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Mr. A. J. Ellis on Plane Stigmatics. [June 14, 



fore a common s. point. They touch when a slight alteration in the con- 

 stants of one equation will make one of their common s. points into several, 

 having their stigmata very near. When they have no common ordinate 

 or s. point, they are parallel or asymptotic, according as the distances 

 between the stigmata corresponding to a common index remain unchanged 

 or, under certain circumstances, diminish infinitely (81). 



It has been shown in the Introductory Memoir that a s. line can be 

 determined by two stigmata corresponding to known indices, or by two 

 points called the direction-point and original stigma"^. Taking either the 

 two first-named or the two last-named points as the index and stigma in 

 a stigmatic relation, we are able to make its expression denote a linear rela- 

 tion, which leads to the conception of enveloped s. curves, and therefore of 

 classes of s. curves (82, 83). 



In the above relations two ordinary geometrical points (^'g. points") in 

 a plane form the index and stigma. If we now take two s. points (that is, 

 two relations of index to stigma), terming owe primary and the other secon- 

 dary, we are able, by means of two equations, to express a relation between 

 these s. points, so that one primary shall correspond to n secondaries, and 

 one secondary to m primaries. This is the bipunctual relation (84), and 

 includes the whole subject of related s. curves, of which related Cartesian 

 curves form a particular case. The most important and elementary cases 

 are, first, those in which the related s. points lie respectively on known s. 

 lines, s. involutions, s. homographies, or s. circles, and there is a stigmatic 

 relation between the stigmata considered as forming two groups, one of 

 indices and one of stigmata ; and secondly, those in which one primary s. 

 point corresponds to one secondary and conversely. In the latter case the 

 relations between one index and stigma may be inferred from known rela- 

 tions between another index and stigma, or from the same index and a new 

 stigma, or, as is most usual, from the same stigma and a new index. Hence 

 follows the general theory of change of coordination (85). nomographic 

 and homologic s. figures (86) are another example, in which one primary 

 corresponds to one secondary s. point and conversely ; but there is gene- 

 rally one solitary s. line in each s. figure such that no s. point taken upon 

 it will have any corresponding s. point in the other figure. 



Corresponding to the bipunctual relation there is a bilinear relation 

 (87) in which one primary s. line corresponds to several secondary s. lines, 

 and one secondary to several primaries, by means of the elements chosen 

 to represent linear relations. To this belongs the biradial relation (87 

 where two pencils of s. rays, being drawn from known s. points, are deter- 

 mined generally by s. points having a bipunctual relation, and exceptionally 

 * The original stigma is the stigma of which the origin is the index. If through the 

 s. point {ii) (that is, a s. point of which both index and stigma are the point I at the 

 further extremity of the axis of reference 01) a s. line be drawn s. parallel to any 

 given s. line, the original stigma of this parallel s. line is the direction-point of the given 

 s. line. Direction-points are of great use iq explaining " imaginary " lines and " imagi- 

 nary " angles. 



