NOTE ON THE HOMOLOGY OF SETS. 
[From the Quarterly Mathematical Journal, vol. I. (1857), p. 178.] 
Let L denote a set of any four elements a, h, c, d, and in like manner A, L x &c. 
sets of the four elements a, ¡3, y, 8; a /} b t , c n d,, &c. ; then we may establish a 
relation of homology between four sets L, L x , L 2 , L 3 , and four other sets A, A 1} A 2 , A 3 ; 
viz., considering the corresponding anharmonic ratios of the different sets, we may 
suppose a relation of homology between these ratios. Thus considering the set to L, 
write 
x = (a — b) (c — d), 
y = (a — c) (d — b ), 
2; ={a — d)(b — c), 
then x + y + z = 0 and the anharmonic ratios of the set are x : y : z—we may, if 
we please, take x : y as the anharmonic ratio of the set. And in like manner taking 
£ : y as the anharmonic ratio of the set a, /3, y, 8, &c., the assumed relation between 
the sets L, L 1} L. 2> L s and the sets A, A 1} A 2 , A 3 will be 
xÇ , 
XT) , 
y£, 
yy 
X 1 7] 1 , 
yifi> 
yiVi 
#2^2) 
X 2 7] 2 , 
y&-> 
y-2V2 
^3^3 > 
XsVs, 
y 3 %3, 
ysVs 
= 0; 
and it is to be observed, that this relation is independent of the particular ratio 
x : y which has been chosen as the anharmonic ratio of the set; in fact, if we write 
x = _ y _ Z} £ = _ v — £ then reducing the result by means of an elementary 
property of determinants, the equation will preserve its original form, but will contain 
the ratios y : zrj : &c., instead of the ratios x : y; | : y, &c. 
5—2