1866.] 



Mr. A. J. Ellis on Plane Stigmatics. 



197 



two lines AB, A'B' may be so placed that the s. homography has only one 

 double stigma O coinciding with the middle point of SZ', and O will then 

 be the centre of a circle of which AB, A'B' are tangents ; the ordinates of 

 the s. points in the s. homography (each of which contains an index and 

 a stigma) will then be tangents to the circle of which the centre (being a 

 double stigma) will also contain an index and stigma. Hence is imme- 

 diately obtained the relation in G. S. 664, and an explanation of the remark 

 there made, that "le centre du cercle tient lieu en quelque sorte d'une 

 tangente."' 



The tangent of the angle subtended by an ordinate at the double stigma 

 may be constantly zero, that is, the ordinates may converge to the double 

 stigma. Then, if through any point E two straight lines be drawn, and 

 through any other point F rays he drawn intersecting the first line in a 

 series of indices A, B, C, and the second in a seines of stigmata A', B', C 

 respectively, the resulting s. points (aa'), (bb'), (cc') will form part of a 

 ^. homography of which E, F are the double stigmata. This shows the 

 nature of the points of issue and intersection in the fundamental proposition 

 of anharmonic ratios, G. S. 14, and completes it so far as "real" rays are 

 concerned. It will be further generalized hereafter. 



Hence ABB'A' is a quadrilateral, of which E, F are the intersections of 

 opposite sides. Consequently if A, B' and h!, B are opposite vertices of 

 a quadrilateral, and E, F the intersections of the opposite sidesj^ah'), (a'b), 

 (ef) are s. points in a s. involution. This completes G. S. 350, for "real" 

 rays *. 



The above considerations determine all the relations of s. homography, 

 but do not suffice to give an explanation of those " imaginary " rays which 

 play so important a part in Chasles's geometry. The requisite theory is 

 very simply obtained by considering the direction-points of two sets of 

 (primary and secondary) s. rays as indices and stigmata in a s. homo- 

 graphy or s. involution (95). In particular it is shown that if the s. tan- 

 (jent of the s. angle between two primary s. rays is equal to that between 

 the two corresponding secondary s. rays, there are two paii^s of parallel 

 s. rays, which are parallel to the asymptotes of a s. circle ; and all the 

 other relations for pencils of ordinary homographic rays are similarly gene- 

 ralized. In this case, if the direction-points of all the rays lie upon the 

 same straight line passing through the origin, the s. rays correspond to 

 those usually termed "real ;" and if any do not, the s. rays belong to the 

 class usually termed "imaginary," and were supposed not to have any 

 geometrical existence. 



When the two s. points of issue of the two pencils are distinct, there 

 will be generally two primary rays which are parallel to their secondaries ; 



* Some of these results of s. involution and s. homography have been previously ob- 

 tained by Mobius (cited in the Introductory Memoir, 59 t) ; but his method and con- 

 ception are perfectly different, and do not admit of the extension which the present pro- 

 positions immediately receive, to include the case of " imaginary" rays, a subject never 

 previously treated as strictly geometrical. 



