30 Prof. Tait on Listing's Topologie. 



E.M.F. and fall of resistance, is the only one which appears 

 to offer any constancy of conditions to speak of. It furnishes 

 values which are not far from those generally accepted, but 

 appears hardly to admit of sufficient certainty for a standard 

 of tension. 



IV. Listing's Topologie. By Prof. Tait*. 

 [Plate II.] 



SOME of you may have been puzzled by the advertised 

 title of this Address. But certainly not more puzzled 

 than I was while seeking a title for it. 



I intend to speak (necessarily from a very elementary point 

 of view) of those space-relations which are independent of 

 measure, though not always of number, and of which perhaps 

 the very best instance is afforded by the various convolutions 

 of a knot on an endless string or wire. For, once we have tied 

 a knot, of whatever complexity, on a string and have joined 

 the free ends of the string together, we have an arrangement 

 which, however its apparent form may be altered (as by 

 teazing out, tightening, twisting, or flyping of individual 

 parts), retains, until the string is again cut, certain perfectly 

 definite and characteristic properties altogether independent 

 of the relative lengths of its various convolutions. 



Another excellent example is supplied by Crum Brown's 

 chemical Graphic Formula?. These, of course, do not pretend 

 to represent the actual positions of the constituents of a com- 

 pound molecule, but merely their relative connexion. 



From this point of view all figures, however distorted by 

 projection &c, are considered to be unchanged. We deal 

 with grouping (as in a quincunx), with motion by starts (as in 

 the chess-knight's move), with the necessary relation among 

 numbers of intersections, of areas, and of bounding lines in a 

 plane figure; or among the numbers of corners, edges, faces, 

 and volumes of a complex solid figure, &c. 



For this branch of science there is at present no definitely 

 recognized title except that suggested by Listing, w r hich I 

 have therefore been obliged to adopt. Geometrie der Lage has 

 now come, like the Geomdtrie de Position of Carnot, to mean 

 something very different from our present subject; and the 

 Geometria situs of Leibnitz, though intended (as Listing shows) 

 to have specially designated it, turned out, in its inventor's 

 hands, to be almost unconnected with it. The subject is one 



* Introductory Address to the Edinburgh Mathematical Society, 

 November 9, 1883. Communicated by the Author. 



