90 PROF. W. K. CLIFFORD ON THE 



and a triad ; for it must contain a triad, by the same rea- 

 soning by which this was proved for a fivefold statement ; 

 and then either all the other four marks are oddly distant 

 from thisj and so form a group by themselves^ or else one 

 of them is evenly distant from the triad and so forms a 

 group with it. If the group is proper, being the proximates 

 to a certain origin, the triad must consist of two mediates 

 and either the origin, the obverse, or another mediate ; and 

 in the latter case the three mediates are distant i i i or 3 3 3 

 from some proximate ; 4 types. If the group is improper, 

 the triad is either aU origins or all obverses, or two origins 

 and an obverse, or an origin and two obverses ; 4 types. 

 In all, 8 types of pure sevenfold statement. 



14. A sevenfold statement with one pair of obverses 

 must consist either of four marks evenly distant from one 

 another and three oddly distant from them; or of five 

 marks evenly distant from one another and two oddly dis- 

 tant from them. In the former case the pair of obverses 

 may be in the four or in the three. If they are in the 

 four, the three form a triad which are proximates to one 

 origin ; and then the pair may be the origin and obverse or 

 a pair of mediates. If the pair are origin and obverse, the 

 other two (at distance 2) are mediates, distance i i, i 3 or 

 3 3 from the proximate which is not in the triad ; if the 

 pair are mediates, the two may be the origin or obverse with 

 a mediate distant i or 3 from that proximate (4 types) , or 

 two mediates distant 11, i 3, 3 3 from it (3 types). If the 

 pair of obverses are in the set of three marks, the four form 

 a group, which may be proper or improper. If proper, the 

 three may be origin and obverse with a mediate, or a pair 

 of mediates with origin, obverse, or another mediate; 4 

 types. If improper, the three must be two origins and an 

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



Five marks evenly distant containing only one pair of ob- 



