134 Professor Baher, On a proof of the theorem of a double six 



With the notation above, denote the respective intersections 

 (6', c), (6, c'), {c', a), {c, a'), {a', h), {a, h'), (a,/'), {bj'), {c,f') by 

 4, A', B, B', C, C, W, r, W. Let/ be the transversal other 

 than e of a', b', c', d', which we may represent by/ = {a', b', c', d')le, 



'f V, 



Pig. 1 



and denote the points (a'J), {b',f), (c'J) respectively bv U, V, W. 

 Similarly let /^ be the transversal other than d of (a'," 6' c' e') 

 which we may denote by/^ = {a', bi, c', e')/d', and let'the points 

 («',/i), (6',/i), (c',/i), be U^, Fi, W,. ^ 



Now take the lines 



a, b, c,f; e 

 a',b',c'J'-d''' 



The two quadric surfaces defined respectively as containing {b, c e) 

 and {b', c', d'), have, both of them, the two generators e and 'd' 

 which are intersecting lines. The other common points of these 

 two quadrics are then coplanar. Such points are A and A' respec- 

 tively {b\ G) and {b, c'), and U' or (a,/') and U or (a'J). Thus U 

 les on the plane A, A', V). So, by considering the quadrics 

 (c, a, e), [c , a , d ), we find that V lies on the plane (B B' V) 

 and by considering the quadrics (a, b, e), (a', b' , d'), that W lies 

 on the plane {C, C , W). By taking the hnes ■ 



^, b, c,f^\ d\ 

 <&',c',/';4' 

 and considering the pairs of quadrics 



{b, c, d), {b\ c', e'); (c, a, d), {c' , a', e'); {a, b, d), {a', b', e') 

 we similarly show that V, V, W„ lie respectively on the planes 

 {A, A, U), [B, B', V), {C, C, W), and therefore coincide 



