RESULTING FROM THE TUEORY OF TUE GYROSCOPE. 



Let a, be the equatorial diameter of the earth. 

 6, be the polar diameter, 

 p, its variable density. 



(7, its moment of inertia about the polar axis. 

 A^ its moment of inertia about an equatorial one, 



e= , earth's ellipticity. 



a 



S, the absolute attractive force of the sun, or its attraction upon a unit of 



mass at a unit's distance. 

 r, the mean distance of centres of sun and earth, x, ij, z being rectangular 

 co-ordinates of any element of the earth's mass, dm ; the origin being 

 the earth's centre, tlie axis of z the polar one, of x an equatorial one 

 in a plane passing through the sun's centre, of y an equatorial one per- 

 pendicular to this plane. 

 The moment of the sun's attractive force upon the earth is shown in various 

 works on precession [vide Mr. Airy's " Figure of the Earth," Encyc. Metropolitana) 

 to be (0 being the angle of earth's axis with line drawn to sun). 



1 8. "3^ rrr p ^^"—^") ^^^ ^-^n ^^^ ^"^ ^ ^^^ ^ 



the integral being taken through the spheroid. 



Tlie quantity under the signs of integration may be written 



p (jx--\-ij") dm — p (y"-|-s') dm, the integral of the first terra of which is C, and of 



the second A. 



Hence the moment of the sun's force (18) 



19. ^{C—A) sin cos 0=L 



Hence the gyration produced upon the earth by the sun's force aoout the line 

 of its direction is (17) 



cos ; and in time df. 



20. 



21. 



r^n C 

 3S C—A 

 C 



cos -0 dt 



n 



Let EST be a great circle in the plane of the ecliptic, 

 EE' an equatorial one, PE'TP' a great circle through 

 the tropics, PSP' one through the sun in any position, 

 C the centre of the cartli, and PCP' its axis. SCP' is 

 the angle 0. If the sun moves in the ecliptic from E 

 (the equinox) towards T with an angular velocity Wj, 

 nj, will be the value of the arc ES. In the spherical 

 triangle P'ST, right-angled at T, Ave have cos P'S (or 

 cos 0)=cos PTcos ra=sin TE' sin SE. 



TE' is the inclination of the equator to the ecliptic ; 

 call this /. Then 



cos fl^sin 1 sin n-ft 



