30 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv. 



namely: AH trains which contain a principal or maximnm succession consisting of 1, 2, 



3^ ,1c,... triads are placed together and numbered consecutively; the subordinate 



arrangement of these is easily observed. In general, the trains are exhibited in full, but in 

 order to save space a few of the longest trains which are <livisible into parts cychcaUy repeated 

 are shown m one part only. These, on account of their greater length, appear toward the 

 end of the series of figures. 



The noncongruence of the remaining 42 systems is shown in the dissimilarity in the number 

 and m the type of trains enumerated mider each system. Systems with different numbers of 

 trains in their classes are noncongruent. The distinctiveness of trains numbered differently 

 is evident, and the fact that no possible confusion of the trains as pictured can arise constitutes 

 the chief merit of this method of proof of the nonconvergence of triad systems. 



The trains for the 44 systems contain 216 different types, among which appear trains 

 with cycles of period 4, 6, 9, 10, 12, IS, 20, 24, .30, and 72. Trains with cycles of period 2 or 

 of period 3 are impossible, but while no cycle of period 5 has appeared among the trams of 

 these systems, there is no evident reason why such a cycle may not occur in the trains of sys- 

 tems not yet investigated. The groups for the 44 systems have been determined and the 

 generators for each of the 20 new groups are exhibited. 



The groups for the 24 systems lA, IB, IC, IIA, IIB, IIC, IID, HE, IIF, HIA, IIIB, IIIC, 

 HID, IVA, IVB, VA, VB, VC, VD, VIA, VIB, VIC, VID, VII have already been determmed,' 

 and for the sake of brevity are omitted. 



Mr. Cole has pointed out an error in the order of the group for the system IIIB. The 

 order of this group is 192, and not 96, as previously stated. 



Trains fok the System IA. 



Three classes of trains terminating in triads of the system: (1) Seven trains, figure 20 

 (2) 7 trains, figure 1; (3) 21 trains, fig-ure 2. 



One class of trains terminating in cycles of period 18: (4) Seven trains, figure 194. 



Trains for the System IB. 



Six classes of trains terminating in triads of the system: (1) Three trains, figure 24; (2) 4 

 trains, figure 20; (3) 3 trains, figure 4; (4) 6 trains, figure 3; (5) 9 trains, figure 2; (6) 10 trains, 

 figure 1. 



Three classes of trains terminating in cycles of period 18: (7) Three trains, figure 195; 

 (8) 3 trams, figure 196; (9) 1 train, figure 197. 



Trains for the System IC. 



Six classes of trams terminating m triads of the system: (1) Six trains, figure 24; (2) 1 

 train, figure 20; (3) 3 trains, figure 4; (4) 12 trains, figure 3; (5) 6 trains, figure 2; (6) 7 trains, 

 figure 1. 



Three classes of trains terminating in cycles of period 18: (7) One train, figure 198; (8) 3 

 trains, figure 199; (9) 3 trains, figure 200. 



Trains for the System IIA. 



Five classes of trains terminating in triads of the system: (1) One tram, figure 5; (2) 12 

 trams, figure 21; (3) 6 trains, figure 15; (4) 4 trains, figure 49; (5) 12 trains, figure 35. 



Trains for the System IIB. 



Eleven classes of trains terminating in triads of the system: (1) Two trains, figure 21; 

 (2) 1 tram, figure 14; (3) 2 trains, figure 15; (4) 4 trains, figure 17; (5) 2 trams, figure 16; (6) 

 4 trams, figure 85; (7) 4 trains, figure 44; (8) 4 trains, figure 77; (9) 4 trains, figure 74; (10) 

 4 trains, figure 39; (11) 4 trains, figure 31. ^___ 



> Cummings, L. D. (Loc. cit.) 



