536 
A FIFTH MEMOIR UPON QUANTICS. 
[156 
vanishes: but this being so, the determinant 
1, a, a', clcl' 
1, /3, /3', /3/3' 
1, Y> 7; 77' 
1, S, S', SS' 
which is equal to 
a, 1, a + a', aa' 
/3, 1, /3 + /3', /8/3' 
7» 7 + 7> 77' 
S, 1, S+S', SS' 
will vanish, or the two sets (a, /3, 7, S) and (o', /3', 7', S') will be homographic; that 
is, if any number of pairs are in involution, then, considering four pairs and selecting 
in any manner a term out of each pair, these four terms and the other terms of 
the same four pairs form respectively two sets, and the two sets so obtained will be 
homographic. 
110. In particular, if we have only three pairs a, a'; /3, /3'; 7, 7', then the sets 
a, /3, 7, a' and a', /3', 7', a will be homographie; in fact, the condition of homography is 
which may be written 
or what is the same thing, 
1, a, 
a', 
act! 
= 0, 
1, /3, 
/3', 
/3/8' 
Y> 
7. 
77' 
1, a', 
a, 
aa' 
a, 1, 
a + a , cia 
= 0, 
/8, 1, 
/3 + /3', /8/3' 
7> !> 
7 
+ Y> 77' 
a', 1, 
a + a', aa' 
a 
1, 
a + a 
, aa' 
/3 , 1, /3 + /3', /3/3' 
7 , 1, 7+7> 77 
a' — a, 0, 0 ,0 
so that the first-mentioned relation is equivalent to 
(a! - a) 
a +a', 
aa' 
/3 + /3', 
/3/3' 
7 +7> 
77