1866.] 



Mr. A. J. Ellis on Plane Stigmatics. 



193 



the index moTes upon a straight line, the ordinate remains parallel tasome 

 other straight line, the relation between index and stigma is that expressed 

 by the relation between abscissa and ordinate in the coordinate geometry of 

 Descartes. When only one index corresponds to one stigma and con- 

 versely, and both indices and stigmata lie always on one and the same 

 straight line, or the indices upon one and the stigmata upon another, the 

 relations between indices and stigmata are those between homologous points 

 in the homographic geometry of Chasles. 



The general expression of the stigmatic relation is obtained by a gene- 

 ralization of Chasles's fundamental lemma in his theory of characteristics 

 {Comptes Rendus, June 27, 1864, vol. Iviii. p. 1175), clinants being sub- 

 stituted for scalars*. It results that in certain forms of the law of coordi- 

 nation, which "coordinates " the stigmata with the indices, there may be 

 solitary indices which have no corresponding stigmata, and solitary stig- 

 mata which have no corresponding indices, and also double points in which 

 the index coincides with its stigma (76)t. The particular case in which 

 one index corresponds to one stigma and conversely, and no solitary index 

 or stigma occurs, is termed a stigmatic line (henceforth written "s. line"), 

 because the Cartesian case is that of a Cartesian straight line in ordinary 

 coordinate geometry, but in the general s. line the figures described by 

 index and stigma may be any directly similar plane figures (77). The 

 investigation of this particular case occupies almost the whole of the In- 

 troductory Memoir. When one index corresponds to one stigm.a and 

 conversely, but there is one solitary index and one solitary stigma, we have 

 s. homography, provided the solitary index is distinct from the solitary 

 stigma (79), and s. involution when the solitary index coincides with the 

 solitary stigma (78), so called because they generalize the relations treated 

 of under these names by Chasles. 



When the relations between index and stigma are expressed by an equa- 

 tion of the second order, the s. curves are termed s. conies, because they 

 include Cartesian conies as a particular case (80). Generally two stigmata 

 correspond to each index, and two indices to each stigma, and there is no 

 solitary index or stigma. 



Two s. curves irdersect when they have a common ordinate, and there- 



* If 01 be any fixed axis of reference, and AB any line upon a fixed plane passing 

 through 01, we may consider that a line equal in magnitude and direction to AB can 

 be formed from 01 by the operation of turning 01 through the Z (01, AB), and altering 

 its length in the ratio of the length of 01 to that of AB. When all such lines AB lie 

 upon the same plane, this operation is termed a clinant and represented by ab to distin- 

 guish it from the line AB. If the lines AB lay on different planes, the operation would 

 be in general a quaternion. But if AB is parallel to 01, whatever be the plane on 

 which it lies, the operation ab is termed a scalar. Clinants and scalars obey the laws 

 of ordinary commutative algebra. Quaternions do not. 



t The figures in parentheses refer to the articles of this " Second Memoir," which are 

 numbered consecutively to those in the Introductory Memoir (Proceedings, April 6, 

 1855, vol. xiv. p. 176). 



