442 [Feb. 26, 



II. " On Clinant Geometry, as a means of expressing the Gene- 

 ral Relations of Points in a Plane, realizing Imaginaries, 

 reconciling Ordinary Algebra with Plane Geometry, and 

 extending the Theories of Anharmonic Ratios." By ALEX- 

 ANDER J. ELLIS, B.A., F.C.P.S. Communicated by ARTHUR 

 CAYLEY, Esq. Received January 28, 1863. 



(Abstract.) 



The serious difficulties presented by " imaginaries " in plane 

 geometry arise from treating the " principle of signs " as a matter 

 of convention, and not as a particular case of a general operation, 

 here termed a clinant, -which consists in altering the length of a line 

 in a given ratio, and rotating it through a given angle. As the 

 calculus of clinants furnishes a geometrical representation for every 

 algebraical result, imaginaries disappear, and there is no longer any 

 apparent disagreement between analysis and geometry. Many theo- 

 ries, as, for example, those of anharmonic ratios, hitherto only 

 established for points on a straight line, are also easily extended 

 by means of clinants to embrace any points upon a plane. The 

 object of the present paper is to establish and illustrate these facts. 

 For this purpose it is divided into three distinct but closely connected 

 parts. 



Part I. shows that clinants obey the same laws of calculation as 

 ordinary algebraical expressions, and explains their notation and 

 geometrical construction. This is illustrated by the solution of the 

 problem of the determinate section generalized, and by a geometrical 

 explanation of "imaginary" trigonometrical functions, applied to the 

 discovery of the " imaginary " double rays in an homography. 



Part II. establishes the theory of stigmatics. An index point, 

 supposed to move from any origin into every point on a plane, is 

 accompanied by one or more satellite points, termed stigmata, the 

 relative position of the stigmata and index at any time being 

 dependent on the relative position of the index and origin, according 

 to some assigned law. The locus of the stigmata, corresponding to 

 each path of the index, forms a stig matic curve. The aggregate of 

 these curves constitutes a stigmatic, which therefore consists of points 

 conjugated with each other according to a characteristic law, ulti- 



