52 On the Triadic Arrangements of Seven and Fifteen Tilings. 



as £/3S, ty= 9 and rfis 9 >jy&) may easily be found ; the system to 

 be presently given of the triads of fifteen things would answer 

 the purpose. And so would many other systems. 



Dropping the consideration of the order of the letters which 

 form a triad, I pass to the case of a system of fifteen letters, 

 a, b, c, d, e,f g > h, i,j, k, I, m 9 n 9 o. It is possible in this 

 case, not only to form systems of thirty-five triads containing 

 every possible duad, but this can be done in such manner that 

 the system of thirty-five triads can be arranged in seven systems 

 of five triads, each of these systems containing the fifteen let- 

 ters*. My solution is obtained by a process of derivation 

 from the arrangements ab. cf.dg .eh and ab.cd.ef.gh as 

 follows; viz. the triads are 



tab jac kaf lad mag nae oak 

 icf jfb kbc Ice mch ncd ocg 

 idg jde Jcdh Igb mbd ngf ofd 

 ieh jhg kge Ihf mfe nhb obe 



and a system formed with i 9 j 9 k, l 9 m, n, o 9 which are then 



arranged — 



Mo ino jmo Urn jln ijk kmn 



iab jac lad nae leaf mag oak 



ncd mdb kbc ocg mch Ice icf 



mef keg ieh jfb obe ofd jde 



jgh Ihf nfg khd idg nhb Ibg 



an arrangement, which, it may be remarked, contains eight 

 different systems (such as have been considered in the former 

 part of this paper) of seven letters ; viz. of the letters ?", j 9 k 9 

 I, m, n 9 o; and of seven other sevens, such as i 9 j 9 k, a 9 b, c,f 

 The theory of the arrangement seems to be worth further in- 

 vestigation. 



Assuming that the four hundred and fifty-five triads of 

 fifteen things can be arranged in thirteen systems of thirty-five 

 triads, each system of thirty-five triads containing every pos- 

 sible duad, it seems natural to inquire whether the thirteen 

 systems can be obtained from any one of them by cyclical per- 

 mutations of thirteen letters. This is, I think, impossible. 

 For let the cyclical permutation be of the letters a, b, c 9 d 9 e,f 9 



* The problem was proposed by Mr. Kirkman, and has, to my knowledge, 

 excited some attention in the form " To make a school of fifteen young 

 ladies walk together in threes every day for a week so that each two may 

 walk together." It will be seen from the text that I am uncertain as to 

 the existence of a solution to the further problem suggested by Mr. 

 Sylvester, " to make the school walk every week in the quarter so that 

 each three may walk together," 



