of the parallelogram is another ellipsoid reciprocal thereto. 

 These two ellipsoids have a common centre, namely, the point 

 A ; and a common mean axis, which is equal to the diameter 

 of the moving sphere. Two sides, AE, AF, of the parallelo- 

 gram AENF, are thus two semidiameters, which .^ay be re- 

 garded as reciprocal to each other, one of the one ellipsoid, and 

 the other of the other. It is, however, to be observed, that they 

 fall at opposite sides of the principal plane, containing the four 

 fixed lines, and that, therefore, it may be proper to call them 

 more fully opposite reciprocal semidiameters ; and to call the 

 points E and F, in which they terminate, opposite reciprocal 

 points. The two other sides, EN, FN, of the same varying 

 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 the two cylinders of revolution which are circumscribed 

 about the first ellipsoid; and the two latter fixed lines, AC, 

 AC, are the two cyclic normals of the same first ellipsoid: 

 while the diagonals LM', ML', of the inscribed quadrilateral 

 in the construction, are the axes of the two circles on the surface 

 of that ellipsoid, which circles pass through the point E, that is 

 through the centre of the moving sphere, and which are also 

 contained upon the surface of another sphere, having its centre 

 at the point i\^: all which is easily adapted, by suitable inter- 

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

 great facility from the quaternion equations above given. 



It may not be out of place to mention, on this occasion, 

 although for the present without its demonstration, another 

 simple geometrical construction connected with a surface of 

 the second order, and derived from the same calculus of qua- 

 ternions. This construction is adapted to determine the cone 

 of revolution which osculates, along a given side, to a cone of 

 the second degree ; but it will perhaps be most easily under- 

 stood by considering it as serving to assign the interior pole 



