TYPES OF COMPOUND STATEMENT. 87 



obverse of one of its marks — or a proximate or ultimate 

 which are of three kinds, at distances 11,13,33 from the 

 two mediates; 4 types. Thus there are 12 fivefold 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 may be evenly or oddly distant from 

 them; 2 types. On the whole we have 12+12 + 3 = 27 

 types of fivefold statement. 



It is to be remarked that there is no pure fivefold 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. 



10. We may now prove, as a consequence of the last re- 

 mark, that a pure sixfold statement either contains a group 

 of four with a pair oddly distant from it or consists of two 

 triads oddly distant from one another. For there must be 

 a pair at distance 2 : if the other four are all oddly distant 

 from these, they form a group ; if one is evenly distant 

 and three oddly distant, we have the case of the two triads ; 

 if two are evenly distant, we again have a group. We must 

 add, then, first to a proper group, and then to an improper 

 group, a pair oddly distant from it. To a proper group con- 

 sisting of the proximates to a certain origin we may add the 

 origin or its obverse with a mediate, or two mediates; 3 types. 

 An improper group is symmetrical ; that is to say, if we 

 substitute for any one of its marks the obverse of that mark, 

 we shall obtain a proper group. In this way we shall get 

 four origins distant 1 1 i 3 from the group, and four obverses 

 distant i 3 3 3 ; if we add to these the obverses of the marks 

 in the group itself, we have described the relation of the 

 twelve remaining marks to the group. To form, therefore, 

 a pure sixfold statement we may add either two origins or 

 two obverses or an origin and an obverse ; 3 types. 



