1863.] 443 



mately expressible by an equation between the clinant of the line 

 connecting the index with the origin and the clinant of the line 

 connecting the stigma and the index. By elimination between two 

 such equations, the common stigmata (systigmata) of two stigmatics, 

 and by the condition of equal roots their coalescent systigmata, or 

 homostigmata, may be determined. These systigmata and homo- 

 stigmata include, as particular cases, the points of "real" and 

 " imaginary " intersection and contact of algebraical curves. 



These generalities are illustrated by a consideration of the general 

 stigmatic straight line and the central stigmatic circle. The stigmatic 

 straight line consists of stigmatic curves similar to the paths of the 

 index, and their systigmata are the "double points" of similar 

 figures. The stigmata of a stigmatic circle are always harmonically 

 conjugated with the extremities of its axis (with which they always 

 lie either on the same straight line, or the circumference of the same 

 circle), and hence form an " involution "of points on a plane. The 

 construction of the systigmata and homostigmata of a stigmatic 

 straight line, and stigmatic circle, furnishes a complete geometrical 

 explanation and realization of the " imaginary intersections " of 

 straight lines, with "real" and "imaginary" circles, "imaginary 

 tangents " to such circles, and their polars and radical axes and 

 common chords. 



Part III. contains an extension of the theories of anharmonic 

 ratios from points on straight lines to any points in a plane, and 

 explains and constructs the homography and involution of such 

 systems of points, with their double points, &c. Constant reference 

 is made throughout this part to M. Chasles's ' Geometric Superieure,' 

 to show how his fundamental theories may be interpreted as conclu- 

 sions in clinant geometry, to explain all cases of " imaginaries," and 

 to establish the fact that " real " and " imaginary " points are only 

 two very particular cases of the general theory of conjugated points. 



The whole memoir forms an introduction to a new and practical 

 geometrical calculus, including and interpreting all analytical in- 

 vestigations on plane geometry. 



