484 ON THE TRIADIC ARRANGEMENTS OP SEVEN AND FIFTEEN THINGS. [82 
in question, to produce the whole system of the seventy-eight duads of the thirteen 
letters. Hence arranging the duads of the thirteen letters in the form 
ab . be . cd . de . ef .fg . gh . hi . ij .jk . kl . lin. ma 
ac .bd.ee . df. eg ,fh. gi . hj . ik .jl . km. la . mb 
ad .be .cf.dg. eh .fi . gj . hk . il ,jm . ka . lb . me 
ae .bf.cg. dh. ei .fj . gk .hi . im .ja . kb . Ic . md 
af .bg . eh . di . ej .fk . gl . hm . ia . jb . kc .Id . me 
ag .bh . ci . dj . ek .fl . gm . ha . ib . jc . kd ,le . mf 
and consequently the duads of each set ought to be situated one duad in each line. 
Suppose the sets of duads are composed of the letters a, b, c, d, e, f, g, h, i, j, k, l, 
it does not appear that there is any set of six duads composed of these letters, and 
situated one duad in each line, other than the single set al, bk, cj, di, eh, fg; and 
the same being the case for any twelve letters out of the thirteen, the derivation of 
the thirteen systems of thirty-five triads by means of the cyclical permutations of 
thirteen letters is impossible. And there does not seem to be any obvious rule for 
the derivation of the thirteen systems from any one of them, or any prima facie 
reason for believing that the thirteen systems do really exist, it having been already 
shown that such systems do not exist in the case of seven things.