270 CELESTIAL MECHANICS. I. Vli. 29. 



obviously be the plane in wbich the force itself acts, and 

 which passes through the centre of gravity : this plane is 

 therefore the invariable plane of rotation. Now calling 

 the distance of the direction of the primitive impulse from 

 the centre of gravity /, and v the velocity communicated 

 to the centre of gravity, m being the mass of the body, 

 the rotatory power of the impulse must have been mfv ; 

 and multiplying this hy^tj the product will be equal to the 

 sum of the areas described during the time t, which has 

 already been found equal to |f v'(p'^ + q"^ + r'-) (355); 

 consequently s/ {p'^ -\- q'^ + r'^') = mfv. Hence if we know 

 the origin of the motion, and the position of the principal 

 axes of the body with regard to the invariable plane, as 

 determining the angles Q and (p, we shall have the values of 

 p\ q\ and r' in the first instance, and consequently those of 

 Py q, and r, whence the values of the same quantities may 

 be found for any other time. 



Now if we imagine any one of the planets to be a homo- 

 geneous sphere,"deriving its rotation and its annual motion 

 round the sun from a single impulse, the radius being R, 

 and the angular velocity of revolution U\ r being the dis- 

 tance from the sun, we shall have v:=.rU: and ify be the 

 distance of the direction of the impulse from the centre, it 

 is plain that the planet will acquire a rotatory motion round 

 an axis perpendicular to the invariable plane. If therefore 

 we consider this axis as the third principal axis zf', we shall 

 have 6=zO, and consequently q'zzO and /z:0, and p'=zmfv, 

 or CpzzmfrU, Now, in the sphere, CzzfmR" (356), con- 



sequently /=|- — • -~> whence we have/", the distance 



of the direction of the impulse from the centre of the 

 planet, which corresponds to the proportion between the 



