339 
of Edinburgh, Session 1876-77. 
regard to the second condition ; and so on : 2(1) means (1) + (2) 
... +(r): 2(12) means (12) + (13) + (2 r) ... + (r - 1, r) ; and so 
on, up to (12 .... r), which denotes the number failing in regard 
to each of the r conditions. 
Thus, in the special problem, the first condition is that the 
letter in the first place shall not be a; the second condition is 
that the letter in the second place shall not be b ; and so on ; 
and taking r = n , we have the known result, No. = 
_ 71 , „ s n.n — 
Tin — jll(?2 - 1 ) + — y~2 — ^ (w - 2 ). + . . =t= 
01 . n — 1 .. 2.1 
“iTF.TTtT' ’ 
= 1.2.3... » I 1-- + A- - ± — I — ] 
1 1 1.2 1.2.3 1. 2. 3. . .» J 
giving for the several cases 
w = 2,3,4, 5, 6, 7, . . 
No. = 1,2,9,44, 265, 1851 . . 
I proceed to consider the following problem, suggested to me by 
Professor Tait, in connexion with his theory of knots : to find the 
number of the arrangements of n letters a b c . . j h , when the 
letter in the first place is not a or b , the letter in the second place 
not b or c, . . . the letter in the last place not h or a. 
Numbering the conditions 1,2,3 . . n , according to the places 
to which they relate, a single condition is called [1] ; two condi- 
tions are called [2] or [1 , 1], according as the numbers are conse- 
cutive or non-consecutive : three conditions are called [3], [2 , 1] or 
[1,1,1], according as the numbers are all three consecutive, two 
consecutive and one not consecutive, or all non-consecutive; and 
so on : the numbers which refer to the conditions being always 
written in their natural order, and it being understood that they 
follow each other cyclically, so that 1 is consecutive to on. Thus, 
n = 6, the set 126 of conditions is [3], as consisting of 3 consecu- 
tive conditions ; and similarly 1346 is [2,2]. 
Consider a single condition [1], say this is 1 ; the arrangements 
which fail in regard to this condition are those which contain in 
the first place a or b ; whichever it be, the other n- 1 letters may 
be arranged in any order whatever; and there are thus 2 II (n— 1) 
failing arrangements. 
