142 Professor Baker, On a jjroof of the theorem of a double six 



1 



which is x{A + A') + y {B + B') + z {C + C) 



and is therefore its intersection with the line e. 



Similarly it can be shown that a general plane meeting the 

 lines a', b', c', d' and e is that joining the points, respectively on 

 a', b', c', represented by 



yB' + zC, zC' + xA, xA' + yB. 

 If the plane, meeting a, b, c, d, joining the points 

 yB + zC, zC + xA', xA + yB' , 

 pass through an arbitrary point of the four dimensional space 

 expressed by 



^A + r^B+W + e^' + ^'S' + ^'C, 

 it can be shown without difficulty that 



X y z 



and also 



iv - D iv - ^ a - n a' - i) ^ a - v) (f - ^) ^ ^ . 



y — z z — X ^ — y ' 



to satisfy the condition it is necessary and sufficient, in fact, that 

 numbers A, /x, v, p should be possible such that 



^ = vx + p, 7) = Xy + p, t= p.z + p, 

 i' = fix-{- p, 7]' = vy + p, t,' = Xz+ p. 



The first two of these equations determine the two planes 

 meeting the lines a, b, c, d, e which pass through the arbitrary 

 point in question. The last equation determines the two points, 

 of the form 



x{A + A') + y{B + B') + z {C + C), 

 or {X -z){A + A') + {y-z){B + B'), 



where these planes meet the line e. As this last equation is un- 

 altered by interchanging f, r], I, respectively with |', -q' , I,' , it shows 

 that the two planes drawn through the arbitrary point to meet 

 a, b, c, d intersect the line e in the same two points as do the planes 

 drawn through this point to meet a', b', c', d' . 



§ 7. Now consider a figure in three dimensions consisting of 

 four lines a, b, c, d, skew to one another, and four other lines, 

 a', b', c', d', skew to one another, each meeting three of the former. 

 As before, with the same notation, the figure consists of a skew 

 hexagon BC'AB'CA', of which one set of three alternate sides is 

 intersected by a line respectively in P, Q, R, and the other set of 

 alternate sides is intersected by another line in P', Q' , R' respec- 

 tively. But here the six points A, B, C, A', B' , C, being in three 



