e PRECESSION' OF THE EQUINOXES AND NUTATION 



equations will be precisely the same, the moments of inertia A and C, referring to 

 principal axes throuirh the centre of gravity, and Mg expressing the mtensity of the 

 resultant of the forces, and y the distance from the centre of gravity of the point 

 throu"-h which the resultant acts. 



In expression (11) Mf/y is the moment of the force with respect to the point 0, 

 dividcHl by the sine of the angle (0) whicli its direction makes with the axis of 

 figure. Denote that moment by L. Then the expression for the velocity of gyra- 



tion (11) becomes, 



L 

 17. • .. ■ „ 



On sm U 



If the body in question, like the earth, is acted upon by forces, the resultant of 

 whicli does not pass through its centre of gravity, its movements about that centre are 

 precisely the same as if that centre were fixed ; in other words, it will gyrate about 

 the line connecting its centre and the origin of the force with a velocity denoted 

 by expression (11). In the case of the earth, however, the direction of the disturb- 

 ing force and its moment are constantly changing, and I have to assume something 

 not proved in what foregoes, viz., that the elementary gyration at each moment of 

 time will be likewise expressed by (11) ; an assumption not (probably) strictly true, 

 since, when the forces are constant in direction and intensity, equation (14) shows 

 (the value of h, ecpiation (15) being substituted) that the gyratory velocity, though 

 its mean is always expressed by (11), varies at each instant unless the value of m 

 has a certain relation to that of Ic. 



Since the integral of these varying elementary displacements shows, under all 

 cir(;umstances of constantly directed force (though these elementary motions of the 

 axis exhibit all possible directions with regard to that of the force), a mean rate of 

 gyration expressed by (11), we may assume that the fact will hold good though 

 the direction and moment of the force chanse.^ 



In the case of the earth there is probably no instant of time at Avhich it is revolv- 

 ing exactly about its axis of figure ; the quantity m has, for it, in all cases, a finite 

 (though exceedingly small) value ; neither observation nor (scarcely) analysis can 

 detect the minute diurnal (nearly) nutations which belong to the diurnal cycloidal 

 movement ; and hence the presumption that the gyration is at all instants perpendicu- 

 lar, or nearly so, to the direction of tlie force, and hence that even its elementary 

 values vary little from expression (11).- 



' Sucli an assumption is mado in all the investigations not, like Laplace's, purely analytical, with- 

 out always giving the true grounds on which it should be based. 



» In reality, if the moment L remains the same for different values of e, the elementary displace- 

 ment produced by the gyration is independent of o, for, though the expression . ^ varies inversely 



. , , ,. Cn sin S 



as sin 9, yet the radius of the small circle on which the displacement takes place increases in like 

 proportion. Again, that a revolving body should gyrate around a given a.xis it is not necessary that 

 the accelerating force .should be always parallel i„ direction to that axis, but that it should remain in 

 the uioying plane through the a.Kis of figure and the given axis. The general equations of rotation 

 wouhl be the same. 



