NO. 2] TRIAD SYSTEMS— WHITE, COLE, CLTMMINGS. 33 



Group for the system 12. — The sets of transitive elements are ah c d e; a fi y 5 e; 1234 5; 

 these with the trains separate the system into 7 nonpernmtahle subdivisions. The group is 

 generated by 



s = (a 6 c <Z f ) (1 2 3 4 5) (a /? 7 3 e) 

 and is of order 5. 



Tkains fok the System III 2. 



Three classes of trains terminating in triads of the system (1) 15 trains, figure 1; (2) 2 

 trins, figure 48; (I!) 18 trains, figure 145. 



One class of trains terminating in a cycle of period 4: (4) 9 trains, figure 182. 

 (rroupfor the system III 2. — The sc^ts of transitive elements are a, a^ a^ hi h^ h^; c, Cj ^3 ^1 <^2 d^ 

 e, fj fj; these with the trains separate the system into 9 nonpermutablo subdivisions. The 

 group is generated by 



s= (a,) («2) (^1 ^2 ^3) (Ci di 61) (Cj cZj e,) {c^ d^ e^) (d^), 

 t=(ai a^) {a^) (6, h^) (63) (c, e^) (c^ e^) (c, e,) W, d,) (d^), 

 10= {Ui 6, a^ 62) (03 63) (c, gj C2 di) (C3) (^2 63 Ci (Z3), 

 d is of order 36. 



Trains for the System III3. 



Four classes of trains terminating in triads of the system: (1) 1 train, figure 179; (2) 3 

 trains, figui-e 2; (3) 3 trains, figure 27; (4) 28 trains, figure 1. 



Three classes of trains terminating respectively in cycles of periods 5, 6, and 6: (5) 1 train, 

 figure 203; (6) 1 train, figure 188; (7) 1 train, figure 185. 



Group for the system III3. — The sets of transitive elements are Oj a^ a^; i, h^ b^; c, c^ Cg, 

 (?, (?2 d^; f , ^2 fit' these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



S=(«l 0,2 ds) (^I ^2 ''3) (Cl C2 ^3) (^1 <^2 ^^3) (^t ^2 ^3), 



and is of order 3. 



Trains for the System Ilia. 



Nine classes of trains terminating in triads of the system: (1) 1 train, figure 106; (2) 3 

 train, figure 12; (3) 6 trains, figure 148; (4) 3 trains, figure 122; (5) 3 trains, figure 6; (6) 3 

 trains, figure 62; (7) 3 trains, figure 2; (8) 3 trains, figure 30; (9) 9 trains, figure 1. 



Two classes of trains terminating in cycles of periods 6 and 4, respectively: (10) 2 trains, 

 figure 189; (11) 3 trains, figure 182. 



(rroupfor the system IIl^. — The sets of transitive elements are a^ a^ a^; 6, h^ h^; Cj ^2 ''3/ 

 di d^ d^ «! ^2 ^3/ these with the trains separate the system into 9 nonpermutable subdivisions. 

 The group is generated by 



s=:{a^) (hi) (Ci) (aj 03) (b^ 63) (c, Cg) (d^ e,) (d^ e^) {d^ e^), 

 /=(fl, O2 «3) (^1 ^2 ^3) (<^i C2 ^3) (di dj dj) (e, e^ e^), 

 and is of order 6. 



Trains for the System III,. 



Eight classes of trains terminating in triads of the system: (1) 1 train, figure 171; (2) 3 

 trains, figure 178; (3) 3 trains, figure 93; (4) 3 trains, figure 162; (5) 6 trains, figure 2; (7) 3 

 trains, figure 6; (8) 13 trains, figure 1. 



Two classes of trains terminating in cycles of period 6: (9) 1 train, figure 184; (10) 1 train 

 figure 187. 



Group for the system IIlj. — The sets of transitive elements are Ui a^ a^; 6, h^ h^; c, fj c,; 

 d, (Zj ds; fi e, e,; these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s^a^ Oj flj) (&, hj 63) (c, Cj C3) (di dj d,) (f i e, e,) , 

 and is of order 3. 



54061°— 19 3 



