and on the Rotation of a Heavenly Body. 211 



Note on the Rotation of a Heavenly Body. 



When a distant body is attracted by a central force^ the result- 

 ant attraction does not in general pass through the centre of 

 gravity of the attracted body, but produces an angular accelera- 

 tion about an axis passing through this point. For instance, 

 let the attracted body be a uniform straight rod, whose dimen- 

 sions are small in comparison with its distance from the central 

 force; and suppose the centre of gravity of the rod to move in a 

 circle about the centime of attraction with a uniform angular 

 motion {?i). Suppose, for simplicity, that the rod is situated in 

 the plane of motion, and that <f) is its inclination to a line drawn 

 through the centre of the circle. If t be the time v/hich has 

 elapsed since the centre of the rod was in the initial line, the 

 equation of angular motion is 



^ = u^sm2{nt-cl>), 



where a is a small constant quantity. The equation of motion 

 is the same for any solid body whatever. 

 If the equation be written in the form 



-^l(^=,Hin2{nt-4>), 



it immediately admits of integration ; 

 d{nt — ({)) 



J =+ »/ c^-\-ct^ cos 2{nt — ^), 



c being an arbitrary constant. In order that the motion may 

 be real for all values of t, it appears that c must not be less than a. 

 When cz=ci., the equation can be completely integrated. In 

 this case we have 



^^ ^ -±oi\^2 COS {nt-4>) ; 



1 1 + sin {ni — cji) 



d<f)_ 2a\^2 



If / be increased indefinitely, this becomes 

 d<b 



P2 



