igof\ Henderson — A Memoir. 121 



derived therefrom is the; most simple and convenient that has 

 yet been discovered for the 27 lines and 45 triple tangent 

 planes. 



Notation. Starting- with the double-six, written 



*.» ^.' ^ 3 » ^ 4 » ^5' ^6 



we are enabled to express the complex and diversified symme- 

 try of the 27 lines and 45 triple tangent planes in unique and 

 simple form. 



Returning to the double-six, written in Salmon's notation, 

 it appears that the lines ab, cb, and eb lie in the same plane 

 and are the only three of the 27 lines that lie in the plane b. 

 In like manner cb, cd, and cf a.\\ lie in the plane c and hence 

 the line that lies in the plane of ab and eb is identical with 

 the line that lies in the plane of cd and cf, viz., the line cb. 



In the new notation, we shall call the third line in the 

 plane of a x , and b 2 which intersect, the line c n and the trian- 

 gle so formed shall be designated by 12. As has been shown 

 above, the side c i2 forms with a 3 and b x a triangle, designated 

 21. Hence we have 15 (=± 6 £ a ) lines c, each of which inter- 

 sects only those four lines a, b the suffixes of which belong to 

 the pair of numbers forming the suffix of c. For suppose c JS 

 should intersect any other line, sa}- a 3 , of the eight lines a 3 , 

 a v fl 5 » a v ^ 3 » ^ 4 » ^ 5 ' &6 m Then c i2 intersecting a x , b x , a 2 and b 2 

 already, c j9 a 3 b x and c n a 3 b 2 form two triangles, and since 

 they have two lines in common, their planes are identical and 

 consequently b t intersects b 2 , contrary to hypothesis. 



Any two c's, the suffixes of which have a number in com- 

 mon, do not intersect. For suppose c i2 , c J3 intersect; they 

 form a plane in which a z and b z lie and therefore a t meets b x , 

 contrary to hypothesis. It may also be shown that any two 

 c's, the suffixes of which have no number in common, do inter- 

 sect. 



