No. 2.] 



TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 



71 



In further illustration of the productiveness of this method in generating new systems 

 from those already known, we exhibit the results of its application to a groupless system 27 

 previously obtained by this same method. The 35 triads of the system arranged in classes 

 according to their thi'ee-column indices are shown in the following table : 



The analysis of this system by two-column indices reveals the existence of 1; 24; 15; 65 

 pairs of columns with the indices 2'; 2, 4; 3^; 6, respectively. The group for this system is the 

 identity and, therefore, it is possible that the appUcation of all the quadrangular and of all the 

 he.xagoual transformations would yield 39 noncongruent systems. Since however a quadran- 

 gular transformation may lead back to a headed system, already completely determined, we 

 apply only the 15 hexagonal transformations. These generate the following 15 systems: 30, 33, 

 21, 18, Yl2,a, 28, 29, III,, 31, 9, 32, 17, 24, 17, V472; 2 of these are congruent, 3 are systems 

 with groups already determined in Part 1, while 11 are new groupless systems, which may be 

 shown to be noncongruent by a comparison either of their indices or of their distinctive sets 

 of trains. 



The application of this method to a number of the systems given in Part 1 yielded the 33 

 noncongruent groupless systems tabulated below. 



We make use of the notation (ai) c d e f to denote a quadrangular transformation which 

 occurs in the pair of columns a, b and which involves the four elements c, d, e, f. Similarly (ab) 

 c d e f g 1 is & hexagonal transformation in the columns a, b which involves the six elements 

 c,d,e,f,g,l. 



The S5'stem 1 may be derived from the system IB by the application of the quadrangular 

 transformation (a 2) b d 5 3, and for the sake of brevity we shall write this in the form 1 = IB, 

 (a 2) bd 53. 



The 33 new groupless systems are derived as follows: 



l = IB, (a 2) bd53; 



3 = IB, (b 8) c/5 2; 



5 = IC, (aS) C€3 5; 



7 = IC, (6 3) eSfif 6; 



9 = IIC, (c3)(Z/7 6 

 11 = lie, (64) (Ze86 

 13 = 7/F, {b2)de7 5 

 15 = IID, (c 3) eg 8 5 



2 = IB, 0)3) ci n; 



4 = 10, (a2) c€ 4 1; 



6 = IC, (a5)/!7l4; 



8=y7<?, (6 3)/^ 1 7; 

 10 = lie, {a 2) 5 6dg; 

 12 = 7/5, (c5) e/4 1; 

 14 = 77(7, (c4) df8 5; 

 16 = 77, I3, (03 dj) Ci 62 61 ffj p, e^. 



In the system 7, 2 we now replace the elements a, /3, y, 5, e, by/, g, 6, 7, 8, respectively. 



17 = 7,2, (/I) 62 57; 

 19 = 17, (6l)a65f2/8 

 21 = 17, (6c)a6e3 14 

 23 = 17, (68)a6/5<'7 

 25 = 17, (6e)8 5/d; 

 27=17, (23)6(77/4e; 

 29 = 27, (e3)6 2a{?58; 

 31 = 27, (2 4)6fil63 8; 

 33 = 27, (a/)665378. 



18=17, ((Z/) a2 4 7 1 3; 

 20=17, (c d) a i 7 b 5 2; 

 22 = 17, (6 5) ffl3 e 8/ 6; 

 24 = 17, (a6) c 4/1 7 8; 

 26 = 25, (c5) 63o4^/; 

 28 = 27, (c 5) o 4 /fir 3 6; 

 30 = 27, (a6) c4 1 /8 7; 

 32 = 27, {2 5) b d e 8 a 3; 



