of Edinburgh, Session 1876 - 77 . 
341 
Those for the preceding cases, n- 3, 4, 5, respectively are 
No. = 6-12+ 9- 2 
No. = 24- 48+ 40- 16 + 2 
No. = 120 - 240 + 210 - 100 + 25 -2 
1 
2 
13 . 
We have in general [1 J = n , [2] = n , [1, 1] = \n(n - 3) ; and in the 
several columns of the formulas the sums of the numbers thus 
represented are equal to the coefficients of (1 + 1) 2 , thus, n = 6 as 
above, the sums are 6, 15, 20, 15, 6, 1. As regards the calculation 
of the numbers in question, any symbol [a, /?, y] is a sum of sym- 
bols [a -a +/3- jS' + y-y'. .], where a' + /3' + y'. . is any partition 
of w — (a + /? + y . .) ; read, of the series of numbers 1, 2, 3 ... ti, 
taken in cyclical order beginning with any number, retain a, omit 
a', retain J3, omit /3 ', retain y, omit y', . . Thus in particular, 
n = 6, [1, 1] is a sum of symbols [1 - 3 + 1 - 1] and [1 — 2 + 1 — 2] ; 
it is clear that any such symbol [a - d + j3 - /3'. .] is —n or a sub- 
multiple of n (in particular if n be prime, the symbol is always 
= n): and we thus in every case obtain the value of [a, /?, y . .] 
by taking for the negative numbers the several partitions of 
n — (a + /3 + y . .) and for each symbol [a - a +/? - J3' + y - y'. .], 
writing its value, ~n or a given submultiple of n, as just men- 
tioned. There would, I think, be no use in pursuing the matter 
further, by seeking to obtain an analytical expression for the 
symbols [a, /?, y . .] . 
For the actual formation of the required arrangements, it is of 
course easy, when all the arrangements are written down, to strike 
out those which do not satisfy the prescribed conditions, and 
so obtain the system in question. Or introducing the notion of 
substitutions,* and accordingly considering each arrangement as de- 
rived by a substitution from the primitive arrangement abccl . . . jk, 
we can write down the substitutions which give the system of 
arrangements in which no letter occupies its original place : viz., 
we must for this purpose partition the n letters into parts, no part 
less than 2, and then in each set taking one letter (say the first 
in alphabetical order) as fixed, permute in every possible way the 
* In explanation of the notation of substitutions, observe that (abode) means 
that a is to be changed into b, b into c, c into d, d into e, e into a ; and simi- 
larly ( ab)(cde ) means that a is to be changed into b , b into a , c into d , d into e, 
e into c. 
