Mr. Graves on the Porismatic Arrangement of Points. 135 



may always bear to each other a given ratio, the locus of the 

 intersection (s) of the lines so drawn will be a straight line. 



2. If a m-lateral, all of whose sides but one pass respect- 

 ively through given points, be inscribed in a given m-lateral, 

 the remaining side of the former will always cut oflF from two 

 fixed points respectively in the two sides circumscribing it of 

 the latter segments which contain a constant rectangle. 



Hence the problem — to inscribe in a w-lateral another m- 

 lateral whose sides shall pass through given points — reduces 

 itself to this; to draw through a given point a straight line 

 which shall cut off from two given points, on two given straight 

 lines, segments containing a given rectangle. 



3. If a 7W-lateral, all of whose sides pass respectively 

 through given points, have m — \ of its angles respectively on 

 given straight lines, the locus of its remaining angle will be a 

 conic section. 



This theorem and its converse cases are very fertile. By 

 assuming conditions which shall cause 3 points of the locus to 

 lie in directo, we get, among others, the two theorems, which 

 have formed the principal subject of the preceding paper. 



By the angles of a ?;2-lateral I understand here those angles 

 only which are contained by the sides taken once round two 

 by two in a pre-determined order, so that no side occurs more 

 than twice. Otherwise a wz-lateral has more angles than sides. 



4. If the angles of a quadrilateral, and the intersection of 

 one pair of its opposite sides lie respectively on 5 given 

 straight lines diverging from a point, the locus of the inter- 

 section of the other pair of sides will be a 6th straight line 

 diverging from the same point. This theorem admits of 

 extension. 



The pretty theorem of Dr. James Thomson of Glasgow, 

 {a7ite, p. 41.) has suggested to me the following more general 

 and fertile proposition. 



5. If straight lines be drawn from the vertices of a triangle 

 to any 2 points in its plane to meet the sides, the 6 points of 

 meeting will be in a conic section. Hence, when 5 of the 

 points lie in a circle, the 6th will lie in the same circle. If 3 

 of them lie in directo, so will the other 3. The intersections 

 of the opposite sides of any hexagon of which they form the 

 vertices, will lie in directo. 



It is an interesting question whether it be not possible to 

 contrive one or more schemes consisting of sets of 6 points, 

 such that, if each set be supposed to lie in a separate conic 

 section and all the sets but one be given, the remaining set 

 shall porismatically follow. From 6 equations of the form 



