188 Proceedings of the Royal Society 
first or second place , ihe third not in the second or third place , and 
so on. 
It being understood tbat U M denotes the said number of arrange- 
ments, the concluding lines of the paper referred to will show to 
what extent the problem was solved. 
“ Hence for the determination of TJ n when U 3l „ 2 , . . . . are 
known we have 
U w = (» - 2)U W _ 1 + (2 n - 4)U W _ 2 + (3n - 6)U M _ 3 \ 
+ (4 n — 10)TJjj_4 4- (5n - 14)TJ 3l _ 5 1 
+ (6w- 20)U w _ 6 + (7w - 26)U w _ 7 l (1) 
where the coefficients proceed for two terms with the common 
difference n - 2, for the next two terms with the common difference 
n - 4, for the next two terms with the common difference n - 6, and 
so on. 
“ And as it is self-evident that U 2 = 0, we obtain 
U 3 = 1U 2 + 1 
= 1 
U 4 = 2U 3 
= 2 
U 5 = 3U 4 + 6U 3 + 1 
= 13 
U 6 = 4U 5 + 8U 4 + 12U 3 
= 80 
U 7 = 5U 6 + 10U 5 + 1 5U 4 + 18U 3 + 1 
= 579 
U 8 = 6U 7 + 12U g + 18U 5 + 22U 4 + 26U 3 
= 4738 
and so forth.” 
What is aimed at now is to reduce the above equation of differ- 
ences and thereafter to obtain the generating function of U. 
2. From (1) we have 
V,-» = (n - 4)U„_ 3 + (2m - 8)U n . 4 + (3 m - 12)TJ„_ 5 
+ (4b - 18)U„_ 6 + (5» - 24)U„_ 7 
+ 
l-(-l)” 
+ 2 
and therefore by subtraction 
- U w _ 2 = (n - 2)U„ -l- (2 n - 4)U„_ 2 
+ (2 n - 2)(U„_3 + TJ m _4 .... 
+ U 3 ) . . (2). 
