. GENERATIN-Q eunctiors in the theory op numbers. 
159 
Art. 65. Hence it has been established that the coefficient of the term 
X ^32 X *''/P rv* ^I i-z y» |‘i 
• 2 \ .32 * • • H/o ... H ;2 5 
in the product, enumerates the permutations of the 'ti quantities in 
'V* 'Y* S 2 
J 
which possess exactly 
% contacts XoX-^, 
O 'y* /V* 
O 32 ,, .^ 3 .^ 2 ’ 
and since the redundant product can assume the appearance derived from the matrix 
( 1 ^21 • • • ) 
1 1 . ■ . K 2 
1 1 ... 1 
we find that the enumeration is identical with that of the permutations which are 
such that the quantity occurs times in places originally occupied by the 
quantity x^, when q > p, and, as before, we take the coefficient of 
X; 
32 
^(/}> /yt 
qj) 
^1 /7* ^2 
«^2 
X 
Hence, an arithmetical correspondence, and, also, the fact tliat the true generating 
function for the enumeration of these permutations is 
_1_ 
1 - - S - 1) xx^ - 2 - 1) - 1) xx^x^ 
... — (Xo]^ 1 ) (X 32 1) • • • 1) . . . x,i_-^ x,i 
The above example is only a solitary one of a large number that might be furnished. 
An advantageous method for procedure appears to be to take some simple interpret¬ 
able redundant product, and to then pass through the condensed form to the general 
redundant product, involving n — 1 undetermined quantities as well as quantities c^y, 
which admit of a choice of values. The assignment of these quantities then leads to 
