156 
l^rAJOR P. A. MACMAHOX OX A CERTAIN CLASS OF 
Art. 63. In this product we may interpret the coefficient of 
/y» f]/y* ^2 ^ 
■ v^.2 • • * • 
From the nature of the condensed form we know that this coefficient is an integral 
O 
function of the quantities X.,^. We may prove that if a portion of the expansion be 
n\ % X ^ 
''■32 • * • ^2 • ■ ’ 5 
the number c indicates the number of permutations of the quantities in 
/y» -y* ^2 /y» 
A'l *^2 • • • 5 
which possess exactly 53 ^ contacts 
•S 32 ,, 
* J? 
^qp j) 
Idegard the above product, as written, as being a square form of n rows and n 
columns involving iv elements. 
Observe that if s ^ the element common to the row and column is 
‘^st 
while the element common to the d'’ row and 6 '*'' column is 
l^si^sh 
and that the product of these two elements is 
/y» /y* 
/y* >yi 
/y» /y* 
tAy qxAJjp 
Now, take a particular permutation of the quantities and observe how it may 
be considered to arise in the multiplication. Let a portion of the permutation be 
rv* \ /yi ryt J />>//> ^/Y>/y» 'V \ 'V 
Jb2 \ I 1 *^5 I • 
divided off by bars into compartments in such wise that in any compartment the 
suffixes are in descending order. 
