ON THE GRAPHIC REPRESENTATION 



OF THE 



Projection of Two Triads of Planes into the Mystic 



Hexagram. 



BY ARCHIBALD HENDERSON, PH.D. 



WITH PLATE I. 



Cayley* has considered the following- question: to find a 

 point such that its polar plane in regard to a given system of 

 three planes is the same as its polar plane in regard to another 

 given system of three planes. 



Let us desig-nate for convenience the first three planes as «, 

 b, and c, the second three as f, g, and h\ The line ab will 

 denote the line of intersection of the planes a and 3, and the 

 point abc will denote the point of intersection of the planes 0, 

 b, and c\ and so in other cases. 



The conclusion reached (1. c.) is that there are four points 



0„ 2 , 3 , and 4 , which fulfil the required conditions. It was 



shown, that from any one of the points 0, it is possible to draw 



a line meeting the lines af X bg X ch [1] 



" ag X bkX cf [2] 



" akXb/Xcg [3] 



" " afXbhXcg [4] 



" agXbfXch [5] 



44 ahXbgX cf [6] 



and moreover that these six lines [1], [2], [3], [4], [5], and 



[6] lie on a cone of the second order. Projecting now the 



figure of the six planes a, b, c,f, g, and h upon any arbitrary 



plane (not passing through one of the lines [1], [2], [3], [4], 



[5], or [6]), we obtain the Pascal configuration. The twenty 



points abc, abf, .... fgh, are as follows, viz. (omitting the 



two points abc,fgk) the remaining eighteen points are the 



* CoUected Math. Papers, Vol. VI., pp. 129-134. 



124 [Dec. 



