272 CELESTIAL MECHANICS. I. vii. 30. 



VZi siQyd^.dr=(— sinyi— v/::~icosydOycl^=:— rVl^ 



y' 1 <y^ 



ydf and r = e : consequently V — r=2>/3 



sin yt :=z e ""^^ — e ^^ , and F + T = 2 cos yf = 



v''— -1 y^ •— v^__-i "v^ ___ 



€ +e . And if we substitute V — iv for 



the indeterminate quantity y, we have 2-v/Zri sin >^ZZ\vt 

 = e — e , and 1^ cos v _ivr = € 4-e .] 



359. Theorem. The permanency of ro- 

 tation round two of the principal axes of 

 every irregular body is stable, and round the 

 third unstable. 



We might deduce the laws of the oscillations of a body 

 turning round an axis very near to the third principal axis 

 from the fluents found in the preceding propositions ; but 

 it is more simple to derive them at once from the differen- 

 tial equations (D) of article 346. The forces acting on 



g ^ 



the body being supposed to vanish, we have dp'z=. ^ 



(J R 4 C 



qYdt=:0, dq' + -—~- r'p'dt^iO, and dr' + -— -- p'q'dt = 



0; and substituting Cp, Aq, and Br, for/?', q', and r', dp 



B—A ,, ^ , C—B . ^ , , A—C 



-i ~ — qrdtzzOy dq-\ — rpdtzzO, and drH — — pq 



C> A. B 



dt=0. 



Now supposing the solid to perform its rotation very 

 nearly round the third principal axis, so that q and r may 

 be very small, their squares and their products may ob- 

 viously be neglected in comparison with the other quan- 

 tities concerned; we shall therefore have dp—0, and if we 

 substitute in the other equations the indeterminate values 

 ^iz/Lc sin (ni-^y), and r—f/ cos (nt-ry) [in order to obtain 

 a particular solution of the problem], we shall have nzzp 



