12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. pol. xiv. 



Of these five only the first and the fifth can be completed to full systems. They give each a 

 unique system. 



System II, I5: afi^a^, bfij)^; System II, l^: afi^a^, hfi^b^; 



'^I^2'"3> ''l''2'^3) ^1^2^3' ^1^2^31 ^fli^Z) ^l^i^it 



(^i^'i&i, diCfls, diC^a^; dyC^a^, d^e^l^, d^e^a^; 



CidJ>3, c^ej)^. CjdJ>2, Cicfis. 



The second of the two possible schedules diverges from the first in its fifth triad and is the 

 following, with seven triads void of subscripts: 



«ia2«3. bfi^h^ ^iCjCj, c,<Z2(?3, diC^e^; 

 6jC,(Zi; adb, adb, ode; 

 ceb, cch, cea; abe. 



On trial only two ways are found for affixing subscripts to the triads still left blank, and these 

 differ only by the interchange of 2 and 3. Hence we have finally in this family Ic only the 

 one additional system: 



System II, 1,: Oia^a^, bfij)^, a^CiCs, c^d^d^, die.j^, b^c^di 

 a^dfi^, a^djji, aid^e^; 



^1^1^3J ^l^2^2J ^1^3^!^ 



Division 2. — In the second principal division of this class there are five triads formed from 

 single cycles of thi"ee letters, and these are necessarily 



a,a2ffl3. ^i&2&3) C1C2C3, d^d^d^, c^e^c^. 



The ten sets of three triads which are to complete the system contain, as we saw, each three 

 (UfTcrent letters; and no combination of three letters can be repeated, as is shown readily by 

 writing the diophantme equations of condition. There are only 10 combinations possible, so 

 that each wiU occur once, i. e., in one set of thi"ee triads. If we attend first to the triads con- 

 taining a, remembering that in each of the five cycles the subscripts ma}" be permuted cychcally, 

 it is found that there are but two distinct kinds of sets. To characterize them, we may best 

 use Cole's term, interlacing; either there is an interlacing in the triads containing sj^mbols a 

 or there is none. In the first case the four symbols concerned may be given index or subscript 

 1, b^c^di^e^, and the four triads may be: 



afiiCi, a^d^e^, flj^i'^i «2fifi- 



In the second case we find, Iiy permutations of letters and of the five cycles independently, 

 that we may denote four triads by 



afiiC^, aidifi, ajb^d^, a^CjC^. 



Each of these is completed, in two equivalent ways, to a full sj^stem. 



System II, 2,: Oi?>,c,, a^d^e^, a^bid^, a2C,e,, afi^ei, a^c^d^, 



biCJs, byC^e^, hid^e^, c^d^e^. 

 System II, Zj: afi^c^, ald^e^, aji^di, a^sC^, a^c^e..^, a^c^di, 

 he J 3, 6,C3fi, h^d.es, c^d^e^. 

 This last system is found to have no interfacings whatever, and so is evidently the exceptional 

 system in Cole's enumeration, the headless cyclical system of Heffter. 



