1866.] 



Mr. A. J. Ellis on Plane Stigmatics. 



361 



From these relations, which are much more general than any hitherto 

 enunciated, the ordinary relations in Cartesian geometry respecting the 

 lengths of conjugate diameters and the areas they contain are readily de- 

 duced. 



By referring s. centrics to conjugate diametrics as new coordinate axes 

 (108), a simple general method for constructing the intersections of a 

 s. line with a s. centric, and for drawing two s. tangents (109) from any 

 s. point (except the s. centre) is obtained. This is illustrated by actual 

 geometrical constructions corresponding to " imaginary " cases in Carte- 

 sian geometry. It is also shown how the chord of contact or polar may 

 have a Cartesian portion when both the s. points of contact are non- Car- 

 tesian and hence "imaginary." 



It is now possible (110) to demonstrate the fundamental anharmonic 

 properties of tangents and asymptotes to a s. centric. If through four 

 s. points in a s. centric, four tangents he drawn, intersecting any fifth 

 tangent, and also four s. chords meeting in any fifth s. point in the s. cen- 

 tric, the anharmonic ratio of the four stigmata of the s. points of intei^- 

 section of the four tangents with the fifth will he equal to the anharmonic 

 ratio of the four s. chords (which generalizes S. C. 2, 5, the demonstra- 

 tion being here direct and therefore widely differing from Chasles's), 

 whence, if two fixed tangents are cut by a moveable tangent the stigmata 

 of its s. points of intersection with them will he the indices and stigmata 

 of s. points in a s. homography. It is now possible to generalize the whole 

 of Chasles's S. C, and make the theories applicable, mutatis mutandis^ 

 to s. conies. 



The s. foci of a s. centric (11 \)are defined as the s. points of intersection 

 of any two tangents which are parallel to the asymptotes of a s. circle. 

 Hence it is shown that there are four s. foci to a s. centric ; so that if 

 o's^=d e^-\-d g'^-=o' z^, the two principal s. foci are {ss) {zz), and the two 

 transverse s. foci are {o's), (o'z). When the s. centric is Cartesian, the 

 two principal s. foci are those usually recognized, and the only two recog- 

 nized by Chasles, S. C. 275, 294. The other two transverse s. foci have, 

 however, been recognized as imaginary " foci by Pliicker (System der a. 

 Geometric, p. 106, 1. 6), and by Salmon (Conies, 4th ed. p. 242). 



Besides the usual properties of the focal stigmata in Cartesian curves, the 

 following entirely new correlative properties of the principal and transverse 

 s. foci are demonstrated. Let there he two conjugate diametrics, and 

 through any s. point in one draw a s. line parallel to the other, and let 

 the stigmata of the s, points of intersection of the conjugate diametrics 

 and the chord, with the s. centric, he respectively E', F' ; G', H' ; Y„ Y^. 

 From E', F' draw straight lines parallel to the bisectors of the angles 

 Yj^SYg, YjZYg respectively, forming two sides of a triangle of which 

 E'F' is the base. Then the lengths of these sides will he mean propor* 

 tionals between the lengths of SY„ SY„ and ZY„ ZY^ respectively. This 

 IS a generalization of the usual property that the sum or difference of the 



s 2 



