93 
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, 
tlie 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 five-fold types. 
9. In a five-fold 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 together 
with (1) two proxi mates, (2) a proximate and an ultimate^ 
(3) two mediates. To the first we may add another proxi- 
mate, an ultimate, or a mediate of 3 kinds, viz., at distances 
11, 13, 33 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, 13 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 obverse of 
one of its marks ; or a proximate or ultimate, which are of 
3 kinds, at distances 11, 13, 33 from the two mediates; 4 
types. Thus there are 12 five-fold types with one pair of 
obverses. With two pairs of obverses at odd distances there 
is only one type, all the remaining marks being similarly 
related to them ; at even distance, the remaining mark 
maybe evenly or oddly distant from them; 2 types. On 
the whole we have 12-M2-f-3 = 27 types of five-fold state- 
ment. 
It is to be remarked that there is no pure five-fold state- 
ment in which all the distances are even ; and that if there 
is only one pair of obverses with all the distances even, the 
type is a proper group together with the obverse of one of 
its marks. 
