RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 9 



according as it is drawn to an isolated point, or to a point already connected 

 with the system. Hence the sum of points and faces is increased by one for 

 every new line. If the closed surface is acyclic, or simply connected/" like that 

 of a solid body without any passage through it, then, if from any point we draw 

 a closed curve on the surface, we divide the surface into two faces. We have 

 here one line, one point, and two faces. Hence, if e be the number of lines, 

 s the number of points, and /the number of faces, then in general 



e — s — f = m 



when m remains constant, however many lines be drawn. But in the case of a 

 simple closed surface 



m = — 2 . 



If the closed surface is doubly connected, like that of a solid body with a 

 hole through it, then if we draw one closed curve round the hole, and another 

 closed curve through the hole, and round one side of the body, we shall have 

 e — 2, s = 1,/— 1, so that n = 0. If the surface is n-lj connected, like that 

 of a solid with n — 1 holes through it, then we may draw n closed curves 

 round the n — 1 holes and the outside of the body, and n — 1 other closed curves 

 each through a hole and round the outside of the body. 



We shall then have 4(^ — 1) segments of curves terminating in 2(n — 1) 

 points and dividing the surface into two faces, so that e = 4(w — 1), 

 s = 2 (n — 1), and/= 2, and 



e — s — f —In — 4 , 



and this is the general relation between the edges, summits, and faces of a 

 polyhedron whose surface is ra-ly connected. 



The plane reciprocal diagrams, considered as plane projections of such 



* See Riemann, Crelle's Journal, 1857, Lehrsatze aus der analysis sitits, for space of two dimen- 

 sions ; also Catley on the Partitions of a Close, Phil. Mag. 1861 ; Helmholtz, Crelle's Journal, 1858, 

 Wirbelbewegung, for the application of the idea of multiple continuity to space of three dimensions ; J. 

 B. Listing, Gottingen Trans., 1861, Der Census Raumlicher Complexe, a complete treatise on the 

 subject of Cyclosis and Periphraxy. 



On the importance of this subject see Gauss, "Werke, v. 605, " Von der Geometria Situs die Leibnitz 

 ahnte unci in die nur einem Paar Geometern (Euler unci Vandermonde) einen schwachen Blick zu thun 

 vergb'nnt war, wissen und haben wir nach anderthalbhundert Jahren noch nicht viel mehr wie nichts." 



Note added March 14, 1870. — Since this was written, I have seen Listing's Census. In his 

 notation, the surface of an rc-ly connected body (a body with n — 1 holes through it) is (2k — 2) 

 cyclic. If 2n — 2 = K 2 expresses the degree of cyclosis, then Listing's general equation is — 



s- ( e - K x ) + (/- K 2 + *r 2 ) - - K 3 + «r a - to) = , 



where s is the number of points, e the number of lines, K a the number of endless curves, /the number 

 of faces, K 2 the number of degrees of cyclosis of the faces, sr 2 the number of periphractic or closed 

 faces, v the number of regions of space, K 3 their number of degrees of cyclosis, sr 3 their number of 

 degrees of periphraxy or the number of regions which they completely surround, and to is to be put 

 = 1 or = 0, according as the system does or does not extend to infinity. 



VOL. XXVI. PART I. C 



