122] 
105 
122. 
ON THE HOMOGEAPHIC TEANSFOEMATION OF A SUEFACE 
OF THE SECOND OEDEE INTO ITSELF. 
[From the Philosophical Magazine, vol. VI. (1853), pp. 326—333.] 
The following theorems in plane geometry, relating to polygons 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 m of the 
sides pass through given points, the (m + l)th 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 conics having double contact with the given 
conic, the (m + l)th 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+ 1) 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 1 .” 
1 See Phil. Mag. vol. xxxv. [1849] p. 200. The form in which the theorem is exhibited by Sir W. Hamilton 
is somewhat different; the surface containing 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 con 
tact 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; and a quadruple contact is the coincidence of four generating 
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. 
C. II. 
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