igos] Henderson— A Memoir. 123 



geometrical proof but he did not give the proof. Salmon* has 

 given a method for geometrically constructing a double-six 

 but I do not understand it to be a proof of the theorem, inde- 

 pendent of the cubic surface. 



In 1868 Cayleyf gave a proof of the theorem from purely sta- 

 tic considerations, making use of theorems on six lines in 

 involution. Again in 1870 Cayley+ verified the theorem, 

 using this time his method of the six co-ordinates of a line. 

 Kasner|| has recently given a proof by using the six co-ordi- 

 nates of a line. 



The method I have adopted in the following is indepen- 

 dent of the theory of cubic surfaces. 



[Note. This proof and a model of the configuration constructed in Nov. 

 1902, were presented by me before the Chicago Section of the Am. Math. 

 Society on April 11th, 1903. I had not at that time seen Kasner's article in 

 the April, 1903, number of the American Journal, an article having points 

 of contact with mine. J 



§7. Proof of the Double- Six Theorem. 



Representing the double-six in the Schlafli-Cayley notation 



/l23456\ 

 \ V 2' 3' 4' 5' 6' / 



it is seen that these 12 lines have the thirty intersections Py \ 



♦Geometry of 8 Divisions, 4th edition, p. 600. 



t"A 'Smith's Prize' Paper; Solutions", Coll. Math Papers, vol. VIII 

 (1868) pp. 430-431. 



$"On the Double-Sixers of a Cubic Surface", Coll. Math. Papers, vol. 

 VII., pp. 316-330; Quarterly Journal of Mathematics, vol. X, (1870) 

 58-71. 



||' 'The Double-Six Configuration Connected with the Cubic Surface, and 

 a Related Group of Cremona Transformations", American Journal of 

 Mathematics vol. XXV, No. 2 (1903), pp. 107-122. 



