124 
MAJOR P. A. MACMAHOX OX A CERTAIX CLASS OF 
1 
[1 — rt;, ... {x„ ...) + {x.^ + x- ...) x^ {x^ + ... + 3 c„_^a;„] 
which may be written 
1 
a 771 
1 P ^ 
Art. 19. Again, if it be necessary to enumerate the permutations of 
f2 
a 2 
X 
in which occurs times in the compartment A^, 
/3i 
5> 
5) 
7i >. 
) > 
A 
A 
25 
35 
5> 
we are led to the true generating function 
_1_ 
1 — — x^ — x^ — ... — Xn + (ff;^ — &i) x^x^ + (rti — x^v^ + ,.. + (rt^ — «j) :i.\x,i ’ 
in which we have to seek the coefficient of 
rc{"^h/'c{''. . . . . . xj\ 
Art. 20. Again consider the general problem of “ Derangements in the Theory of 
Permutations.” 
In regard to the permutations of 
W* 'Y* ^'2 xyt ^II 
.4^ cto ... 
it is necessary to determine the number of permutations such that exactly 9n of the 
symbols are in the places they originally occupied. 
We have the particular redundant product 
(ax^ + cco + . . . + x„y^ (Xj + oXjj + .. . + x„y-^ . .. (ajj + + . . . + 
in which the number sought is the coefficient of 
. . . xj\ 
The true generating function (i.e., condensed form) is derived from the coaxial 
minors of the detei’ininant of order oi :— 
