218 Sir W. Rowan Hamilton on Quaternions. 



to BD' (because these are the two equal diagonals of the qua- 

 drilateral in the construction), must vary inversely as BD (by 

 an elementary property of the sphere), we are to seek the 

 difference of the squares of the extreme values of BD, or of 

 B'D, because the square of the perpendicular BB' is constant 

 for the section. But the longest and shortest straight lines, 

 B'D], B'Dg, which can thus be drawn to the auxiliary circle 

 round C, from the fixed point B' in its plane, are those drawn 

 to the extremities of that diameter DiC'D2of this circle which 

 passes through or tends towards B' ; so that the four points 

 Di C Dg B' are on one straight line, and the difference of the 

 squares of B'Dj, B'Dg is equal to four times the rectangle 

 under B'C and C'D^, or under B'C and C'A. We see therefore 

 that the shortest and longest semi-diameters AEj, AEg of the 

 diametral section of the ellipsoid, are perpendicular lo each 

 other,because (by the construction above-mentioned)they coin- 

 cide in their directions respectively with the two supplementary 

 chords ADi, ADg of the section of the auxiliary sphere, and 

 an angle in a semicircle is a right angle; and at the same time 

 we see also that the difference of the squares of the reciprocals 

 of these two rectangular semlaxes of a diametral section of the 

 ellipsoid varies, in passing from one such section to another, 

 proportionally to the rectangle under the projections, B'C and 

 C'A, of the two fixed lines BC, CA, on the plane of the vari- 

 able section. The difference of the squares of these recipro- 

 cals of the semi-axes of a section therefore varies (as indeed it 

 is well-known to do) proportionally to the product of the sines 

 of the inclinations of the plane of the section to two fixed dia- 

 metral planes, which cut the ellipsoid in circles; and we see 

 that the normals to these two latter or cyclic planes have 

 precisely the directions of the sides BC, CA of the generating 

 triangle ABC, which has for its corners the three fixed points 

 employed in the foregoing construction : so that the auxiliary 

 and (liacentric sphere, employed in the same construction, 

 touches one of those two cyclic planes at the centre A of the 

 ellipsoid. If we take, as we are allowed tO do, the point B 

 external to this sphere, then the distance BC of this external 

 point B from the centre C of the sphere is (by the construc- 

 tion) the semisum of the greatest and least semiaxes of the 

 ellipsoid, while the radius CA of the sphere is the semidiffer- 

 ence of the same two semiaxes: and (by the same construc- 

 tion) these greatest and least semiaxes of the ellipsoid, or 

 their prolongations, intersect the surface of the same diacen- 

 tric sphere in points which are respectively situated on the 

 finite straight line BC itself, and on the prolongation of that 

 line. The remaining side AB of the same fixed or generating 



