122 Journal of the Mitchell Society. [Nov. 



These facts may be briefly put as follows: 



e*j intersects flj, b _?•; aj, b i "] 



^ intersects c kl (i i k I — 1 2 ^ 



dj does not intersect c ik V Kh J\ *\ 7 ~ ,V /• '.: ' ? } 



3 K *, /, k, /, all distinct 



Cij = Cji " 



Ay is not equal to A j? 



We see then that there are triangles of the form c ra , c^ c s6 , 

 which may be briefly represented by 12 • 34 • 56. Hence 

 there are thirty ( 6 P 3 ) triangles of the type 12 and fifteen of 

 the type 12 • 34 • 56. The latter arises from the fact that, 

 if we fix our attention upon 12, the other two sets may be 

 written in only three ways. 



§6. History of the Theorem. 



In 1858 Schlafli (/. c) proved the double-six theorem inci- 

 dentally in connection with his investigations on the 27 lines 

 on the cubic surface. He enunciated the theorem in the fol- 

 lowing- form: — 



Given Jive lines a, b, c, d, e, which meet the sa?ne straight 

 line X: then may any four of the jive lines be intersected by 

 unoiher line. Suppose that A, B, C, D, E are the other lines 

 intersecting (b, c, d, <?), (<:, d, e, a), (d, e, a, $,), (g, a, b, c,) 

 and (#, b, c, d) respectively. Then A, B, C, D, E will all be 

 met by one other straight line x. 



The double-six in this case is written 



(a, b, £, d, e, x \ 

 A. B, C, D, E, X ) 



Schlafli then proposes the question — "Is there, for this ele- 

 mentary theorem, a demonstration more simple than the one 

 derived from the theory of cubic forms?" 



Sylvester* states that the theorem admits of very simple 



•Note sur les 27 droites d'une surface du d e degreV' Comptes Rendus, 

 vol. LII (1861) pp. 977-980. 



