TYPES OF COMPOUND STATEMENT. oy 



an ultimate. The two proximates may be distant from 

 the two mediates i i, i 3, or i i, 3 3, or i 3, i 3, or i 3, 3 3 ; 

 4 types. The proximate or ultimate must not be respec- 

 tively distant i i^ 3 3^ or 3 3, i i ; for then they would form a 

 pair of obverses ; there remain the cases i i with i i or 1 3, 



1 3 with 1 3, and 3 3 with i 3 or 3 3 ; 5 types. In the fourth 

 case we have an origin, obverse, and three mediates distant 



2 from one another ; the remaining mark must be distant 

 either i or 3 from these mediates ; 2 types. This makes 

 twenty-two types of sixfold statement with one pair of 

 obverses. 



12. If a sixfold statement contains 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 

 may be both evenly distant ; taking one of them as origin, 

 it is associated with 5 mediates, so that there is i type 

 only. Or both oddly distant ; here there are two cases, ac- 

 cording as the distances are i i, 3 3, or i 3, 13. Or one oddly 

 and one evenly distant ; the latter is any one of the four 

 remaining mediates, and then the former is distant i or 3 

 from it ; 2 types. If the two pairs of obverses be oddly 

 distant they form an aggregate which is related in the same 

 way to all the remaining twelve marks ; viz. any one of 

 these being taken as origin, we have a pair of mediates and 

 a proximate with its obverse ultimate. The thing to be 

 considered, therefore, is the distance between the two marks 

 to be added, which may be 1,2, or 3, and each in two 

 ways; 6 types, 



A sixfold statement with three pairs of obverses is one 

 of two types only; viz. these are all evenly distant when 

 they are the mediates to one origin, or two evenly distant 

 and one oddly distant from both of them. 



1 3. A pure sevenfold statement must consist of a group 



