34 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol.xiv. 



Trains for the System VI 7. 



Three classes of trains terminating in triads of the system: (1) 6 trains, figure 9; (2) 1 

 trains, figure 6; (3) 17 trains, figure 1. 



One class of trains terminating in a cycle of period 72 : (4) 1 train, figure 204. 



Group for the system VI y. — The sets of transitive elements are A; B C; Ot a^ a^ &i 5, 63 

 r, c, C3 di 6,2 d^; these with the trains separate the system into five nonpermutable subdivisions. 

 The group is generated by 



s = {A) {B (7) (a, di 63 c, a^ d^ 6, fi a^ d^ h^ c^), 

 and is of order 12. 



Trains for the System V4(31. 



Nine classes of trains terminating in triads of the system: (1) 3 trains, figure 160; (2) 3 

 trains, figure 180; (3) 9 trains, figure 2; (4) 3 trains, figure 138; (5) 1 train, figure 139; (6) 3 

 trains, figure 3; (7) 3 trains, figure 27; (8) 3 trains, figure 164; (9) 7 trains, figure 1. 



Group for the system V40^ ■ — The sets of transitive elements are A; B; C; a^ a^ a^; i^ 62 ^3^ 

 c, C2 c^; di d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s= (A) (B) iO (a, 02 a,) (6, h i,) {c, c^ c,) (d, d^ d,) 

 and is of order 3. 



Trains for the System V4/32. 



Seven classes of trains terminating in triads of the system: (1) 3 trains, figure 134; (2) 3 

 trains, figure 10; (3) 3 trains, figure 29; (4) 6 trains, figure 2; (5) 1 train, figure 95; (6) 3 trains, 

 figure 7; (7) 16 trains, figure 1. 



Two classes of trains terminatmg in cycles of periods 9 and 6 respectively: (8) 1 train, 

 figure 191; (9) 3 trains, figure 190. 



Group for the system V4P2. — The sets of transitive elements are A; B; C; a, Oj O3; &, &2 ^3/ 

 c, C2 c^; (Z, (Z, d^; these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s= (A) (B) (C) («! 0.3 a.3) (6, 62 63) (c, fj C3) ((?, (?2 J3) 

 and is of order 3. 



Trains for the vSystem V47I. 



Thirteen classes of trains terminating in triads of the system: (1) 3 trains, figure 107 

 (2) 3 trains, figure 42; (3) 3 trains, figure 100; (4) 3 trains, figure 8; (5) 3 trains, figure 36 

 (6) 3 trains, figure 41; (7) 3 trains, figure 78; (8) 3 trains, figure 150; (9) 3 trains, figure 97 

 (10) 3 trains, figure 117; (11) 1 train, figure 95; (12) 3 trains, figure 63; (13) 1 train, figure 1 



Group for the system, V-j-yl . — The sets of transitive elements are A; B; C; a, a, a^; 6, b^ is,' 

 Ci C2 C3; di d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s={A) (B) (0) (a, 03 a,) (b, K b,) (c. c, c,) (d, d, d,) 

 and is of order 3. 



Trains for the Systems V472. 



Ten classes of trains terminating in triads of the system : (1 ) 3 trams, figure 161 ; (2)3 trains, 

 figure 71; (3) 3 trains, figure 8; (4) 3 trains, figure 3; (5) 3 trains, figure 68; (6) 3 trains, fig- 

 ure 168; (7) 3 trains, figure 158; (8) 3 trains, figure 155; (9) 1 train, figure 140; (10) 10 trains, 

 figure 1. 



Group for the system V4y2. — The sets of transitive elements are A; B; C; a, aj (^3! ^1 ^2 ^3/ 

 c, Cj C2! ^1 <^2 ^3' these with the trains separate the system into 13 nonpermutable subdivisions. 

 The group is generated by 



s={A) (5) (6*) (a, a^ a^) (6, b^ b^) (c^ c^ c^) (d^ d^ d^) 

 and is of order 3. 



