OF THE MOTIONS OF A SOLID BODY. 953 



§ 27. Of the principal axes of rotation. In general 

 a body has only one system of principal axes. Of 

 [7^otatory'\ inertia. The greatest and least inertia belong 

 to the principal axes, and the least of all belongs to one of 

 the three principal axes passing through the centre of 

 gravity. Casein which there is an infinity of principal 

 axes. P. 73. 



347. Theorem. Every material body has 

 at least three principal axes of rotation, at 

 right angles to each other. 



In order to determine the situation of the axes x" y\ 

 and z"y in such a manner as to agree with the conditions 

 of the definition (344) we have (324 Cor.) 

 x"-=.x' (cos d sin \|/ sin (p + cos -^ cos (p) 



+ y' (cos 5 cos -^ sin 9— sin >^ sin <p)— z'sin 5 sin <p\ 

 y"'=.z! (cos d sin -^ cos (p — cos \J/ sin (p) 



-f y'(cos 6 cos -^^ cos (p-f sin -^ sin (p)—2! sin 6 cos ^; 

 z"—x' sin Q sin -^-{-y' sin Q cos x|/ + s' cos 5. Hence we 



readily obtain 

 x" cos (p — y" sin (p^zx' cos -^—y sin ^ 

 af' sin (p-\-y" cos (pz=.x' cos d sin %^+y cos 5 cos -^ — z' sin Q. 



If we now put S/^om =z a2^ Sy^B?n=b', Sz'^dtwizc^ ; 

 Sx'yDm:=zf Sx'z'Dm:=:g, Sy'zfDmzzh, we shall have Sx'V 

 Dm. cos (p — Sy'z"Dm sin ^=[S {x'^ sin 5 sin cos ^^-\-x'y* 

 sin 5 cos ^^l^+x'z' cos d cos 4/ — x'y sin 5 sin ^4^ — f^ sin 9 sin 

 cos A'—y'^ cos 5 sin ^^) D/wn:] (a^ — 6-) sin 6 sin cos ^l' +y* 

 sin d (cos2%|/ — sin -•4') +cos fi (<; cos%{.— A sin ^f/); S arVom. 

 sin (p •\-^y" 2!' nm. cos ^[=z S (x'^ sin cos fi sin^ -vj/ + x'y' sin 

 cos fi sin cos i|/ + a;V cos^ 9 sin 4'+^y sin cos 5 sin cos 4' + 

 y'2 sin cos 5 cos 2^^ + y'z cos *fl cos \^ — xV sin 25 sin -^ — 

 yV sin ^ cos ^— z'^ sin cos 5) D?w]=:sin cos 6 (a^ sin^ 4' + 



