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XLIX. On the nomographic Transformation of a Surface of the 

 Second Order into itself. By A. Cayley, Esq * 



THE following theorems in plane geometry relating to poly- 

 gons of any number (odd or even) of sides are well known. 



"If there be a polygon of (m + 1) sides inscribed in a conic, 

 and in of the sides pass through given points, the m + lth side 

 will envelope a conic having double contact with the given conic." 

 And "If there be a polygon of (m + 1) sides inscribed in a 

 conic, and m of the sides touch conies having double contact 

 with the given conic, the m + 1th side will envelope a conic having 

 double contact with the given conic." The second theorem of 

 course includes the first, but I state the two separately for the 

 sake of comparison with what follows. 



As regards the corresponding theory in geometry of three 

 dimensions, Sir W. Hamilton has given a theorem relating to 

 polygons of an odd number of sides, which may be thus stated : 

 "If there be a polygon of (2m-f-l) sides inscribed in a surface 

 of the second order, and 2m of the sides pass through given 

 points, the (2m + l)th side will constantly touch two surfaces of 

 the second order, each of them intersecting the given surface of 

 the second order in the same four lines f." 



The entire theory depends upon what may be termed the 

 transformation of a surface of the second order into itself, or 

 analytically, upon the transformation of a quadratic form of four 

 indeterminates into itself. I use for shortness the term trans- 

 formation simply ; but this is to be understood as meaning a 

 homographic transformation, or in analytic language, a trans- 

 formation by means of linear substitutions. It will be conve- 

 nient to remark at the outset, that if two points of a surface of 

 the second order have the relation contemplated in the data of 

 Sir W. Hamilton's theorem (viz. if the line joining the two points 

 pass through a fixed point), the transformation is, using the 

 language of the Recherches Arithmetiques, an improper one, but 



* Communicated by the Author. 



t See Phil. Mag. vol. xxxv. p. 200. The form in which the theorem is 

 exhibited by Sir W. Hamilton is somewhat different ; the surface contain- 

 ing the angles is considered as being an ellipsoid, and the two surfaces 

 touched by the last or (2m+l)th side of the polygon are spoken of as 

 being an ellipsoid, and a hyperboloid of two sheets having respectively 

 double contact with the given ellipsoid : the contact is, in fact, a quadruple 

 contact at the same four points ; real as regards two of them in the case of 

 the ellipsoid, and as regards the other two in the case of the hyperboloid 

 of two sheets j and a quadruple contact is the coincidence of four genera- 

 ting lines belonging two and two to the two series of generating lines, 

 these generating lines being of course in the case considered by Sir W. 

 Hamilton, all of them imaginary. 



