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I.—On Reciprocal Figures, Frames, and Diagrams of Forces

Published online by Cambridge University Press:  17 January 2013

J. Clerk Maxwell
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Extract

Two figures are reciprocal when the properties of the first relative to the second are the same as those of the second relative to the first. Several kinds of reciprocity are known to mathematicians, and the theories of Inverse Figures and of Polar Reciprocals have been developed at great length, and have led to remarkable results. I propose to investigate a different kind of geometrical reciprocity, which is also capable of considerable development, and can be applied to the solution of mechanical problems.

Type
Transactions
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 26 , Issue 1 , 1870 , pp. 1 - 40
Copyright
Copyright © Royal Society of Edinburgh 1870

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References

page 9 note * See Riemann, , Crelle's Journal, 1857Google Scholar, Lehrsätze aus der analysis situs, for space of two dimensions; also Cayley on the Partitions of a Close, Phil. Mag. 1861; Helmholtz, , Crelle's Journal, 1858Google Scholar, Wirbelbewegung, for the application of the idea of multiple continuity to space of three dimensions; Listing, J.B., Göttingen Trans., 1861Google Scholar, Der Census Räumlicher 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 und in die nur einem Paar Geometern (Euler und Vandermonde) einen schwachen Blick zu thun vergönnt 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 n-ly connected body (a body with n − 1 holes through it) is (2n − 2) cyclic. If 2n − 2 = K2 expresses the degree of cyclosis, then Listing's general equation is—

where s is the number of points, e the number of lines, K1 the number of endless curves, f the number of faces, K2 the number of degrees of cyclosis of the faces, ϖ2 the number of periphractic or closed faces, v the number of regions of space, K3 their number of degrees of cyclosis,ϖ3 their number of degrees of periphraxy or the number of regions which they completely surround, and w is to be put = 1 or = 0, according as the system does or does not extend to infinity.

page 11 note * On the Bending of Surfaces, by Maxwell, J. Clerk. Cambridge Transactions, 1856.Google Scholar

page 11 note † This has been shown by ProfessorJellett, , Trans. R.I.A., vol. xxii. p. 377.Google Scholar

page 11 note ‡ On the Equilibrium of a Spherical Envelope, by Maxwell, J. C.. Quarterly Journal of Mathematics, 1867.Google Scholar

page 27 note * Phil. Trans. 1863.