198 



Mr. A. J. Ellis on Plane Stigmatics. 



[June 14, 



and if the two s. points of issue coincide (96), these parallel rays will 

 become double rays. Both are found by finding the double points of the 

 homography of direction-points. When the double rays are s. perpen- 

 dicular*, the sum or difference of the s. tangents of the two s. angles made 

 by the primary and its corresponding secondary with the double rays will 

 be zero. If the direction-points form a s. involution, the s. rays form 

 an involution, and there will be always two s. rays s. perpendicular and 

 easily constructed. 



The stigmata of the s. points in which the primary and secondary s. rays 

 of two pencilsy making the s. tangent of the s. angle between such rays 

 constant, intersect two given s. lines, are indices and stigmata respectively 

 of s. points in a s. homography, of which the stigma of the point of issue 

 is a double stigma (97). Whence it follows that, in any s. homography, if 

 the indices he considered as the stigmata of s. points on a known s. line, 

 and the stigmata as the stigmata of s. points on some other known s, line, 

 and a second index be given to one of the double stigmata so as to form a 

 s. point which shall lie upon neither of those given s. lines, and pairs of s. 

 rays be drawn from the s. point thus formed to pairs of corresponding s. 

 points in the two s. lines, the s. tangent of the s. angle between any two 

 corresponding rays will be constant. 



Supposing each primary s. ray to coincide with its secondary, and 01 to 

 be the axis of reference (98), the stigma of the s. point of issue of a pencil 

 of s. rays, and the stigmata of the intersection of these s. rays with a given 

 s. line, form the indices, and the extremity I of the axis of reference 01, and 

 the direction-points of the same s. rays, respectively form the stigmata of a 

 series of s. points in a s. homography. Whence it follows that the anhar- 

 monic ratio of any four stigmata of intersection is the same as that of the four 

 corresponding direction-points. This completely generalizes the fundamental 

 proposition of anharmonic ratios, including the case where all the lines are 

 *' imaginary," and can be made the starting-point of a series of propositions 

 precisely analogous to those in G. S., but with a much wider signification. 

 Thus the property of the complete quadrilateral shows, on being generalized, 

 that if E, A, B, C, D, E', A', B' be any eight points upon a plane, and two 

 other points C, D' be assumed so that the triangles E'A'C, E'B'D' shall be 

 directly similar to the triangles EAC, EBD respectively , and then two 

 additional points F', F be determined so that the triangles F'A'B', F'C'D' 

 shall be directly similar to the triangles FAB, FCD, the point F will be 

 constant, if the first five points E, A, B, C, D remain unchanged, whatever 

 the next three, E', A', B', may he, and will lie so that if SM he drawn bisect- 

 ing the angles ASD, BSC and in length a mean proportional between the 

 lengths of SA, SD, and also of SB, SC, then SM will also bisect the 

 angle ESF, and be in length a mean proportional between SE and SF. 



* If A and B are the direction-points of two s. lines, the latter are stigraatically per- 

 pendicular when oa.oh=\, and the direction-points are then harmonically situate with 

 respect to I, V, where Or=IO. 



