390 
Proceedings of the Royal Society 
and thence 
u = 
s 2 " 1 --(■*+£) /* ^ 
a* J(x+ 1) 2 
dx , 
which is the value of the generating function 
u — u 3 + u 4 x + u b x 2 + &c. 
But for the purpose of calculation it is best to integrate by a 
series the differential equation for Q : assume 
Q= - - q 4 x b - q b x 6 - . . . 
then we find 
q± = 4 ? 3 - 2 > 
q b = 5g 4 +q 3 +3, 
? 6 =6 25 + ?4 " 4 > 
q 7 = 7q Q +q b + 5, 
Qn = W «- 1 +^»-a + ( -D W_1 ( w ~ 2 ) • 
We have thus for g 3 , g 4 , q 5 . . . the values 1, 2, 14, 82, 593, 4820, 
and thence 
u = (1 — a? 2 )(l + 2x+ 14a? 2 + 82a? 3 + 593a; 4 + 4820a? 5 + . . .) , 
viz., writing 
1 2 14 82 593 4820... 
-1-2-14 -82 
the values of u 3 , u 4 . . . are 1 , 2 , 13 , 80 , 579 , 4738 .. . 
agreeing with the results found above. 
In the more simple problem, where the arrangements of the n 
things are such that no one of them occupies its original place, if 
u n he the number of arrangements, we have 
u 2 = 1 =1 
u 2 = 2u 2 , = 2 
u 4 = 3 (u 3 + u 2 ) , = 9 
u b = 4 K + %) > = 44 
1 n(yi n + u n _i) , 
