176 Mr. Ellis on Plane Siigmatics. [April 6, 



found for the aromatic series exist between the marsh-gas hydrocarbons 

 and the alcohol radicals*. But Fittig's isomeric hydrocarbons belong to 

 the class of metamers having a different chemical structure f, whilst only one 

 kind of structure can be given for the saturated hydrocarbons of the 

 formula ^H^+2- 



I am still pursuing these researches, and hope soon to obtain more 

 definite results. 



III. " Introductory Memoir on Plane Stigmatics." By ALEXANDER 



J. ELLIS, F.R.S. Received March 23, 1865. 



(Abstract.) 



If from every point in a plane curve parallel straight lines be drawn 

 cutting a given straight line in another series of points, the first set of 

 points, which for convenience may be termed stigmata, will be coordinated 

 with the second set of points, which may be termed indices, in the same 

 manner as by the system of ordinates and abscissae in ordinary Cartesian 

 coordinate plane geometry. 



Now, the writer remarked that the essence of this coordination consisted 

 iu the relation of the two sets of points to each other forming two related 

 figures, and that the circumstances of the ordinates being parallel, and the 

 indices all lying upon one straight line, were accidents. Moreover, he 

 observed that these accidents were not regarded in the ordinary Cartesian 

 equations, where there was nothing to point out that the ordinates were 

 parallel or the abscissae coincident lines, nor any mention made of the 

 direction of the ordinates and abscissae. It seemed to him that all the 

 anomalies which occurred in analytical geometry under the name of 

 " imaginaries," were traceable on the one hand to these restrictions in the 

 figure, and on the other to the absence of any indication of their existence 

 in the equations. He therefore thought that it would be possible to gene- 

 ralize plane coordinate geometry as the expression of the law which connects 

 two or more plane figures, point for point, indices with stigmata. These 

 relations would certainly include all those of ordinary geometry, and would, 

 apparently, explain all anomalies hitherto encountered. 



It was necessary, in the first place, to form a conception of such a gene- 

 ralized relation between indices and stigmata. Now, in the Cartesian 

 straight line, the lines connecting any three stigmata are proportional to 

 the lines connecting the three corresponding indices, and any pair of the 

 first lines are in the same or opposite directions, according to the relative 

 directions of the corresponding pair of the second lines. If the stigma 

 figure and index figure were no longer straight lines, this could be gene- 

 ralized by saying that the triangle formed by three stigmata was directly 



* Kekule, " Sur la constitution des substances aromatiqnes," Bull. Soc. Chim., Fevrier 

 1865, p. 98. f Annal. Chemie und Pharm. pp. 133, 222. 



