No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 13 



The preceding system, however. No. II, 2,, is found hy quite obvious indications to be 

 equivalent to System II, 1 , and the one is transformed into the other by the folk)wing substitution: 

 II, li: Giflja, 61^362 C1C2C3 dicZjtij 61^263. 

 II, 2i: bib-fi^ (tid^a^ <^i'h''i <'i<'2<^3 di^'i^i- 

 Further, the two systems II, I3, II, I4 are equivalent by the interchange of letters 

 (ffj «,) (6162) (Cs C2) W3 ^:) (^1^3)- In conclusion, therefore, this class II contains not more 

 than seven triad systems that are essentially distintit, Nos. 1,, I2, I3, I5, Is, I7, 23. 



§5. TRIAD SYSTEMS INVARIANT UNDER AN OPERATION OF TYPE (1)(7)(7). 



There are three distinct systems, and no more, wliich admit a substitution of period 7. 

 The proof is almost intuitional, no long analysis being needed. Indicate the 1+7 + 7 letters, 

 and the substitution, thus: 



S={A) [a h c d efg) (12 3 4 5 6 7). 



By diophantine equations of condition it is found that there must be seven triads — a system 

 A7, constituted within the one cycle, e. g., the cycle of letters {abed efg); seven triads like 

 Aal, and 21 triads composed of one letter and two digits. 



Take S to be such a power of the cyclic operation that the included A? consists of the triad 

 ahd and its six conjugates. Of the 21 now remaining to be determined, thi-ee must contain the 

 letter a. Indicate on a circle in their order of sequence the seven digits at equal intervals. Since 

 a and 1 are already together hi the triad Aal, we have now to connect in pairs, by tlirce chords, 

 all six digits 2, 3, . . ., 7. No two of these chords can be of equal length, since then the rotation 

 effected by the substitution S would produce a repetition of a pair, contrary to the definition of 

 a triad system. Trial shows at once that chords 23 and 25 would lead necessarily to equality 

 of at least two chords, hence these are excluded; while 2i, 26, and 27 lead to one solution each, 

 as represented in figure 1. 



Fig. 1. 



Accordingly the three possible systems are given by the following, with the triads conjugate to 

 them imdcr the operation S. 



System III, 1: Aal, abd, 024, a37, a56. 



System ni, 2: ylal, abd, a26, a34, a57. 



System III, 3: Aal, abd, a27, a36, a45. 

 The fu'st of these is evidently the one ordinarily constructed from the A^ by the method 

 for passing from n to 2n+l; viz. by substitution of coiTesponding elements from the second 

 cycle in triads of the first, there are formed from abd, for example, three others: a24, Ibi, I2d. 

 The 21 of this structure, the original 7, and the 7 like Aal make up the complete set. It is, 

 prima facie, the Kirkman system (No. III^ of Miss Ciuumings's dissertation). 



§6. NO TRIAD SYSTEM CAN EXIST THAT ADMITS A SUBSTITUTION OF TYPE (1)(2)'. 



If a triad system can be invariant under an operation of the type (1) (2)', denote the element 

 in the unit cycle by A, and in each duad cycle denote one element by a letter, the other by a 

 digit: 



S=U) (a 1) (6 2) (c 3) (d 4) (e 5) (/6) (g 7). 



