6 PRECESSION OF THE EQUINOXES AND NUTATION 







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

 principal axes through the centre of gravity, and Mg expressing the intensity of the 

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

 through which the resultant acts. 



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

 divided by the sine of the angle (0) which 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. 



Cn sin 



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

 which 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 u, equation (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 k. 



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

 circumstances 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 change. 1 



In the case of the earth there is probably no instant of time at which 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 the force, and hence that even its elementary 

 values vary little from expression (II). 2 



1 Such an assumption is made 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 0, for, though the expression . varies inversely 



Cn sins 



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

 proportion. Again, that a evolving body should gyrate around a given axis it is not necessary that 

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

 the moving plane through the axis of figure and the given axis. The general equations of rotation 

 would be the same. 



