OF THE MOTIONS OF A SOLID BODY. 271 



two velocities. V\^itli regard to the earth, —^ being 



R 



=366.25638,and — . the sun's parallax, = .000042665, / 



is found very nearly pL_ R^ [or about 25 miles]. 



Scholium. The planets not being homogeneous, they 

 may here be considered as formed of concentric spherical 

 strata of different densities, and in this case we have C =:-| 



y^KM/e .^... , n , p f^R^dR , .. 



"^flFdlV (''^^ "'^"^^ f=^ VU' 'jfiiMR'' ^"' ^'' ^^ 



it is natural to suppose, the strata nearest the centre are 



fpR^dR 

 the densest, the quantity ■^^. will be less than |-R«, 



and the value of f will be less than for a homogeneous 

 body. 



§ 30. Of the oscillations of a hodi/ which turns very 

 nearly round one of the principal axes. Stability of the 

 motion round the principal axes of which the rotatory in- 

 ertia is the greatest and the least: instability with re- 

 spect to the third axis, P. 85. 



[358. Lemma. The cosine of an imagi- 

 nary arc may be expressed by a real expo- 

 nential quantity : thus we have sin -s/"!!! 4 = 



and COS n/ — i v^ 



2V-1 2 



If r =: cos yt + /n/Zi sin y#, dFn: — sin yt . ydt + -y/ZI 



cos yt ,ydt :=. T ^/^\ ydt, and — - — dhir = V^iydf, 

 whence r=e , if hle=:l. Again, ifr=: cosy* — 



