and its meaning in hyper geometry. 
477 
serve equally to define twelve lines in space of three dimensions ; 
and when two points such as A x and B. 2 lie each in the tangent 
plane of the other, the corresponding lines intersect. Putting this 
interpretation on the last result we obtain a known method of 
building up a double-six; viz. If a x be a line in space, and b 2 , b s , 
i 4 , b 5 , b a any five* lines meeting a x , five further lines a 2 , a 3 , u 4 , 
a 5 , a 6 may be found each meeting four of the five lines b. 2 , b 3 , 
b 4 , b 5 , b 6 ; and a twelfth line b } must exist which meets the five 
a 3 , a 3 , a 4 , a 5 , a$. Clearly the formulae obtained above can be 
applied to prove this without mention of hypergeometry. 
I have not succeeded in finding any extension or analogue of 
the above theorem to a geometry of other than five dimensions ; 
it is remarkable if the theorem belongs to space of five dimensions 
and to no other. In conclusion, the theorem may be stated in a 
purely algebraic form : 
If in a symmetrical determinant of six rows the six elements 
in the leading diagonal all vanish, and the first minors of five of 
these elements also vanish, the minor of the remaining element must 
vanish. 
* No two of the five should intersect and no four should lie on a conicoid. 
