But for the purpose of calculation it is best to integrate by a series the differential 
equation for Q: assuming 
Q = - q#? - q^ - q 5 af - ..., 
we find 
We have thus for q 3 , q 4 , q- 0 , ... the values 1, 2, 14, 82, 593, 4820, ..., and thence 
u = (1 — a? 2 ) (1 + 2x + 14a? 2 + 82x 3 + 593X 4 + 4820a; 5 + ...), 
viz. writing 
the values of u 3 , u i} ... 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 be the number of arrangements, we 
have 
u 2 = 1 =1, 
u 3 — 2 u 2 = 2, 
w 4 =3 (u 3 + u 2 ) = 9, 
u B =4 (u 4 + u 3 ) = 44, 
and writing 
we find 
that is, 
or, what is the same thing, 
whence 
u n +i — ^ (^n 4" u n —i), 
U = U 2 + U& + U 4 X 2 + ..., 
u = 1 + (2a? + 3a? 2 ) u + (a? 2 + a? 3 ) u ; 
(_ i + 2x + 3a? 2 ) u + (a? 2 + a? 3 ) u' = - 1, 
,/31) 1 
U + \x~x*) U - a? 2 (l + a?) ’ 
But the calculation is most easily performed by means of the foregoing equation of 
differences, itself obtained from the differential equation written in the foregoing form, 
(- 1 + 2x + 3a? 2 ) u + (a? 2 + a? 3 ) u' = — 1. 
32—2