A THEOREM CONCERNING POLYGONIC SYNGRAPHY

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 5 (1853), p. 474–475.)

Edited by David R. Wilkins A Theorem concerning Polygonic Syngraphy. By Sir William R. Hamilton.

Communicated June 13, 1853.

[Proceedings of the Royal Irish Academy, vol. 5 (1853), p. 474–475.]

Professor Sir William Rowan Hamilton exhibited the following Theorem, to which he had been conducted by that theory of geometrical syngraphy of which he had lately submitted to the Academy a verbal and hitherto unreported sketch, and on which he hopes to return in a future communication.

Theorem. Let A_{1}, A_{2}, … A_{n} be any n points (in number odd or even) assumed at pleasure on the n successive sides of a closed polygon BB_{1}B_{2} … B_{n−1} (plane or gauche), inscribed in any given surface of the second order. Take any three points, P, Q, R, on that surface, as initial points, and draw from each a system of n successive chords, passing in order through the n assumed points (A), and terminating in three other superficial and final points, P′, Q′, R′. Then there will be (in general) another inscribed and closed polygon, CC_{1}C_{2} … C_{n−1}, of which the n sides shall pass successively, in the same order, through the same n points (A); and of which the initial point C shall also be connected with the point B of the former polygon, by the relations

\frac{ael}{bc} \frac{βγ}{αϵλ} = \frac{a′e′l′}{b′c′} \frac{β′γ′}{α′ϵ′λ′} , \frac{bfm}{ca} \frac{γα}{βζμ} = \frac{b′f′m′}{c′a′} \frac{γ′α′}{β′ζ′μ′} , \frac{cgn}{ab} \frac{αβ}{γην} = \frac{c′g′n′}{a′b′} \frac{α′β′}{γ′η′ν′};

where

\begin{aligned}a & = QR, & b & = RP, & c & = P Q, \\ e & = BP, & f & = BQ, & g & = BR, \\ l & = CP, & m & = CQ, & n & = CR, \\ a′ & = Q′R′, & b′ & = R′P′, & c′ & = P′Q′, \\ e′ & = BP′, & f′ & = BQ′, & g′ & = BR′, \\ l′ & = CP′, & m′ & = CQ′, & n′ & = CR′; \end{aligned}

while α β γ ϵ ζ η λ μ ν, and α′ β′ γ′ ϵ′ ζ′ η′ λ′ μ′ ν′, denote the semidiameters of the surface, respectively parallel to the chords a b c e f g l m n, a′ b′ c′ e′ f′ g′ l′ m′ n′.

As a very particular case of this theorem, we may suppose that P Q′RP′QR′ is a plane hexagon in a conic, and BC its Pascal’s line.