1873.] 



of a Body about a Fixed Point. 



235 



4. Since the body is turning about 1 as instantaneous axis, it is evi- 

 dent that the motion of Gr in the body is perpendicular to the great circle 

 I Gr. Hence I Gr produced is a normal to the invariable ellipse. 



5. If w be the angular velocity of the body, the angular velocity of O Gr 



A 



is bi sin G-I. But the resolved angular velocity of the body about O Gr is 

 T 



constant, and = _ by a theorem due to Lagrange. Hence the velo- 

 Gr 



T A 



city of the invariable line along its ellipse is ^ tan GrI. Also the re- 



T 1 



sultant angular velocity of the body is ^ 



6. Let the great circle GrI cut the axis major in N, so that GKN" is a 

 normal to the invariable ellipse. Referring the lines to the principal axes 

 at the fixed point as coordinate axes, the direction cosines of OGr, ON 

 are easily seen to be proportional to 



Ao) v Bw 2 , Cw 3 , 

 (A-CK, (B-C> 2 , 0. 



Hence 



cogQN- A(A- CK + B(B-C)< 

 Gr -V (A— C) 2 ^ 2 + (B - C) 2 w 2 2 

 Substituting for w x , w 2 their known values in terms of w, 



tanGN=CyGV=T 2 . 



GP-CT 



But (o cos Grl= q_ ; .-. tan Grl= ^ — — . 



It follows at once that the ratio ^ an !^ is constant throughout the motion 

 q_2 q/ji tan (xJN 



and = —jjji — 



It also follows that the extremity of the invariable line moves along its 

 ellipse ivith a velocity tvhich bears a constant ratio to the tangent of the length 

 of the normal to the ellipse intercepted between the curve and either axis. 



7. Let us find the motion of the invariable axis referred to one focus, 

 S, of the ellipse described by it. 



The velocity of Gr resolved perpendicular to the focal radius vector S Q 

 is, by what precedes, 



G 2 -CT 



CG- 



tan GrN . cos SGN. 



It may be shown that in any spherical ellipse the projection of the 



normal on either focal radius is constant. If this constant be called Z, 



t . 7 tan 2 b 



we have tan I— 



tana 



u2 



