196 



Mr. A. J. Ellis on Plane Stigmatics. 



[June 14, 



their corresponding four stigmata, and the anharmonic ratio of any three 

 indices and stigma is the same as that of the corresponding three stigmata 

 and index in a s. involution. If the index move on the characteristic circle 

 whose centre is S and radius SE, the corresponding stigma moves in the 

 opposite segment of the same circle, and in an opposite direction of rotation. 

 Generally if the index move on any circle passing through E, F, the stigma 

 moves on the segment of the same circle lying on the other side of EF, and 

 in the opposite direction of rotation. If the index moves on a straight 

 line passing through S, the stigma moves on another straight line also pass- 

 ing through S. If the index move on a straight line not passing through 

 S, the stigma moves on a circle passing through S and conversely. If the 

 index moves on a circle not passing through S, the stigma moves on another 

 circle also not passing through S, and in an opposite direction of rotation. 



If the whole group of stigmata be removed without disturbing their 

 mutual relations, and so that the solitary index S no longer corresponds 

 with the solitary stigma Z', a s. homography results (93), in which the 

 relations of index and stigma may be deduced from those in a s. involution. 

 These relations are expressed by the equation sx . z'y = sa . s'a, where (^y), 

 {aa) are s. points in the s. homography. When three s. points are given, 

 the solitary index S, and solitary stigma 7J , and double stigmata E, F can 

 be determined by an elementary geometrical construction in all cases. 

 These double stigmata possess important angular properties with respect 

 to the indices and stigmata of the system (94). Let {aa), {bh'), (cc) be 

 any three s. points in a s. homography. Then if ABC, A'B'C are straight 

 lines (in which case the former contains S, and the latter Z'), 



tan AEA' = tan BEB'; 

 that is, if two intersecting straight lines EA, EA' revolve about their point 

 of intersection E, and cut two given straight lines AB, A'B', determining 

 an index A upon one, and a stigma A' upon the other, so that the ordinate 

 AA.' is seen under an angle whose tangent is constant"^, the s. points will 

 form part of a s. homography, of which the given point E is one double 

 stigma, the solitary index S lying upon the index line AB, and the solitary 

 stigma Z' upon the stigma line A'B'. If Obe the middle point of%21 and 

 F the other double stigma, FO = OE. If the two Hues AB, A'B' coalesce, 

 then the two points E, Y,from which the ordinates xlA', BB', CC of three 

 or more s. points, of which both indices and stigmata lie upon one straight 

 line, are seen under an angle whose tangent is constant, are the double 

 stigmata of the s. homography determined by these three s. points. This 

 shows the nature of the points named in G. S. 1/1 f, and completes it, so 

 far as ''real rays" are concerned. It is further generalized hereafter. The 



* It is customary, but erroneous, to say in such cases that the angles are equal and in 

 the same direction. It is evident that one angle may be the supplement of another mea- 

 sured in the opposite direction. 



t Chasles's Geometrie Superieure and Sections Coniquee are referred to by the letters 

 G. S. and S. C, followed by the number of the article. 



