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between the equations of a surface of the second order and of 

 its tangent plane, as found by Sir William R. Hamilton in 

 his geometrical applications of the Calculus of Quaternions, 

 that Mr. Graves was led to investigate the theorem now com- 

 municated to the Academy. 



The following theorem respecting ellipsoids, obtained by 

 the method of quaternions, was communicated by Sir William 

 Rowan Hamilton, in a note to the Secretary of Council : 



" On the mean axis of a given ellipsoid, as the major axis, 

 describe an ellipsoid of revolution, of which the equatorial 

 circle shall be touched by those tangents to the principal sec- 

 tion of the given ellipsoid (in the plane of the focal hyperbola), 

 which are parallel to the umbilicar diameters. In this equa- 

 torial circle, and in every smaller and parallel circle of the new 

 ellipsoid thus constructed, conceive that indefinitely many 

 quadrilaterals are inscribed, for each of which one pair of op- 

 posite sides shall be parallel to the given umbilicar diameters, 

 while the other pair of opposite sides shall be parallel to the 

 asymptotes of the focal hyperbola. Then the intersection of 

 the first pair of opposite sides of the inscribed quadrilateral 

 will be a point on the surface of the given ellipsoid. 



" I may remark that the distance of either focus of the 

 new ellipsoid from the common centre of the new and old 

 ellipsoids, will be equal to the perpendicular let fall from 

 either of the two points, which were called t and v in a recent 

 note and diagram, on the umbilicar semidiameter au, or on 

 that semidiameter prolonged ; while the distance of the um- 

 bilic u from the foot of either of these two perpendiculars, 

 that is, the projection of either of the two equal tangents to 

 the focal hyperbola, tu, uv, on the umbilicar semidiameter 

 AU, or on that semidiameter prolonged, will be the minor 

 semiaxis, or the radius of the equator, of the new ellipsoid (of 

 revolution). 



2d2 



