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4. There is clearly only one type of simple statement. 
But of two-fold statements there are four types ; viz. the 
distance may he 1, 2, 3, or 4, and so in general with n 
classes there are n types of two-fold statement. 
5. A compound statement containing no pair of obverses 
is called 'pure. In a three-fold statement there are three 
distances ; one of these must be not less than either of the 
others. If this be 2, the remaining mark must be at odd 
distance from both of these or at even distance from both ; 
thus we get the types 1, 1, 2, and 2, 2, 2. If the not-less 
distance be 3, the remaining distances must be one even and 
the other odd ; the even distance must be 2, the odd one 
either 1 or 3, and the types are 1, 2, 3 ; 2, 3, 3. Thus there 
are 4 pure three-fold types ; with a pair of obverses, the re- 
maining mark must be at odd or even distance from them ; 
1, 3, 4 ; 2, 2, 4. In all six three-fold types ; observe that 
there is necessarily one even distance. 
6. A fortiori, in a four-fold statement there must be 
one even distance. In a pure four-fold statement this dis- 
tance is 2. From this pair of marks let both the others be 
oddty distant ; then they must be evenly distant from one 
another, i.e. at distance 2, obverses being excluded. The 
odd distances are 1 or 3, and it will be easily seen that the 
following are all the possible cases : 
ijl 1|1 ill l\3 113 313 
1|1 113 3|3 3|1 3|3 3|3 
• ••••* 
In these figures the dots indicate the four marks, the cross 
lines indicate distance 2, and the other figures the distances 
between the marks on either side of them. Next, from the 
pair of marks at distance 2 let one of the others at least be 
evenly distant, i.e. at distance 2. Then we have three 
marks which are all at distance 2 from one another, and it 
is easy to show that they are all proximates of a certain 
other mark. For select one of them as origin; then the 
