234 



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



It is clear that the three points I, Gr, H describe spherical ellipses on the 

 sphere of radius unity. We know that — lies between the greatest, A, 



and the least, C, of the principal moments of inertia ; let us suppose that 

 it is also greater than the mean principal moment B. In this case the 

 concavities of the three ellipses will be towards the point A, i. e. the 

 point where the axis of greatest moment cuts the sphere. This point is 

 therefore to be regarded as the common centre of the three ellipses. If 

 Gr 2 



— <B, the common centre would be the intersection of the least axis of 



moment with the sphere. 



In the figure the eye is supposed situated on the axis of greatest 

 moment, viewing the sphere from a considerable distance. All great 

 circles on the sphere are represented by straight lines. 



The semiaxes A K, A L of the eccentric ellipse are evidently given by 



cotan a=4 / G2 ~ BT and cotan S= x / ^ 2 ~ GT . If („', b'Y (a, b) be 



V AT — Gr 2 V AT — Gr 2 V 



the semiaxes of the instantaneous ellipse and invariable ellipse respec- 

 tively, we have 



tan a 



_ tana' 



tan a 



B 



A 



Vab 



tan b 



_ tan b' 



_ tan/3 



C 



A 



~ vTc 



Thus it appears that the ratios of the tangents of the semiaxes are inde- 

 pendent of the initial conditions. 



If we put 1 — e 2 = s | n 2 we may define e to be the eccentricity of the 



invariable ellipse. If e be the eccentricity of the eccentric ellipse, we easily 

 find 



2 _ A B-C 2 _ B-C , 

 6 BA-C' 6 A-C ; 



so that both these eccentricities are independent of the initial conditions. 



