NOTICES AND ABSTRACTS 



OF 



MISCELLANEOUS COMMUNICATIONS 

 TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



On a General Geometric Method. By Charles Graves, F.T.C.D. 



Mr. Graves was led to the views he was about to explain, from ob- 

 serving the use in the doctrine of Conic Sections of a theorem given 

 by M. Chasles, in his "Histoire de la Geometric," viz., that ''the en- 

 harmonic relation of four lines drawn from four fixed points in a conic 

 section, to any fifth point in the curve, will remain invariable" Mr. 

 Graves explained the term "enharmonic relation," as employed by 

 M. Chasles, to mean the ratio of Sin. (a, d). Sin. {h, c) to Sin. (a, b) 

 Sin. (c, d) ; a, b, c, and d, being right lines diverging from the same point. 

 He insisted on the importance of M. Chasles's theorem, as a kind of 

 geometrical characteristic of the conic sections, defining them like an 

 equation ; and showed how it might be advantageously applied in the 

 determination of loci, and also in the invention, proof, and generaliza- 

 tion of theorems relating to the conic sections. In ascertaining whether 

 the plane curve described by a point, subject to a certain condition, is 

 a curve of the second degree or not, the general method that suggests 

 itself is, to find four particular positions of the point, and to draw from 

 these points right lines to any fifth point in the locus. If the enhar- 

 monic relation of these four lines be invariable, the curve will be a 

 conic section, and not otherwise. Among several exemplifications of 

 this method, Mr. Graves discussed the problem of finding the locus of 

 the centres of all the conic sections passing through four given points. 

 The middle points of the sides of the quadrilateral, at whose angles ar^ 

 the given points, being evidently situated on the locus, it was sufficient 

 to show that the enharmonic relation of lines drawn from them to any 

 VOL. VII. 1838. B 



