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Note on the equation connecting the mutual distances of four points in a plane

Published online by Cambridge University Press:  20 January 2009

Thomas Muir
Thomas Muir*
Affiliation:
Bishopton, Glasgow
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If the distances (12), (13), (14), (23), (24), (34) between four points 1, 2, 3, 4 on the circumference of a circle be denoted by a, b, c, d, e, f respectively, then a certain relation (A) is known to connect a, b, c, d, e, f. The same four points, however, being points in a plane, there subsists between their mutual distances another relation (B). Now, it occurs to one that from these two relations some deduction ought to be possible regarding the mutual distances of four points on a circumference, and the problem is suggested of making the said deduction.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 3 , February 1884 , pp. 34 - 38
Copyright
Copyright © Edinburgh Mathematical Society 1884

References

* Cambridge Mathematical Journal, II., pp. 267–271.Google Scholar

* This theorem seems to have been first enunciated by Salmon in the 1st edition of his Lessons Introductory to the Modern Higher Algebra, p. 124 (Dublin, 1S59). Hesse gave it in Crelle's Journal, LXIX., p. 321 (1868), with the foot-note—“ Den Satz (8) findet man von Herrn Weierstrass bewiesen in dem ″ Monatsberichte der Königl. Akademie der Wissenschaften zu Berlin, 4 ″ März 1858, p. 211. Denn seine von einem Faotor abgelöste Function ″ welche er als das Quadrat einer. linearen function ″ darstellt, ist, wenn man ″ setzt, gerade die symmetrische Determinante A deren Unterdeterminante ″ verschwindet.”