154 MAJOR P. A. MACMAHON ON A CERTAm CLASS OE 
and herein we see that the coefficient of 
X ®'lX ®J)X fly fi^' fa 
''•21 ''•31 ^33 •'•I *^3 *^-3 
represents the number of permutations of the symbols in 
which possess exactlj^ 
-y fl y fa y fa 
^1 .-tg , 
S.21, x.2X^ contacts 
53 J, a’s^’i 
■%2> x^x .2 ,, 
Here is an entirely new interpretation and we see that the true generating function 
for the enumeration of the indicated permutations is 
__ 1 _ 
1 !X'2 ^'2 ^'3 (^31 ^ (^31 1) '^4^3 (^33 1^ ^3'^3 (^31 1) (^33 — 1) '^r^3'^'3 
a result which does not lie by any means on the surface. 
The arithmetical correspondence should also be noted. 
Art. 62. For the order n we have the matrix 
*13 
‘-13 
13 
^11 
n 
/3i«Ahi 
^12^1 
b3 
bi 
1 
'^23^33 
/^24^43 
1 
1 
'■'‘ 34^43 • 
/3gnX,2‘i 
— 
. 
«33 
1 
1 
«34 
^•in 
Cgtt 
P-ln 
/ 3 sii 
^in 
1 
and we obtain another form by writing for 
Moreover (?/ > x) we have 
Cyx - 1 j 
-1^ 
■y- 1: 1/ - U 
. . X 
■X + 1, a: 5 
where 
