ACTED ON BY NO EXTEBXAL FOBCES. 



759 



Op, and rj) perpendicnlar to Op, and make Or=r, Op=p ; then by virtue of what has 

 been remarked above, r, R, lie in a hyperbola, of wliicli OpP, is an f* i 



asymptote, and the rotation about the instantaneous axis OK is repre- 

 sented by L.P.Oll,, and may be resolved into L.P.O;-' about 0/ and 

 L.PyR, about Op. 

 But 



L.P.Or'=L.P.r.g^'=L.r.P.^=L.r.P.^=L.r.^, 



and 



L.P.E,r=L.P(OP,-P,R,tanr,0^;) 



=L.p(0P-P.R.^f) 

 ==L.p(p-^.f) 



i«-X' A*-A' c*-X 



, we derive the 



=L(P='-jp^)=lA; 



for if a, (3, y be the angles which OP, Op make with the axes of the ellipsoid, 



P==«2(cos a)H*'(cos /3)=+(r'(cos y)\ 



2)^=(a' -X)(cos uy+(b^-K)(cos (5)-+(c'-X){cos y)*, 



V^-p'-z=X{(cos ay +(cos (3y+(cosyy\ =X. 



Observing, then, that tlie motion has been resolved into a variable rotation Lpr about 

 Or, and a uniform rotation LX about Op, and that accordingly the motion of a free body 



whose moments of inertia are as -j ; r^ ; i differs only by the uniform rotation lA from 



that of another one whose moments of inertia are as 



following theorem : — 



If the reciprocals of each of the moments of inertia of any number of Hgid bodies 

 B, B„ Bo, B3, ... differ from one another by constant quantities, say those of the second, 

 tliird, fourth, &c., from those of the first by \, X^, X3, . . ., and these bodies be arranged 

 with their corresponding principal axes parallel and be set in motion by an impulsive 

 couple L given in magnitude and direction, then, after the lapse of any interval of time t, 

 the principal axes of all the bodies will remain egually inclined to the axis of the given 

 couple, and moreover the parallelism of the axes may be restored by turning B,, Bj, B3, . . . 

 about the axis of the couple through angles proportional to the time, viz. lA,t, LTut, LXjt, 

 . . . respectively. 



It may be further noticed that if at any moment of time a, u are the angular velocities 

 of JB, Bi about their respective instantaneous axes, 



«»-»?=L'(P».R*-/.r') 



= L'(P'(R'-P)-i>'(r'-y)) + L'(P-p*) 



5l2 



