the curved surface of the fluid, X, Y, Z the compo- 

 nents parallel to the rectangular axes of #, y, and z, it appears 

 that we shall have at the fluid surface, when nearly spherical, 



x=M2 [ 2 S cosa+ (S + £) cos ^ + (S + £) cos 'i']' 



™*[d+D«"+ ■'j-'tg+j).-*], 



where u, v, w are components of velocity parallel to the 

 coordinate axes, and where k is a coefficient depending on 

 friction and viscidity. If no viscidity and no friction exists 

 we must have £ = 0, and hence also 



X^O, Y=0, Z = 0. 



Now as X, Y, and Z are the effective components with 

 which the nearly spherical mass of fluid acts at its surface 

 when each of them is separately equal to zero, it follows that 

 the fluid can do no work at the surface, and the motions of the 

 shell would take place quite independently of the contained 

 mass of fluid when the latter is totally devoid of friction 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 de- 

 termined by the differential equations 



d±_ 1 dV 



dt~ Cnsin0d0' 

 d0_ _1 dV 



dt Cn sin dy{r' 



V is the potential of the rotating solid, C its maximum moment 

 of inertia, and -yjr direction-angles of the axis of rotation. In 

 the case of the Earth has a particular value when it becomes 

 the obliquity of the ecliptic, and ty the longitude of the first 

 point of Aries. It follows that the determination of ty and 

 at any time depends upon C and V. 



