NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 35 



Trains for the System V451. 



Nine classes of trains terminating in triads of the system: (1) 3 trains, figure 128; (2) 3 

 trains, figure 170; (3) 3 trains, figure 6; (4) 3 trains, figure 172; (5) 3 trains, figure 154; (6) 1 

 train, figure 66; (7) 3 trains, figure 147; (8) 3 trains, figure 137; (9) 13 trains, figure 1. 



Group for the system V451 . — The sets of transitive elements are A; B; C; ay a^ a^; &, h^ h^; 

 c, Cj c^; (/, d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s={A) (B) (C) (a, a., a^) (6, i^ h) (<^i c^ Cj) (J, d^ d^) 

 and is of order 3. 



Trains for the System VIlj/S. 



Nine classes of trains terminating in triads of the system: (1) 2 trains, figure 176; (2) 2 

 trains, figure 2; (3) 2 trams, figure 128; (4) 4 trains, figure 142; (5) 4 trains, figure 152; (6) 4 

 trains, figure 62; (7) 4 trains, figure 6; (8) 4 trains, figure 107; (9) 9 trains, figure 1. 



One class of trains terminating in a cycle of period 4: (10) 1 train, figure 182. 



Group for the system VIIS- — Tlie sets of transitive elements are A; B C; Oj 6, a^ h^; 

 Cj 62 03 63; a^ 6, flj 65/ these with the trains separate the system into 11 nonpermutable subdi- 

 visions. The group is generated by 



s= (A) (B C) (a, a^ b^ b^) (a^ 63 b^ a^) (a^ 65 b^ a^) 

 and is of order 4. 



Trains for the System ¥12^0. 



Eleven classes of trains terminating in triads of the system: (1) One train, figure 105; 

 (2) 1 train, figure 11; (3) 4 trains, figure 3; (4) 1 train, figure 149; (5) 2 trains, figure 8; (6) 

 4 trains, figure 2; (7) 2 trains, figure 181; (8) 2 trains, figure 6; (9) 1 train, figure 28; (10) 2 

 trains, figure 175; (11) 15 trains, figure 1. 



One class of trains terminating in a cycle of period 24: (12) One train, figure 202. 



Group for the System F/^^a. — The sets of transitive elements are A; B; C; a, &,; a^ h^; 

 ds ^3/ O'i iiJ ^5 ^5/ (^e K- These with the trains separate the system into 21 nonpermutable sub- 

 divisions. The group is generated by 



s=(A) (B) (0) {a, b,) {a, b,) (a, b,) (a, &J (a, b,) (a, bj, 



and is of order 2. 



Trains for the System YI2^S. 



Eighteen classes of trains terminating in triads of the system: (1) One train, figure 18 

 (2) 2 trains, figure 130; (3) 1 train, figure 132; (4) 2 trains, figure 133; (5) 2 trams, figure 153 

 (6) 1 train, figure 38; (7) 2 trains, figure 174; (8) 2 trains, figure 126; (9) 2 trains, figure 37 

 (10) 1 train, figure 8; (11) 2 trams, figure 62; (12) 2 trains, figure 120; (13) 3 trains, figure 2 

 (14) 2 trains, figure 6; (15) 2 trains, figure 156; (16) 1 train, figure 109; (17) 2 trains, figure 

 112; (18) 5 trains, figure 1. . 



One class of trains terminating in a cycle of period 4: (19) One train, figiu-e 182. 



Group for the System VI2^5. — The sets of transitive elements are A; B; C; o, b^; a^ b^; a^ b^ 

 a^ bj flj 65/ Co \. These with the trains separate the system into 21 nonpermutable subdi- 

 visions. The group is generated by 



s = (A) (BC) (a, 6,) (a.2 b^) (a, 6,) (a, b,) {a^ JJ (a, J,), 



and is of order 2. 



Trains for the System VI2<«. 



Fifteen classes of trains terminating in triads of the system: (1) One train, figure 144; 

 (2) 1 train, figure 8; (3) 2 trains, figure 167; ^4) 2 trains, figure 139; (5) 1 train, figure 165; 

 (6) 2 trains, figure 157; (7) 4 trains, figure 2; (8) 2 trains, figure 62; (9) 1 train, figure 30; (10) 



1 train, figure 6; (11) 2 trains, figure 115; (12) 1 train, figure 114; (13) 2 trains, figure 27;' (14) 



2 trains, figure 166; (15) 11 trains, figure 1. 



