86 PROF. W. K. CLIFFORD ON THE 



with the distances i 3^, i 3 ; i 3*, 3 i ; 2 2% 2 2 ; i 3', 2 2 ; 

 I 3', 2 2. With two pairs of obverses they must be either 

 at odd or even distances from one another ; two types. 

 Altogether 12 + 5 + 2=19 fourfold types. 



8. In Zi pure fivefold statement there is always a triad of 

 marks at distance 2 from one another. For there is a pair 

 evenly distant ; if there is not another mark evenly distant 

 from these^ the remaining three are all oddly distant^ and 

 therefore evenly distant from one another. First, then, let 

 the remaining two marks be both oddly distant from the 

 triad. In regard to the origin of which these are proxi- 

 mates, the two to be added must be either two mediates, 

 like (of two kinds) or unlike, or a mediate of either kind 

 with the origin or the obverse; 7 types. Next, if one of 

 the two marks be evenly distant from the triad, it must 

 form with the triad either a proper or an improper group 

 of four. To a proper group we may add the origin, the 

 obverse, or a mediate ; to an improper group, the origin or 

 the obverse (the mediates give no new type) , 5 types ; or, in 

 all, 12 pure fivefold types. 



9. In a fivefold statement with one pair of obverses 

 there must be another pair of marks at distance 2. We 

 have therefore to add one mark to each of the following 

 three types of fourfold statement, — a pair of obverses to- 

 gether with (i) two proximates, (2) a proximate and an 

 ultimate, (3) two mediates. To the first we may add an- 

 other proximate, an ultimate or a mediate of three kinds, viz. 

 at distances i i, i 3, 3 3 from the two proximates ; 5 types. 

 To the second if we add a proximate or an ultimate, we fall 

 back on one of the previous cases ; but there are again 

 three kinds of mediates, at distances 11,33, i 3 from the 

 proximate and ultimate ; 3 types. To the third we may 

 add another mediate, whereby the type becomes a proper 

 group together with the obverse of one of its marks, which 

 is the same thing as an improper group together with the 



