EEV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. 291 



I shall proceed to draw these lines by the linear construc- 

 tion of the aconic functions in their equations. Putting 

 0—3/ iyx—yx) — (y—y) (yx—yx) 



1 8 45 54 i 5 12 2 1 __ ^IJ 



y — y . X — X — y — y x — x 



IS45 4512 



and .2:" for what this becomes when yi is exchanged for as^ 

 &c. ; the area of the Pascalian triangle, having its angles at 

 the intersections of the opposite sides of the hexagon, 

 123458 is, 



A = y . (x—x) + y (x—x) -j- y (x—x) zz C : D, 



13 23 84 23 34 12 34 13 23 

 45 58 81 68 81 45 81 45 68 



C being the aconic function [123458] and D being the factor 

 (y-y.x-x — y-y.x-x)(.y-y.x-x — y-y.x^(:,y-y.x-x — y-y.x^) 



1246 4612 3368 6823 3481 8134 



which vanishes whenever A becomes infinite, i. e., when 

 any pair of opposite sides of the hexagon 123458 are pa- 

 rallel. ITiat C rz is the equation' to the conic when this 

 hexagon lies in one, is evident from the consideration that it 

 is of the second degree in (xgy^), and thafit vanishes if for 

 (^sy«) you put any of the other five, («,y,) (^23/2) (x^ys) ... If 

 (arsyg) is {xiyi) or (x^y^), this is instantly seen ; and if (xsy^) is 

 (xiys), the triangle A has all its angles in (12) ; if (x^g) is 

 (xtVi)i it has all its angles in (54) ; and if {x%y^ is (x^y^, two 

 of its angles coincide with the point 3, in none of which 

 three cases does D become zero. 



Wherefore, if the points (1245), (23*58), and (34-81), be 

 a, c, e, 



[123458] =: \y ' x—x -\-y . x—x -f- y . x—xl D, 



La o e o e a a a cJ 



ri23457l — \y ■ x—x +y . x—x + y . x—xl D^, 



i-a c; e,' o, «, a ' «/ • ojJ 



where c^ e, Z>, differ from c e D only by the exchange of 8 

 for 7. Neglecting the common factor of D and Dj, which 

 is free from 8 and 7, we have to draw the line 



{ 



(y . x-a; + y . x-x + y . x-x) (y-y . x-x — y-y . x-x) {y-y . x-x — y-y x-x) . (58) i 

 «eeoeaaae2368 68233481 8134 ( 



. — — — — r-o 



— (» i X-X + y . x-x + » , jT-ar) (y-y ■ x-x — y-^ . x-x) {jf-y • *-a' — V-y n^^) . (57) V 

 a G/ e, e, •>»•/> e, ^8 367 6783 3471 7134 J 



