1866.] 



Mr. A. J. Ellis on Plane Stigmatics, 



195 



by their direction-points only, so that the relation between the rays can be 

 expressed by a stigmatic relation between the direction-points considered 

 as indices and stigmata. "When this s. relation between the direction- 

 points is a s. homography, we have pencils of homographic s. rays. The 

 recipi'ocal relation, in which a s. point corresponds to several s. lines, and 

 a s. line to several s. points, follows immediately (88). 



Finally, the multindidal relation (89) shows how several indices may 

 be made to correspond as one group to a single stigma or a group of stig- 

 mata, and thus leads to the geometrical expression of the algebraical theo- 

 ries of several independent variables. The means by which a theory of 

 solid stigmatics, where the indices and stigmata are taken anywhere in space, 

 may be expressed by quaternion equations is briefly pointed out (89 e). 



The systematic explanation and precise expression for all these relations, 

 together with some convenient notations relating especially to s. angles * 

 (90), and to anharmouic ratios (91), occupy the first part of the present 

 memoir. The remaining two divisions are devoted to a somewhat fuller 

 consideration of s. homography (including s. involution) and s. conies. 



It results (92) from the conception of s. involution^ already given, 

 that if S be the coincident solitary index and solitary stigma, and X, Y a 

 corresponding pair of index and stigma, that is, if {xy) is a s. point on the 

 s. involution, there will be two points, E, F (called double stigmata),'m the 

 same straight line with S, in which index and stigma coincide, forming the 

 double points (ee), iff), and then se-=sf^=sx . sy, which implies that 

 ZXSE= ^ESY, and ZXSF= ZFSY, and also that each of the lengths 

 of SE, SF is a mean proportional between the lengths of SX and SY. It 

 is readily shown that the two cases of involution of points in a line, usually 

 acknowledged, are but the particular cases of XY lying on the line EF, or 

 on a line through S perpendicular to EF. In the former case the double 

 points lying on the line containing XY were readily found, and hence 

 termed "real;" in the latter, as the method of investigation did not suf- 

 fice to determine the double points, they were termed "imaginary." If A, 

 B, C, D be any four indices, and A', B', C, D' their corresponding stig- 

 mata in a s. involution, then 



ab . ed a'b' . c'd' ^ ab . cc' a'b' . c'c ^ 

 ad . cb~ ad' . c'b'' ac . cb ~~ a'c . c'b' ' 

 that is, the anharmonic ratiof of any four indices is the same as that of 



* If A, B be the direction-points of two s. lines, and the origin, then — is 



' \ — oa.oh 



defined to be the s. tangent of the s, angle between these s. lines, and written tan aib. 



t The equality of anharmonic ratios is here taken to impl}% not only the same relation 

 of magnitude which would be included in the term if the four points A, B, C, D lay upon 

 one straight line, and their corresponding points A', B', C, D'upon another straight line, 

 but also an angular relation which is expressed by the notation of clinants, and which in 

 the first of these equations is Z BAD-f Z.DCB=:ZB'A'D'+ZD'C'B'. It is readily seen 

 that this relation reduces to the usual relation of direction when the points lie on 

 straight lines. 



