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



ptotes are 1, I'. But if both oa and become =0, the s. cyclic is a s. 

 homography or s. involution. 



The general homographic properties of all s. conies are then deduced 

 (101). First, a s. conic is the locus of the s. points of intersection of 

 the primary and secondary s. rays of two pencils of homographic s. rays 

 with different s. points of issue, and passes through these points, which, 

 with slight modifications, holds when a band of parallel s. lines is sub- 

 stituted for either or both of the pencils. Secondly, the anharmonic ratio of 

 four s. chords in a s. conic drawn from a variable s. point to four fixed s, 

 points, hears a constant ratio to the anharmonic ratio of the stigmata of 

 those four fixed s. points ; and this constant ratio is one of equality when 

 the s. conic is a s. circle. In order to establish the latter part of this pro- 

 position, it is shown that in a s. circle, the s. tangent of the s. angle formed 

 by any two s. rays drawn from a variable s. point to any two fixed s. points 

 in the s. circle is constant (which is a generalization of Euc. iii. 21), 

 whence is deduced the condition that four s. points should lie upon a s. 

 circle. The classification of s. centrics is founded (102) on the first of the 

 equations (5). After showing how the two stigmata can be generally found 

 from the index, the name of the s. centric is made to depend on the name 

 of the locus of the stigma when the index describes the principal diameter 

 in whole or in part. This locus is called the characteristic curve. Accord- 

 ing as o'e^-f-o'^^ is a negative scalar, — 1, a positive scalar, +1, or any 

 clinant, the s. centric is a s. ellipse, circle, hyperbola, equiradial (equila- 

 teral hyperbola), or hyperellipse (for which the characteristic is a confocal 

 ellipse and hyperbola). The mode of finding the two indices correspond- 

 ing to each stigma is then given (103), and the circumstances under which 

 a s. line and a s. centric may have common points investigated (104). The 

 mode of determining s. points of intersection, when there is no intersection 

 in Cartesian geometry ("imaginary" intersections) is illustrated, and 

 shown to lead to the same results as the homographic method. 



The radical axis (105) is shown to be the Cartesian portion of a s. line 

 which is the common s. chord of a series of s. circles, so situated that the 

 extremities of their principal diameters are the indices and stigmata of a 

 s. involution, of which the "real" and "imaginary" circles of Chasles are 

 two particular cases. 



The properties of asymptotes, and the nature of those in a s. ellipse, are 

 then considered (106), and their geometrical properties examined. Conju- 

 gate Diametrics and their relations to the asymptotes are next investigated, 

 (107). In especial it is shown that if (u'e'), (v'g) are two of the s. points 

 in which two conjugate diametrics intersect a s. centric, and if the s. line 

 drawn from (u'e') s. perpendicular to the diametric connecting (oV)and 

 {v'g') meet the latter in a s. point whose stigma is D, we shall have 



oV . w'e' + oV .«?y = 0, . . (c) 



o'u'^ + o'v"=o'e'\ u'e" + v'g'^=o'g\ (<0 



whence o'e'^ + o'g'^=o'e'^-\-o' g^, ^ W 



and de' , (p'v-^v'g')=^±o'e , o'g • • (/) 



