192 



the style at the point of intersection of its straight portions 

 will trace out another geodetic line upon a confocal surface. 



And, if it be kept stretched, being partly in constrai?ied 

 contact with the surface along a line of curvature, and partly 

 free in space, the style will trace another line of curvature 

 upon a confocal surface. 



June 26, 1848. 



REV. HUMPHREY LLOYD, D. D., President, 

 In the Chair. 



Sir W. R, Hamilton stated the following additional theo- 

 rems respecting certain reciprocal surfaces, to which his own 

 methods have conducted him. 



If a plane quadrilateral ABCD be inscribed in a given 

 sphere, so that its four sides may be constantly parallel to four 

 given straight lines ; and if E, F be the two points of meet- 

 ing of the two pairs of opposite sides, namely, E the meeting 

 of the sides AB, CD, and i^ the meeting oi BC, DA (pro- 

 longed if necessary) ; then the locus of the point E will be 

 one ellipsoid, and the locus of the point F will be another 

 ellipsoid reciprocal thereto. 



And other pairs of reciprocal surfaces of the second degree 

 may be generated in like manner, by changing the sphere to 

 other surfaces of revolution of the second degree. 



For instance, a pair of reciprocal cones of the second de- 

 gree may be generated as the loci of two points E, F, which 

 are, in like manner, the points of meeting of the opposite sides 

 of a plane quadrilateral ABCD, inscribed in a circular section 

 of a right-angled cone of revolution, with their directions in 

 like manner constant. And a pair of reciprocal hyperboloids 

 (whether of one or of two sheets) may, in like manner, be ge- 

 nerated from an equilateral hyperboloid of revolution (of one 

 or of two sheets). 



