1866.] 



Mr. A. J. Ellis on Plane Stigmaiics, 



203 



Imaginaries arise whenever an algebra is used which Is more general 

 than the geometry to which it is referred. Ordinary commutative algebra 

 is an algebra of clinants, which corresponds to a geometry where not only 

 the relative lengths, but the relative angular positions of straight lines are 

 regarded. Now the geometry hitherto associated with it has been scalar, 

 that is, has considered relative lengths, but only identity or opposition of 

 direction. Thus in Cartesian geometry the abscissse and ordinates were 

 necessarily supposed to be parallel to given lines ; and in homographic geo- 

 metry, where the ordinates were not taken into account, their extremities 

 were supposed to lie on one or two given lines. But the algebra employed 

 in either case took no notice of these restrictions, and hence led to results 

 which could not be interpreted — that is, "imaginaries." The solution of 

 the problem of imaginaries has, therefore, consisted in that stigmatic con- 

 ception which removes these restrictions, freely introduces relative angular 

 position, and makes the geometry coextensive with the algebra. In the 

 course of these memoirs every fundamental case in which imaginaries occur 

 has been separately examined and shown to be fully explained by this con- 

 ception. 



Second. The coordinate geometry of JDescarteSy and the homographic 

 geometry of Chasles, are only particular cases of the writer's stigmatic 

 geometry, identical in nature, and eccpressible hy identical equations ; 

 for both Cartesian and homographic geometry consist in the relation of 

 index to stigma with various restrictions, which may or may not be re- 

 garded in stigmatic geometry. 



Third. The general theories of plane curves, as treated by ordinary co^ 

 ordinate geometry, hy the systems of coordination introduced hy Pliicher, 

 or the modes of investigation more recently developed hy Chasles, and all 

 the theories derivable from these, hold, in their integrity, for stig- 

 matic curves alone. 



For in all these theories clinant algebra is associated with scalar geome- 

 try, for which the stigmatic conception enables us to substitute that clinant 

 geometry which perfectly agrees with the algebra employed ; and the fun- 

 damental propositions of these theories have been extended accordingly in 

 the course of these memoirs. 



The instrument by which the writer has been enabled to bring these in- 

 vestigations to a successful issue, has proved so serviceable and manageable, 

 embracing old and new properties in a single equation, often rendering a 

 general investigation more easy to conduct than the former special re- 

 searches, and keeping the geometrical operations indicated clearly before 

 the mind, that he would add as a subsidiary result of his theory : — 



Fourth. A calculus and a notation have been invented and developed, 

 which enable magnitude and direction upon a plane to he expressed hy a 

 single symbol, obeying the laws of ordinary algebra, closely resembling in 

 appearance {not in theory) the notation employed by Chasles to represent 

 magnitude and direction upon a straight line, and as easy of manipulation. 



