374 Sir W. Rowan Hamilton on Qjinternions. 



rate a system of two reciprocal ellipsoids, by means of a moving 

 sphere. 



61. Let then a sphere of constant magnitude, with centre E, 

 move so that it always intersects two fixed and mutually inter- 

 secting straight lines, ab, ab', in four points L, M, l', m', of 

 which L and M are on ab, while l' and m' are on ab' ; and let 

 one diagonal lm', of the inscribed quadrilateral lmm'l', be 

 constantly parallel to a third fixed line ac, which will oblige 

 the other diagonal ml' of the same quadrilateral to move pa- 

 rallel to a fourth fixed line ac'. Let n be the point in which 

 the diagonals intersect, and draw af equal and parallel to en ; 

 so that aent is a parallelogram : then the locus of the centre v. 

 of the moving sphere is one ellipsoid, and the loats of the oppo- 

 site corner f of the parallelogram is another ellipsoid reci2^rocal 

 thereto. These two ellipsoids have a common centre a, and 

 a common mean axis, which is equal to the diameter of the 

 moving sphere, and is a mean proportional between the great- 

 est axis of either ellipsoid and the least axis of the other, of 

 which two last-mentioned axes the directions coincide. Two 

 sides AE, af, of the parallelogram aenf, are thus two semi- 

 diameters which may be regarded as mutually reciprocal, one 

 of the one ellipsoid, and the other of the other: but because 

 they fall at opposite sides of the 2)ri?icipal plane (containing 

 the four fixed lines and the greatest and least axes of the ellip- 

 soids), it may be proper to call them, more fully, opjwsite re- 

 ciprocal semidiameters; and to call the points e and f, in which 

 they terminate, opposite reciprocal j^oints. The two other sides 

 EN, FN, of the same variable parallelogram, are the normals to 

 the two ellipsoids, meeting each other in the point n, upon 

 the same principal plane. In that plane, the two former fixed 

 lines, AB, ab', are the axes of Itsoo cylinders of revolution, cir- 

 cumscribed about die first ellipsoid; and the two latter fixed 

 lines, AC, ac', are the tioo cyclic normals of the same first ellip- 

 soid : while the diagonals, lm', ml', of the inscribed quadri- 

 lateral in the construction, are the axes of the tvco circles on 

 the surface of that first ellipsoid, which circles pass through the 

 point E, that is through the centre of the moving sphere; and 

 the intersection n of those two diagonals is the centre of an- 

 other sphere, which cuts the first ellipsoid in the system of 

 those two circles: all which is easily adapted, by suitable in- 

 terchanges, to the other or reciprocal ellipsoid, and flows with 

 facility from the quaternion equations above given. 

 [To be continued.] 



