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MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XIV. 



must leave unchanged the totality of trains connected with the system. The totality of com- 

 plete trains (cycles with their appendices) forms accordingly an arrangement of triads invariant 

 imder those substitutions on the 15 elements that transform the triple system into itself and 

 facihtates the determination of the group of the system. 



Example: The tnple system VI 3i,y on 15 elements. — For convenience the system is trans- 

 formed by the substitution 



^/ABC a J), a^h, afi^ afi^ aj>, a J) A 

 ^=\b acl234defg7S65j' 



and is exhibited in the following 15 by 7 array: 



The transforming process is simple and may be shown in its application to a triad 458 which is 

 extraneous to this system. Its pairs 45, 48, 58 transform, respectively, into a, I, g, giving the 

 transformed triad alg. The triad alg transforms into the triad of the system 76/ which repeats 

 indefinitely. These three triads form the type of train which is exhibited graphically in figure 

 3. This system applied as an operator on the 455 triads yields the following set of covariants 

 (trains) : 



Trains for the System VISiT. 



Six classes of trains terminating in triads of the system: (1) 11 trains, figure 1; (2) 4 trains, 

 figure 2; (3) 12 trams, figure 6; (4) 2 trams, figure 206; (5) 2 trains, figure 210; (6) 4 trains, 

 figure 211. 



One class of trains terminating in a cycle of period 4: (7) 1 train, figure 182. 



Two classes of trains terminating in cycles of period 6: (8) 5 trains, figure 183; (9) 1 train, 



figure 213. 



Twoclassesof trains terminating in cycles of period 12: (10) 1 train, figure 214; (ll)4trains, 



figure 215. 



Determination of the Group for the System VI347. 



The trains for this system separate the 35 triads into 6 distinct classes and every operation 

 of the group that leaves the system invariant must transform any train into itself, or into 

 another tram of the same class. Since only those elements may be permuted which occiu- the 

 same number of times in a class, the enumeration of the appearances of each of the 15 elements 

 in the 6 classes of trains, as m the following table, shows the possible sets of transitive elements. 

 An examination of the triads of the system belonging to class (1) shows that the 15 elements do 

 not enter symmetrically as members of the triads of the class ; for example, in these 1 1 triads 

 the element c appears 7 times but no other element appears 7 times. 



The possible systems of transitivity for the group are therefore a; b; c; d, e,f, (/; 1 , 2, 3, 4 ; 5, 6, 7, 8. 



The sots of possible transitive elements subdivide the classes into sets of triads which are 



not transformable into one another by operations of the group of the system; the subdivisions 



