45^ Sir W. Rowan Hamilton on Quaternions. 



Such, then, is the expression for the square of the distance 

 of either focus of the new or derived ellipsoid of revolution, 

 which has A^ for its varying vector, from the common centre 

 of the new and old ellipsoids, which centre is also the common 

 origin of the vectors X, and p : while these two foci of the new 

 ellipsoid are situated upon the mean axis of the old one. 

 There exist also other remarkable relations, between the 

 original ellipsoid with three unequal semiaxes a, b, c, and the 

 new ellipsoid of revolution, of which some will be brought 

 into view, by pursuing the quaternion analysis in a way which 

 we shall proceed to point out. 



80. The geometrical construction already mentioned (in 

 articles 64, 71, &c.), of the original ellipsoid as the locus of 

 tlie circle in which two sliding spheres intersect, shows easily 

 (see art. 72) that the scalar coefficient^, in the equations (146.) 

 of that pair of sliding spheres, becomes equal to the number 2, 

 at one of those limiting positions of the pair, for which, after 

 cutting, they touch, before they cease to meet each other. In 

 fact, if we thus make 



5=2, (217.) 



the values (145.) of the vectors of the centres will give, for 

 the interval between those two centres of the two sliding 

 spheres, the expression 



T{ix,-X<)=gT{n-Q) = 2b; .... (218.) 



this interval will therefore be in this case equal to the diameter 

 of either sliding sphere, because it will be equal to the mean 

 axis of the ellipsoid : and the two spheres will touch one an- 

 other. Had we assumed a value for g, less by a very little 

 than the number 2, the two spheres would have cut each other 

 in a very small circle, of which the circumference would have 

 been (by the construction) entirely contained upon the surface 

 of the ellipsoid; and the plane of this little circle would have 

 been parallel and very near to that other plane, which was 

 the common tangent plane of the two spheres, and also of the 

 ellipsoid, when g received the value 2 itself. It is clear, then, 

 that this value 2 of ^ corresponds to an umbilicar point on the 

 ellipsoid ; and that the equation 



S.(9->,)p=S2_,,2, .... (219.) 



which is obtained from the more general equation (148.) of 

 the plane of a circle on the ellipsoid, by changing^ to 2, re- 

 presents an umbilicar tangent plane, at which the normal has 

 the direction of the vector *j— 5. Accordingly it has been 

 seen that this last vector has the direction of the cyclic normal 

 «; in fact, the expressions (131.), for yj and 5 in terms of » and 



