134 vs. T. T. WILKINSDH ON 



(6.) If Qm, Q M', be drawn parallel to D a, D A, 

 cutting S c, S C, in m, M'; — the locus of the 

 points m, M', &c., will be a line parallel 

 to TV. 

 Hence (1.) — In every complete quadrilateral, each dia- 

 gonal is conjugately divided by the two others. 

 (Carnot's Essai sur la Theorie des Transver- 

 sales. Theor. VI. Geom. de Position, p. 282, 

 Theor. VIII.) 

 (2.) In every harmonic pencil, if a transversal be drawn 

 parallel to any one line, it will be bisected by 

 the other three. (Maclaurin's Gen. Prop. § 20. 

 Davies's Hutton, vol. i., Theor. 99.) 

 (3.) The point D isjixedfor all quadrilaterals having 

 one diagonal S V constant, and the other a 

 variable portion of T Q. (Garnier's Reci- 

 proques, p. 104. Davies's Trans'cersals, Prop. 

 IX.) 

 (4.) and (5.) correspond to Carnot's Essai, Theor. VI. 

 and VII. ; also Davies's Hutton, vol. i., (2.) 

 Theor. 99. (6.) is also a property worthy of 

 r^tice. 

 The properties of the complete quadrilateral have been very 

 successfully cultivated by the late Professor Davies. 13i\s first 

 paper "On the Trapezium" was published in the Philosophical 

 Magazine for August, 1826, and he afterwards resumed the 

 subject in Clay's Scientific Receptacle, and in the sixth volume 

 of Leybourn's Mathematical Repository.* In the latter work, 

 taking for granted the property that " circles described about 

 the cor/iponent triangles STC, SAN, VAT, VNCj intersect 



* For a detailed account of tnost of the published writings of the late Pro- 

 fessor Davies, reference may be made to my article " On English Malhemutical 

 Literature'"'' in the Westmimier R«viev> for April, 1 851, and also to the Educational 

 Timea, No. 43. 



