1865.] Mr. Ellis on Plane Stigmatics. 177 



similar to the triangle formed by the three corresponding indices. Again, 

 in the circle referred to rectangular coordinates, the ordinate is a mean 

 proportional between the segments of the diameter to which it is perpen- 

 dicular, that is, the angle between which segments it bisects. It was easy 

 to generalize this by supposing two lines to be drawn from the index, 

 wherever it might lie on a plane, to the extremities of the same diameter, 

 and the ordinate to bisect the angle between these lines, and to be a mean 

 proportional between their lengths. Other curves were generalized in a 

 similar manner. 



It was then necessary to have a notation which should express the rela- 

 tions of both magnitude and direction in one symbol. The ordinary nota- 

 tion was found ill adapted for the purpose. The following was therefore 

 chosen. Capital letters were used to represent geometrical points, and two 

 capital letters to represent a geometrical line in length and direction. The 

 operation of changing one such directed line into another, on the same 

 plane, which the writer had already introduced under the name of clinant*, 

 was represented in the fractional form, the changed line being written below 

 and the other above, but instead of capital letters the corresponding small 

 letters were employed, to show that we were dealing with operations and 

 not with quantities ; and when the changed line was the axis of reference 

 itself, it was not expressed. The notation thus introduced closely simulates 

 that employed in M . Chasles's ' Geometric Superieure,' but it is totally 

 different in principle. It has the advantage of clearly showing the geo- 

 metrical operation indicated by each algebraical change, and of perfectly 

 obeying the laws of ordinary algebra, while it not only generalizes but fre- 

 quently abridges the operations of analysis. By means of these clinants it 

 became easy to express the relations between the stigma figure and index 

 figure by equations which are of exactly the same character as the Cartesian 

 equations, and from which the latter, with all their results, can be strictly 

 deduced. 



In the present introductory memoir the writer has confined himself to 

 the investigations connected with the stigmatic straight line, explaining its 

 equation and direction, the intersections of two such lines, the angles be- 

 tween them, and their distances from stigmatic points. These preliminary 

 propositions being given with the requisite detail and illustrated by de- 

 ducing from them the ordinary Cartesian formulae, the rest of the memoir 

 is occupied with the generalization of the fundamental theories necessary 

 for the successful application of the stigmatic theory to plane geometry ; 

 such as those relating to the stigmatic triangle, an harmonic ratio of geo- 

 metrical points anywhere situate on a plane and of stigmatic rays, pencils 

 of such rays with their homography and involution, and the complete 



* "On the Laws of Operation, and the Systematization of Mathematics," 'Proceed- 

 ings,' May 20, 1859, vol. x. p. 89, at bottom. " On Scalar and Clinant Algebraical 

 Coordinate Geometry," ihid. March 22, I860, vol. x. p. 420. The notation in the pre- 

 sent memoir is new. 



