191 



The solution of these equations will give the values of tv l} w 2) and w 3 , 

 and these being known, the value of iv = a/ w i 2 + w 2 2 + or the value 

 of the velocity about the actual or instantaneous axes of rotation is 

 known also ; and if any one of the quantities iv u a> 2 , or iv 3 , contain any 

 term which increases with the time, and is not periodic, there will be 

 a permanent change. The body is supposed to be rotating very nearly 

 about the axis of %, about which u> 2 is the velocity, and where u> u tv 2 , w 3 , 

 are the angular velocities about the principal axes, A, B, C, the moments 

 of inertia about the same ; also x x , y u z l} are the co-ordinates of any par- 

 ticle m referred to the principal axes, the origin being at the centre of 

 gravity of the disturbed body, and x, y, %, are the co-ordinates of the 

 disturbed body, the plane of x, y being that of the orbit, the intersec- 

 tion of the planes of x u y ]} and x, y, the axis of x, and the origin as 

 before. 



Let % be the angle between the planes of x, y, and x lf y x ; 0 the lon- 

 gitude of the disturbed body measured from the axis of x on the plane 

 of the orbit, 0 the right ascension of the axis of x l measured on the 

 plane of x u y, that is, the angular distance of the axis of x x from that 

 of x ; r the distance of the disturbing body from the centre of the dis- 

 turbed, and r x the distance of any particle m from the origin. Then we 

 shall have by spherical trigonometry, 



x 1 + cos i , , _ 1 - cos i . 



- = 1 — cos (0 - 6) + — cos (0 + 0) 



y 1 + cos i . , , ^ 1 - cos i . 



^ = — sm(0-0) — sin (0 + 0) 



% = - sin i sin 9. 



Equations which are usually given in the form 

 x 



- = sin 6 cos 0 + cos t cos 0 sin 0, &c. 



r 



But the form given above will be much the most convenient for the 

 present purpose. The equations (A) can only be solved by successive 

 approximation. The first approximation will be when the right hand 

 member is 0, that is, when there is no disturbing force, or when the dis- 

 turbed body is spherical. The next will be when the bodies are supposed 

 to be spheroids of revolution. This very nearly represents the case of 

 the heavenly bodies ; but inasmuch as they have a variety of irregu- 

 larities both of form and density, will not accurately do so ; and it be- 

 comes therefore necessary to examine what will be the general effect of 

 the said inequalities of surface ; and specially to see whether there will 

 be any permanent alteration in the velocity of rotation arising from 

 them. Supposing such to exist, it is manifest that in consequence of 

 the bodies being so nearly spherical, it will take place very slowly ; 

 but the ultimate amount of alteration will be none the less than if the 



