296- Sir W. Rowan Hamilton <?« Qiialerm'ofis. 



the first equation (122.) shows that the three vectors X^, ju,/, ^^, 

 which are all drawn from one common point, namely the 

 centre a of the ellipsoid, all terminate on one straight line ; 

 since otherwise the quotient of their differences, fJ-, —^i and 

 ^i~^p would be a quaternion*, of which the vector part would 

 not be equal to zero : and in like manner, the second equation 

 (122.) expresses that the three lines A/, |«,y, ^^ all terminate 

 on another straight line. The four-sided figure L,M^L/My' is 

 therefore &. jdane quadrilateral, inscribed [genexoWy) in a small 

 circle of the mean sphere, and having the point h for the in- 

 tersection of its second and fourth sides, m^l/ and m/l^, or of 

 those two sides prolonged. And these two sides, having re- 

 spectively the directions of hm^ and hl^ or of the vector- 

 differences yt6y — ^y and \— ^y, are respectively parallel, by 

 (118.), to the two fixed vectors, » and x; or (by what was 

 shown in former articles), to the two cyclic normals, ac' and 

 AC, of the original ellipsoid. The plane of the quadrilateral 

 inscribed in the mean sphere is therefore constantly parallel 

 to the principal plane c\d of that ellipsoid, namely to the 

 plane of the greatest and least axes, which contains those two 

 cyclic normals. The first and third sides, l^m^ and l/m/, of 

 the same inscribed quadrilateral, being in the directions of 

 ja.^— Xy and jw,/— A/, are parallel, by (118.), (119.)> ^o two 

 other constant vectors, namely < — x and <' — x', or to the axes 

 AB, ab', of the two cylinders of revolution which can be cir- 

 cumscribed about the same ellipsoid. And the point of inter- 

 section of this other pair of opposite sides of the same inscribed 

 quadrilateral is, by (123.), the extremity of the vector p, or 

 the point e on the surface of the original ellipsoid; while the 

 point H, which has been already seen to be the intersection of 

 the former pair of opposite sides of the quadrilateral, since it 

 has, by (1 16.), its vector ^^=— i\ is the reciprocal point, on 

 the surface of that other and reciprocal ellipsoid, which was 

 considered in article 61 ; namely the point which is, on that 

 reciprocal ellipsoid, diametrically opposite to the point which 

 was named f in that article, and had its vector ■=hS. 



67. Conversely it is easy to see, that the foregoiiig analysis 

 by quaternions conducts to the following mode o^constructing-\, 

 or generating, geometrically, and by a graphic rather than by 



* A Quaternion, geometrically considered, is \}ci& 'product, or the quotient, 

 of any two directed lines in space. 



-f This construction, of two reciprocal ellipsoids from one sphere, was 

 'communicated to the Royal Irish Academy in June 1848; together with an 

 extension of it to a mode of generating two reciprocal cones of the second 

 degree from one rectangular cone of revolution ; and also to a construction 

 of two reciprocal hyperboloids, whether of one sheet, or of two sheets, 

 from one equilateral liyperboloid oi revolution, of one or of two sheets. 



