112 
MAJOK P. A. MACMAHON OX A CEPTAIX CLASS OF 
c^jo'Syri, 
«io. 9 iXj, 
,-)^S 
— 1, WooVb. 
(if 0*1 SoOCo . 
ol 0 0? 
Oon.S'g.Tg, 
a^oS.^Xo 1, 
1 -^jX,. 
0, 
0 , 
0, 1. 
— SnXo, 
0 , 
0, 
0, 
^1 X^) 
1 
1 - sXh 
~ b 
1 - .,X, ’ 
1 - 8,Xd ’ 
fS'o 
1 - S.X3 ’ 
1 - 83X3 “ 
1 - S3X0 ’ 
1 - .3X, ’ 
1 - SgX.s ’ 
1 . - .8..X 
0 
and is very easily established. 
An instantaneous deduction of the general theorem is the result that the generating 
function for the coefficients of . . . rr/" in the product 
... X/' 
is 
1 /V„. 
Tlie expression V,; involves the several coaxial minors of the determinant of the 
linear functions. Thus 
Vo 1 — — Ogo.a’o + 1 I •'^r'*"2 + ! ''bVtBS | •'^r'^3 “ ! I •'^' 2^3 
*** I ^^2 f I QC-yiL:)0C^, 
The theorem is of considerable arithmetical importance and is also of interest in 
the algebraical theories of determinants and matrices. 
The product 
Xi^X/^ ... X/', 
often aj^pears in arithmetic as a redundant form of generating function. The theorem 
above supplies a condensed or exact form of generating function. 
Ex. gr. It is clear that the number of permutations of the symbols in the 
product 
7 » Sl^ 7 » 
2 fi 
which are such that ev'ery symbol is displaced, is obviously the coefficient of 
in the product 
(Xg + . . 
^2 /v» 
.C o ... 
+ (Xj + ■'«3 + • • • + • • - (•'■^1 + •'^■2 + • • • + , 
