1866.] 



Mr. A. J. Ellis on Plane Stigmatics, 



199 



When pairs of separate s. rays cutting one s. line are considered, it is 

 shown (99) that if from any point pairs of ordinary rays he drawn which 

 are the two linibs of a constant angle cutting any straight line in points 

 forming the indices and stigmata of s. points in a s. homography of which 

 the point of issue is a double stigma, and we furnish all the points lying on 

 the given straight line, and the point of issue, with Cartesian indices, and 

 hence consider all the s. rays as Cartesian, they will form part of an homo- 

 graphy of s, rays, of which the double rays drawn from the Cartesian point 

 of issue will be two non-Cartesian s. lines parallel to the asymptotes of a s. 

 circle, and therefore independent of the magnitude of the constant angle. 

 This furnishes a complete geometrical explanation of G. S. 171, 172, 181, 

 651, S.C. 293, &c. 



Generally the whole of the propositions in G. S. respecting the homo- 

 graphy of ordinary points and rays may be transferred to the homography 

 of the stigmata of s. points and of s. rays, forming a series of propositions 

 of much greater generality, in which all the s. points and s. rays will be 

 perfectly real, and can be constructed by means of elementary geometry. 



The last division of this memoir is devoted to s. conies, which embrace 

 the ordinary Cartesian conies as particular cases, together with all their 

 so-called " imaginary " points and lines. The general equation is assumed 

 (100) in the form 



oa . ox^-\-2o^ ,ox . oy-{-o'y .oy--{-2ol ,ox-\-2oe .oy-\-o^=^Q ', . (a) 

 and it is shown that if o/3^ = oa,.oy the equation may represent a single 

 s. line, or two parallel s. lines, or a s. acentric, which, by changing , the 

 origin and index, may be made to depend on the equation x'y^ = 4«'o' . o'x'; 

 but if o/3^ is not =oa . oy, the equation may represent two intersecting s. 

 lines, or a s. centric — that is, a s. curve which when cut by any s. line passing 

 through a given s. point (the s. centre) has the index and stigma of that 

 s. point in the middle of the lines joining the two indices and two stigmata 

 of the s. points of intersection respectively. It is also shown that if the 

 origin and index are changed, the s. centric may be represented by the 

 equations 



^ + ^ =1, or oV'.^'V = oVr, , . . , (b) 



in which case the s. lines o'x' = 0, x'y = are indeterminate, but form pairs 

 of s. rays in involution, which are all conjugate diametrics^, with the 

 exception of the double rays oV = 0, x"y = 0, which are asymptotes. In 

 the particular case that oa=0, the general equation can, by changing ori- 

 gin and index, be made to depend on the equation oV^—a?y = o'e'. Such 

 curves are called s. cyclics. In the cyclic proper, the values of oy may 

 be any whatever; in the s. circle o/3 + oy = 0, in which case all the con- 

 jugate diametrics are s. perpendicular, and the direction-points of the asy 



* A diametric is a s. line passing througli the s. centre and intersecting the s. curve in 

 two s. points. A diameter is the straight /me joining the two stigmata of these last s. 

 points. A radius is half a diameter. These distinctions are important. 



VOL. XV. S 



