Sir W. Rowan Hamilton on Quaternions. 59 



the locus of all the old pairs of curves, obtained in the first 

 mode of description. 



63. The ellipsoid (with three unequal axes), thus generated, 

 is therefore the common locus of the four curves, described by 

 the four points ee'e"e'"; of which four curves, the first and 

 third may be made to coincide with any arbitrary curves on 

 that ellipsoid', but the second and fourth become determined, 

 when the first and third have been chosen. And in this new 

 system of two connected constructions for generating an ellipsoid, 

 as well as in that other construction* which was given in ar- 

 ticle 61 for a system of' two reciprocal ellipsoids, the two former 

 fixed lines, ab, ab', are the axes of two cylinders of revolution, 

 circumscribed about the ellipsoid which is the locus of the 

 point E; while the two latter fixed lines, ac, ac', are the two 

 cyclic normals (or the normals to the two planes of circular 

 section) of that ellipsoid. The common (internal and external) 

 bisectors, at the centre a, of the angles bab', cac', made by 

 the first and second, and by the third and fourth fixed lines, 

 coincide in direction with the greatest and least axes of the 

 ellipsoid ; and the constant length b, of the side of either 

 rhombus, is the length of the mean semiaxis. The diagonal 

 lm' of the first rhombus is the axis of a first circle on the ellip- 

 soid, of which circle a diameter coincides with the second 

 diagonal ee' of the same rhombus; and, in like manner, the 

 diagonal l'm of the second rhombus is the axis of a second 

 circle on the same ellipsoid, belonging to the second (or sub- 

 contrary) system of circular sections of that surface : while the 

 other diagonal e"e'", of the same second rhombus, is a dia- 

 meter of the same second circle. In the quaternion analysis 

 employed, the first of these two circular sections of the ellip- 

 soid corresponds to the equations (1 13.) ; and the second cir- 

 cular section is represented by the equations (114.), of the 

 foregoing article. 



64. We may also present the interpretation of those qua- 

 ternion equations, or the recent double construction of the 

 ellipsoid, in the following other way, which also appears to be 

 new; although the writer is aware that there would be no 

 difficulty in proving its correctness, or in deducing it anew, 

 either by the method of co-ordinates, or in a more purely geo- 

 metrical mode. Conceive two equal spheres to slide within two 

 cylinders (of revolution, whose axes intersect each other, and of 

 which each touches its own sphere along a great circle of con- 

 tact), in such a manner that the right line joining the centres of 

 the spheres shall be parallel to a fixed right line ; then the locus 



* See Phil. Mag. for May 1848; or Proceedings of Royal Irish Academy 

 for November 1847. 



