the Solid Crust of the Earth. 285 



another axis OP', such that P' is in the great circle GP, and 

 sin PP' : sin P'G : : angular velocity about G : to angular velocity 

 about P. Hence PP' being extremely small, and n being the 

 velocity of rotation, 



PP ; = 3//./sin /sin co V\ — sin 2 / sin 2 codt -*- n. 



The instant the mass tends to revolve round a new axis OP', 

 close to OP the principal axis of greatest moment (as the earth's 

 axis is), the centrifugal force of the unsymmetrical parts will 

 cause OP to move up to OP' ; that is, not merely is there a new 

 axis in space about which the earth revolves, but the earth's own 

 axis of figure moves up to it. 



Now resolve this small space PP' perpendicular to and along 

 PQ, and integrate, to get the whole effect during one revolution 

 of the sun (that is, of the earth about the sun), by which l=n!t, 

 n' being the earth's mean motion about the sun. Hence, since 



sec ty = \/\ — - sin 2 / sin 2 co -*- sin co cos /, 



space moved over along PQ 



= JPP'cos QPP'= - jPP'cos^-r- sin / 



= — 3 fj,f§ cos I sin 2 co dl -r nn' = 0, from 1=0 to /=27r. 



Hence, though there will be nutation of the axis, the value of 

 PQ or the obliquity will, in a whole revolution, be unaltered. 

 Space moved over by P at right angles to PQ 



= JPP' sin QPP'=JPP' sini/r sin /=3^/sin co cos co^mHdl+nri 



= S7r/jf sin co cos co -r-nn! = Sirfn 1 sin co cos co-^-n, .'. jjl = n n . 



This, measured along the ecliptic, = Sirfn' cos co -r- n, and coin- 

 cides with the usual expression for solar annual precession. An 

 exactly similar expression is true for the moon's action, supposed 

 to move in the ecliptic ; n 11 being the mean motion of the moon 

 about the earth, the lunar monthly precession will be 



37rfn" cos co -r n. 

 Hence the whole 



Annual Precession =3/ , 180°. 



J nn 1 



8. The calculations of the Figure of the Earth* make 

 /=O003136; also n! -r ?i=365-26, n" -r »' = 36526 -r 2732. 



* See my ' Mechanical Philosophy/ chapter on Figure of the Earth. 



