OF THE MOTIONS OF A SOLID BODY. ^73 



AC—JXC—BJ , , AC- A A - , . 



^- -j^ , and /^=: — ^V-g^— ^,/^andybeing 



two constant independent quantities ; and the angular ve- 

 locity of rotation, which is '\/{p^ + q'^-\'r^), will be reduced 

 simply to p, by neglecting the squares of q and /-, so that 

 this velocity may be considered as constant, and the sine 

 of the minute angle formed by the momentary axis of rota- 



tion with the third principal axis will be — -L i. [For 



P 

 the value of dq, being, according to the substitution, fx, cos 



{nt-{-y)mU, and that of dm — // sin {nt-\-y)ndt, we have/* 



cos {nt + y)n-\-2Jfjt! cos {nt-\-y) ——-=0, or y^n-^pf^ 



A A 



=0 and — f/ sin {nt + y)n+p /u sin (nt-\-y) ~~~zzO, or — 



A~C ^ , , A A-C 



/xn±pfx —-—- — [); whence fx=: — f^n — ; TT—py- * 



i> p{t — B) nB 



j^ fi ^ 



and n — — — —= P — —-, consequently w^J^^ziw^ (C—A) 

 p{C~-B) uB 1 ^ z - / 



.(C-i^);and^z._.;, V -^ 'cZrB^~^ ^ 



JffQ J\^\ 



— — ; — — .] Now if, at the beginning of the motion, qzz 

 Jj{L — n) ^ 



0, and rzzO, that is, if the momentary axis of rotation coin- 

 cides with the principal axis, wc shall have /u=:0, f^—O, 

 and q and r will always remain =zO, the axis of rotation 

 always coinciding with the thin! principal axis ; whence it 

 follows that if the body begins to turn round one of the 

 principal axes, it will continue to turn uniformly round the 

 same axis. This remarkable property, belonging to the 

 principal axes, has caused them to be denominated axes of 

 permanent rotation, and it belongs to them exclusively ; 

 for if the momentary axis of rotation be supposed invariable 



