Sir W. Rowan Hamilton on Quaternions. 439 



inscribed in a circle of the derived ellipsoid of revolution, are 

 parallel respectively to the vectors >j+fl, >)~* + ^"^ or to the 

 two umbilicar vectors co, «', of the original ellipsoid, with the 

 semiaxes abc. At the same time, the equations 



wbzh^Oy V^i~^=0, . . . (235.) 



hold good, and show that the two other mutually opposite 

 sides of the same inscribed quadrilateral, namely the sides L2L3, 

 L4L1, are respectively parallel to the two vectors >), $, or to the 

 axes of the two cylinders of revolution which can be circum- 

 scribed about the same original ellipsoid. Hence it is easy to 

 infer the following theorem, which the author supposes to be 

 new : — If on the mean axis 2b of a given ellipsoid^ abc, as the 

 major axis^ and isoith ixm foci Fi, Fg, qf>which the common di- 

 stance Jrom the centre a is 



- - K/{a^-h^s/{h^-c^) . 



we construct an ellipsoid of revolution ; and if in any circular 

 section of this new ellipsoid, we inscribe a quMdrilateral^iu^Li^^^i 

 of which the two opposite sides LjLg, L3L4 are respectively parallel 

 CO the two umbilicar diameters of the given ellipsoid ; while the 

 two other and mutually opposite sides, i-gLg, L4L1, of the same 

 inscribed quadrilateral, are respectively farallel to the axes of 

 the two cylinders of revolution which can be circumscribed about 

 the same given ellipsoid; then the point of intersection E of the 

 Jirst pair of opposite sides (namely of those parallel to the um- 

 bilicar diameters), will be a point upon that given ellipsoid. It 

 seems to the present writer that, in consequence of this re- 

 markable relation between these two ellipsoids, the two foci 

 Fj, Fg, of the above described ellipsoid of revolution, which 

 have been seen to be situated upon the mean axis of the ori- 

 ginal ellipsoid, of which the three unequal semiaxes are de- 

 noted by a, b, c, may be not inconveniently called the two 

 MEDIAL FOCI of that original ellipsoid : but he gladly submits 

 the question of the propriety of such a designation, to the 

 judgement of other and better geometers. Meanwhile it may 

 be noticed that the two ellipsoids intersect each other in a 

 system of two ellipses, of which the planes are perpendicular 

 to the axes of the two cylinders of revolution above mentioned; 

 and that those four common tangent planes of the two ellip- 

 soids, which are parallel to their common axis, that is to the 

 mean axis of the original ellipsoid abc^ are parallel also to its 

 two umbilicar diameters. 



[To be continued.] 



