95 
(1) the four marks are all oddly distant from the obverses ; 
(2) one is evenly distant and three oddly distant ; 
(3) two are evenly distant and two oddly ; 
(4) three a,re evenly distant and one oddly. 
In the first case the four marks form a group. If this is a 
proper group, the pair of obverses must be either the origin 
and obverse of the group, or a pair of mediates; 2 types. 
If the group is improper, the pair must be an origin and an 
obverse; 1 type. In the second case, we have an origin, 
an obverse, and a mediate, to which we must add 3 marks 
taken out of the proximates and ultimates. We may add 3 
proximates distant respectively 118 or 133 from the mediate ; 
2 types, or 2 proximates distant respectively 11, 13,33 from 
the mediate and with each of these combinations an 
ultimate distant either 1 or 3; 6 types. To interchange 
proximates with ultimates clearly makes no difference ; so 
that in reckoning the cases of 1 proximate and 2 ultimates or 
3 ultimates, we should find no new types. In the third case 
we have an origin an obverse and two mediates, distant 2 
from each other, and to these we have to add either two 
proximates or a proximate and an ultimate. The two proxi-* 
maTes may be distant from the two mediates 11, 13, or 11, 33, 
or 13, 13, or 13, 83; 4 types. The proximate and ultimate 
must not be respectively distant 11, 33, or 33, 11, for then 
they would form a pair of obverses; there remain the 
cases 11 with 11 or 13, 13 with 13, and 33 with 13 or 33; 
5 types. In the fourth case we have an origin obverse and 
3 mediates distant 2 from one another ; the remaining mark 
must be distant either one or three from these mediates ; 2 
types. This makes 22 types of six -fold statement with 1 
pair of obverses. 
12. If a six-fold statement contain two pairs of obverses 
these must be either evenly or oddly distant. If they are 
evenly distant, we have an origin obverse and two obverse 
mediates, to which two other marks are to be added. These 
