250 
NOTE ON MR Muir’s SOLUTION OE A “ PROBLEM OF ARRANGEMENT.” [670 
where u' denotes ^, we have 
iC 3 
n — u 
(1—a?) 2 1—sc 
1 
1 — X‘ 
+ (u 3 + u 4 x + u 5 x 2 + ...) (2a? + 6a? 2 + 12a? 3 + 18a? 4 + ...) 
+ u (2a? + 6a? 2 + 12a? 3 + 18a? 4 +...); 
or, what is the same thing, 
u — u 
, + u 
2a? 
2a? 4 
that is, 
(1 — a?) 2 — 1 — a? 2 |(1 — a?) 3 (1 - x) 3 (1+ a?)j ’ 
2a? 4 
a? 2 , 1 
u — u = 
(1 — a?) 3 ~ (1 — a?) 3 (1 + a?)) (1 - a;) 2 1 - a? 2 * 
This equation may be simplified: write 
u = 
then 
and the equation is 
1 — a? 2 ^ 2 1+a? 
1 - a: 2 
a? 4 
'4 2 
a? 4 a? 2 > 
1 — a? 2 , 
4 1 2 
+ 
a; 4 a? (1 + a?) 2 (1 - a;) 2 a? (1 - a?) 2 a? (1 - a?) 2 ) ^ (1+a?) a? 2 1 
Q + n 
1+a? 
Q' = 
,.2 ’ 
that is, 
2 2,2 
+ «TTi ~v» + 
2 1 q+^ l± ^«'= 1 
a? 4 + a? 2 a? (1 - a?) 2 a? 2 (1 - a?) 2 a? (1 - a?) 2 (1 - a?) 2 ) ^ (l-a?)a? 2 ^ 1-a? 2 ’ 
viz. this is 
L(l^ + (J^_2 + 2 + 2_ 2 j 1-^ 
) a; 4 a? 2 a, 13 a? 2 a? j a? 2 1+a? 
that is, 
I 
or 
or finally, 
giving 
and thence 
1 — a? 2 n 1 — a? 
1 Tx’ 
(1 ~ ^ Q , 1 ~ Q' = 1 . 
a^ a? ^ 1 + a?’ 
a; 2 ) ^ — (1 + a?) 2 ’ 
= A* + Dj-*.. * x+1 * 
Q = e 
(a? + l) 2 
cia?, 
_ e + + i) f «* + + s) 
(a? + l) 2 
which is the value of the generating function 
u = u 3 + u 4 x + u 5 x 2 + &c. 
dx,