( 354 ) 



A. An 5(4,3), n = 8 consists of f J:/ 1=14 quadruplets, 



so it is a schematic Cf. (8., 14 4 . II' we a<l<l to the triplets of an 

 S(S, 2), n = 7, that is of a Cf. (7,) a new letter, and if out of I lie 

 quadruplets tlni> formed we choose one, then in the completion 

 with seven new quadruplets sought for the pairs of the selected 

 triplets must appear still twice; so a new quadruplet remains, com- 

 plementary to the one selected. 



This holds for each quadruple! ; the whole completion is thus 

 complementary and only possible in one way l ). 



B. It* in an 5(4,3), n = 10, that is a Cf. (10 ls , 30J, we choose 

 an arbitrary quadruplet 1 2 3 4, each pair of these letters appears in 

 three more quadruplets; so there are 18 more such quadruplets. 

 Then each single letter appears two times more, completed with 

 lri-plel> out of 5,6,7,8,9,0, which triplets form thus together a 

 Cf. (6 4 , 8 8 ). Of the whole system only three quadruplets out of 

 5,0,7,8,9,0 remain, which in pairs may have at most only two 

 letters in common. Such systems of three exist however only in one 

 type a i, as : 



5 7 8: 5 6 90 ; 7 8 9 0, 

 with which the system to be formed must commence. The eight 

 triplets of letters missing here form of necessity the Cf. (6 4 ,8 8 ), which 

 breaks up only in one way into four pairs completing each other. 

 If we completi 1 these pairs respectively with L, 2, 3 and 4 (in which 

 order is irrelevant) the following quadraplets are formed: 



1 5 7 9 ; 2 5 7 ; 3 5 8 9 ; 4 5 8 



1 s o ; 2 8 i) ; 3 7 O ; 4 7 9. 



The entire further completion i> now determined and must run 

 as follows: 



1 4 5 : 12 5 8 : 2 3 7 9 ; 



2 3 5 : 12 7 : 2 3 8 ; 

 13 7 8 : J 3 5 : 2 4 5 9 ; 



2 4 7 8 ; 1 :; 9:24 ; 

 L 2 9 : 14 7 : 3 4 5 7 ; 



3 4 9 : 14 8 9:34 8 . 



Now thai with this the existence of only one. type is assumed, we 

 can give it a simpler form; we shall do this in two ways. 



1 ) The scheme (8 7 , 14,) indicates in the measure-polytope IK the vertices, forming 

 with point zero and its opposite vertex together a cross-polytope (\ [Nieuw 

 Arclüef v. Wish. 2»d Series, VII, p. 255). 



~) The complementary type of a division of six elements into three pairs 



