440 
Proceedings of Boy cd Society of .Edinburgh. [july 18, 
consists of the symmetric function %a r ~ 3 b q ~ 2 and the like functions 
succeeding it , the coefficient in every case being 1 (hi.) 
For example, 
a ( 0 1 3 7)-a(0 2 3 6)+a(l 2 3 5) 
= %a^b + 'ZPb 2 + %as>bc + %a 2 b 2 c + %a?bcd , 
or, as we may for shortness’ sake write, 
= [3a 4 & + ] . 
Seeking first for the coefficient of 2,a a bP we find that it is deter- 
mined for a(0 1 q r) by considering 
fO, 1,0 + 2 
JO, 2,0 + 1 
the cluad 0, 1 along with the triads -<J 0, 3, (3 
i 
lo, q-1, fi-q+i; 
that it is determined for a( 0, 2, g, r— 1) by considering 
(0, 2,/J + l 
i 0, 3, /3 
the duad 0, 1 along with the triads -[ 
[0, q-l, (3- q + 4: 
fl 2 0 
. I 1 3 (3-1 
and the triads 
[12-1 Z^-2 + 3; 
and that it is determined for a(l, 2, q, r- 2) by considering 
fl 2/5 
I 1 3 (3-1 
the duad 0, 1 along with the triads -J 
[ 1 q-1 (3-q + 3. 
Flow if the number of dovetailings in the first case be x, and in 
the last case y, it is clear that in the second case they must be 
( x — Y) + y: hence the coefficient of every term of the form in 
a(0, 1, q , r) - a(0, 2, q, r - 1) + a(l, 2, q, r - 2) is 1 . 
Next, the coefficient of %a a b^(^ is determined for a(0, 1, q, r) by 
considering 
