238 Mr. E. J. Routh on the Motion [Mar. 20, 



ellipse may be found by dividing this by cos AIN', where IN' is a normal 

 to the instantaneous axis cutting the major axis in N'. By a property 



o£ the spherical ellipse proved above, we have cos AIN' = ,. 



tan Al . tan n 



Hence, substituting, 



velocity of I 1 a Gr 2 -CT , , nT 



i M- n ■ h = — — f- — tanw.cosGU, 



along its ellipse J T AB 



where n' is the length of the normal to the instantaneous ellipse inter- 

 cepted between the curve and the major axis. 



12. If we compare this formula with the corresponding formula for 

 the velocity of Gr, viz. 



velocity of a \ Gr 2 — GT , 

 along its ellipse J = CO ' 



we see that for every theorem relating to the motion of Gr in its ellipse 

 there is a corresponding theorem for the motion of I. For example, if 

 S' be a focus of the instantaneous ellipse, we see that the velocity of J. resolved 

 perpendicular to the focal radius vector ST bears a constant ratio to cos Grl. 



This constant ratio is = JL A /(AT- Gr 2 )(Gr 2 -BT^ 

 CTV AB 



13. From the two formulae for the velocities of Gr and I we may easily 

 deduce 



angular velocity of I about A __ A 2 /cot AI\ 2 

 angular velocity of Gr about A ~~ BC \cot AGr/ ' 



14. We may apply the same kind of reasoning to find the motion of 

 H along the eccentric ellipse. Beginning with tan d" = * an 0> we 

 find 



velocity of II = — m tan n". 



Vabct 



where n" is the length of the normal to the eccentric ellipse intercepted 

 between the curve and the major axis. 



Hence the velocity of the eccentric line along its ellipse varies as the tangent 

 of the normal to its path. 



15. It may be proved that in any spherical ellipse 



tan 2 n = — ^ n ~ . (cos 2 r— cos 2 a cos 2 b), 

 cos 6 sin a 



where r is the central radius vector, and n the length of the normal. By 

 using this formula we easily find 



(velocity of H) 2 — (velocity of Gr) 2 = constant. 

 This constant is _ (^-AT)(^--BT)(^-CT) - 



