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



Two classes of trains te.rminatmg in cycles of periods 10 and 4, respectively: (16) Two 

 trains, figure 192; (17) 1 train, figure 1S2. 



Group for the System VI2it. — The sets of transitive elements are^; B; C; a^ 6,; a^ \; a^ 63 

 a^ bj Oj Sj/ Oo &«. These with the trams separate the system into 21 nonpermutable subdi- 

 visions. The group is generated by 



s={A) (B) (C) (a, 6.) (a, h,) (a, b,) (a, b,) (a, b,) (a, b,), 



and is of order 2. 



Trains for the System VlSja. 



Seven classes of trains terminating in triads of the system: (1) Two trains, figure 143; 

 (2) 2 trains, figure 110; (3) 4 trains, figure 2; (4) 4 trains, figure 61; (5) 4 trains, figure 91; (6) 

 4 trains, figure 136; (7) 15 trains, figure 1. 



Two classes of trains termmating in cycles of periods 18 and 20, respectively: (8) Two 

 trains, figure 193; (9) 1 train, figure 201. 



Group for the System VlS^a. — The sets of transitive elements are A; B C; a^ b^ a^ h^; a^ 63 

 ^e ^6/ c-i h C"o ^5- 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^ b^ a^ 63) K "s ^4 ^5)) 



and is of order 4. 



Trains for the System VI337. 



Six classes of trains terminating in triads of the system: (1) Four trains, figure 177; (2) 2 trains, 

 figure 79; (3) 4 trains, figure 173; (4) 6 trains, figure 27; (5) 2 trains, figure 92; (6) 17 trains, 

 figure 1. 



One class of trains terminating in a cycle of period 4: (7) One train, figm-e 182. 



Group for the System VlS^y. — The sets of transitive elements are A; B C; a^ &i a, \; a^ 63 a^ b^; 

 «4 ^4 «5 ^5> these with the trains separate the system into 1 1 nonpermutable subdivisions. The 

 group is generated by 



s = {A) (B C) (a, Oj 6, b^) (a^ b^ b^ a^) (a^ a^ 6, 65), 

 and is of order 4. 



Trains for the System VISjS. 



Ten classes of trains terminating in triads of the system: (1) Two trains, figure 58; (2) 4 trains, 

 figure 87; (3) 4 trains, figure 141; (4) 4 trains, figure 76; (5) 4 trains, figm-e 151; (6) 4 trains, 

 figure 99; (7) 4 trains, figure 34; (8) 2 trains, figure 81; (9) 6 trains, figure 27; (10) 1 train, 

 figure 1. 



One class of trains terminating in cycle of period 4: (11) One train, figm-e 182. 



Group for the System VlS^h. — The sets of transitive elements arc A; B C; a, J, a^ b^; a^ b^ a^ b^; 

 «4 ^4 «5 ^5/ these with the trains separate the system into 1 1 nonpermutable subdivisions. The 

 group is generated by 



s=(A) (B C) (a, a^ &, b^) (a^ b^ 63 aj (a^ a^ 64 65), 



and is of order 4. 



The trams show that 20 of the 71 systems obtained in Part 1 are new systems and the 

 remaining 51 systems are each congruent to some one of the 44 systems thus far derived. The 

 substitution which transforms each of these 51 systems into its congruent system is given below 



T -.—TrTTi. —/'a b c d e 1 2345a/373 A 



I,l=Vnby s^(4 7 b e I a d g 3 6 f 2 5 S c) 



I, 2 anew syetem. 



T— TTTA 1, _/a b c d e 1 2 i -i n a ft y 5 e\ 



I^IIAby s=(^ g 5 8 e c 6 I 2 7 b -l d / ^) 



