3 REPORT 1864. 



On Stigmatics. By Alexander J. Ellis, F.R.S. 



In ordinary analytical geometry, a point M, moving along the axis OM, is con- 

 j ugated by means of certain equations, with one or more points P, P' . . . so situated 

 that RIP, MP' ... are all parallel to a given line. In stifimatics the point M, called 

 the index, maybe situated a7iy2che7-e upon a plane, and the points P, P' . . . , called the 

 stigmata, may be so situated that the angles OMP, OMP' . . . are any whatever con- 

 sistent with certain conditions. The position and length of MP, MP' . . . , with re- 

 spect to those of OM, are determined by a certain law for each particular case. 

 The locus of P, P' . . . for a given locus of M is a stif/matic patJi. The aggi-egate 

 of all possible groups of conjugated points forms a stifpnatic. Stigmatics are the 

 general geometrical representatives of algebraical equations, and comprehend as 

 particular cases all possible and imaginary results of ordinary algebraical geometry. 

 If H and K be fixed stigmata, having the indices A and B, and the triangles 

 HPK, AMB be always similar and similarly situated, M is the index and P the 

 stigma of a stigmatic sti-aight line, the theory of which embraces the whole theory 

 of similar figures and of rays (real or imaginary) in involution. If E and F be 

 fixed points, and the triangles EMP, PMF be similar and similarly situated, then 

 M is the index, and P the stigma, of a stigmatic circle, the theory of which com- 

 prehends that of radical axes, and geometrical involution and homogi'aphy of points 

 on a plane. The mode of calculating the relations of stigmatics is by means of 

 clinants. The clinant ah is the operation of timiing the axis of reference, 01, through 

 the angle (01, AB), and altering its length in the ratio of the length of 01 to that 



of AB, so that ABi=rt6 . 01. The clinant —^is the operation of turning the sti-aight 



line CD through the angle (CD, AB), and altering its length in the ratio of that 



of CD to that of AB, so that -— . CD=AB. These clinants completely obey the 



laws of ordinaiy algebra. The clinant equations to the stigmatic straight line and 



circle, as just defined, are, therefore, v— = y/-, — =; —. respectively, whence all their 



properties may be deduced. If 01, OX, Y be radii of a unit circle, then, in ordinary 

 analytical geometry, if OM, MP be th6 abscissa and ordinate of any point, P, re- 

 ferred to the lines OX, Y as axes, we shall have 



om . OI=OM = x . OX=a: . ox . 01, 

 and 



mp . 0I=MP=y . 0Y = ?/ . 01/ . 01, 



sothata;= — ,y=— , and thus the ordiuarj' algebraical equation to a curve, 



/(.r, ?/) = 0, is converted into the clinant equation to a stigmatic /( — » —j = 0, a 



which is its general form, comprehending both the real and imaginary results of 

 the former as particular cases. The constants of such an equation should also be 

 transfomied into clinants, so as to make the equations homogeneous. Thus the 



eq-uation to the straight line -4-~l, becomes 



a 



om . oa.mp . oh _-, . om .mp _-, 



' — -r — r-^-:- 1, or 1 j 1, 



ox ox oy oy oa ob 



which can be shown to be identical with that already obtained. The equation to 

 tbe circle referred to rectangular coordinates, in which case o.i-+oiy-=0, is x'+y"^ 

 = 0^, whence 



— 5 + — i^ = -- or om"- - mp- = oa; 

 ox^ oy- ox^ 



which is identical with the former if/e = 2oa. Stigmatics, therefore, furnish the 

 required complete generalization of algebraical plane geometiy, comprehending all 

 the results already obtained, explaining all the "impossibilities" hitherto encoun- 

 tered, and developing many new properties of plane figures. 



