OF THE MOTIONS OF A SOLID BODY. ^65 



this determination relates only to the situation of the mo- 

 mentary axis of rotation with regard to the principal axes, 

 and to the angular velocity of rotation. In order to ascer- 

 tain the true motion of the body with respect to a quies- 

 cent space, the position of the principal axes, witli regard 

 to that space must be known ; and for this purpose three 

 new independent quantities are required, and three more 

 integrations, which united to the former, afford the com- 

 plete solution of the problem. The equations (C), of article 

 345, include three independent quantities, N, N\ and N", 

 but they are not altogether distinct from H and k, for if 

 we add together the squares of the first members of the 

 equations (C), we have/>'«+9'*+/*»=iV'^+N'2 + N"2=K 

 [For these equations are q' sin 9 sin ^ -f r sin Q cos <p—p' 

 cos Ozz. — N; (q* cos sin (p-^r' cos cos (p-\-p' sin 6) cos -^ 

 + (r' sin (p—q' cos (p) sin 4^= — N', and — (q' c6s fl sin ^ + i^ 

 cos 6 cos ^ -^ p' sin 0) sin \H-) /sin f> — 5^ cos (p) cos 4.= — 

 N'\ which may be called (a -f /3 — 7), (S' +£ +^) cos^/ + »> 

 sin 4^, and — (S + £+(^) sin -^-{-n cos ^ ; now the sum of the 

 squares of the two latter quantities is (5-f-£ + ^)' + ir, and 

 the whole becomes (a +^— y)^+ (S 4-£+<r)^ + ''^ which, 

 since here (a + |3) . y z= (g + c) 4:, is equal to (a + ^)2+y^ + 

 (S + £)2+4:2-f »»2: now (a + ^y = q^ sin »6 sin ^^ -f- r'" sin^d 

 cos ^p -h 2qy sin ^9 sin cos <p, and (8+ e) «= q'^ cos ^ sin ^(p 

 +r * cos H cos ^<p+2q'r' cos ^fi sin cos (p\ their sum being ^^'^ 

 sin 2^ 4- /^ cos 2^ + 2<yV sin cos <p, to which adding 7^^+^ 

 +«* or jp'2 cos 264. 4-^'2 sin 25^/ 2 sjj, 2^^.^'2 cos ^(p—2qV 

 sin cos ^, we have ^na\\y p' ^-^q'^-^r^zz'N^-{.N'^ + N'\'] 

 The constant quantities, N, N\ and N'', correspond 



XQ^if i/dx 



to c, c', and c" of article 323 [c being there 2m — ^j^ — , 



and N here S'^J^2!-J^dm]; and the quantity ^t >/(;?'* + 

 d^ 



