[ HI ] 
IV. A Certain Class of Generating Functions in the Theory of Numbers. 
By Major P. A. MacMahon, R.A., F.R.S. 
Received November 3.—Read November 24, 1893. 
Intkoductoey Abstract. 
The present investigation arose from my “ Memoir on the Compositions of 
Numbers,” recently read before the Royal Society and now in course of publication 
in the ‘ Philosophical Transactions.’ The main theorem may be stated as follows :— 
If X„ X3,. . ., X„ be 
matricular relation 
linear functions of 
quantities x.^, . . 
. ., given by the 
(Xi, Xo, . ... 
• x„) = ( 
^13 
- . 
(Xi, .Xj, . . . 
. 
a.22 
• • Chn 
Cb'2 
n'i 
that portion of the algebraic fraction 
_ 1 _ 
(1 — sjKf (1 — S3X3)-(1 — s„X«) 
which is a function of the products 
only, is 1/V„, where (putting = 1) 
cqi 
l/.r^, cqo, 
■ ^\n 
V„ = (—.. 
0 0^ 1 j 
- 
efjnn 
The proof of this theorem rests upon an identity which, for order 3 , is 
7.5.94 
