1873.] of a Body about a Fixed Point. 239 



16. The invariable axis OG being fixed in space, it will be useful to 

 find the motion of I relatively to G. 



Let p be the radius of curvature of the spherical ellipse described by 

 G-, and let GI = z. Then clearly 



velocity of I resolved 1 _ /velocity \ sin (p + z) 

 perpendicularly to GI J ~~ ^ of Gr J' smp 



T 



= — tan z . (cos z + cot p srn z). 

 Gr 



But it may be proved that in any spherical ellipse 



, tan 3 n 



tan p = — : 



v tan 2 I 



where n is the length of the normal, and 21 the length of the latus 

 rectum. Substituting for tanp, and remembering that the ratio 



tanz = G 2 -CT 



tann~ CT ' 



we get 



angular velocity \ T , T /G 2 -CTy / tan 2 b \2 

 of I about G I = a + Gr\rcr-J \-^J COt *' 



where a and b are the semiaxes of the ellipse described by Gr. If we 

 substitute for tan a, tan b their values, we get 



angular velocity of 1 T (AT - G 2 )(BT - G 2 )(CT - G 2 ) 2 

 I in space about Gr J ~~ Q- ABCG-T 2 C ° Z * 



which agrees with Poinsot's results. 



17. We may put this result into another form. Substituting in art. 15 



T 



for the velocity of Gr, viz. tan 2, we get an expression for the velo- 



Gr 



city of H in terms of z. Comparing it with the above result, we get 



angular velocity of I G cQt3 _ f 

 I m space about Gr J T 



18. 27i0 angles described in space round Gr by the arcs GA, GS m«y 

 6e represented geometrically thus : — 



Let ^ be the angle described by GrA ; then, as proved in books on 

 Eigid Dynamics, 



dt Gr T AG 



Let f) be the angle GAD, then it has been proved above (art. 9) that 



«Z0 AT — G 2 



