120 



Mr. T. S. Davies's Properties 



Hence from (a) and (b) we find 



IC.FE:IK.FD::CL.LE:KL.LD; and thence 

 (Converse of cor. 2. Pr. 3.) that IF or ST passes through L. 



Q. E. D. 



The geometrical reader will readily perceive that C and K 

 may interchange their places, so as to throw L without the 

 trapezium whilst a similar mode of proof obtains. This will 

 be rendered obvious from constructing the figure. 



Schol. From this curious property a great number of beau- 

 tiful porisms may be deduced ; and it is probable that very 

 many linear loci may be brought within the reach of actual 

 demonstration by the efficiency of this single theorem. I gave 

 a demonstration of it from other principles in the Monthly 

 Magazine for July 1 825. This property is not, however, yet 

 stated in its most general or its most interesting form ; for in- 

 stead of being posited in the sides of the angle KAD, the 

 points V and W may range in the periphery of any Conic sec- 

 tion whatever, the points ECKD being in the same periphery. 

 The demonstration depends upon a system of investigation 

 something different from that employed in this paper : I shall 

 therefore defer the proof till I have completed a paper upon 

 which I am now employed, in which I trace a number of re- 

 lated properties of that class of curves. I shall, however, here 

 set down two or three corollaries from Pr. 6. ; for such they 

 may be called, as they flow from it without the slightest re- 

 ducing analysis. 



Fie. 4. 



Cor. 1. Let BDPEC (fig. 4) be a pcntalpha*, formed by 



pro- 



* This very appropriate name was given to the figure, I believe, by 

 J. C. Hobhouse, Esq. M.P.; at least I do not recollect to have met with it 

 except in his "Illustrations of the Fourth Canto of Childe Harold." This 



figure, 



