OF THE COMPOSITIONS OF NUMBERS. 
901 
I have established that that portion of the expansion of this fraction, which is a 
function of products of powers of s^x^, s.^x^ only, is represented by the fraction 
1_ 
N 
where 
N = 1 — a-^s-yX-^ — 
+ ! I hh^\^2 + i I + I ^3^3 I hh^2^^ 
the notation being that in use in the Theory of Determinants. 
The coefficients of N are the several co-axial minors of the determinant of the 
matrix defining X^, X^, and X 3 , viz.:— 
Ci/Y 
&i 63 63 
Cl Co Co 
The result is immediately deducible from the identity 
^1 (s^\^ Xj) 
1 - ^ 1 X 1 
- 1 , 
1 - %X3 ’ 
1 - 63 X 3 ’ 
1 - ’ 
So (^->^2 Xini 
1 - 5 . 1 X, 
- 1 , 
CoS-^X.. 
1 - S 3 X 3 ’ 
1^1 
1 - .^ 1 X 1 
i-jS^Xo 
1 — SjXo 
^3 (C 3 X ’3 X 3 ) 
1 *' 3 X 3 
X 
• -^iX^, 
0 , 
0 , 
0 , 
1 - 53 X 3 , 
0 , 
0 
0 
1 '^ 3 X 3 
a^s-^x-^ — 1 , 
h-^s^x^, 
C-^S^o, 
Ci/:)S'yX/-^ y 
^2^ 2^ 2 1 5 
^2'®3^3j 
^^3^ 1^3 
hoSoXo 
t 
the determinant last written being, with changed sign, the value of N. 
The theorem for the case of n variables x-^, Xo, . . . x^ will be completely manifest 
from the above. 
It appears to be one of considerable importance with regard to the generating 
functions which present themselves in this domain of the Theory of Numbers. 
The results of its further investigation I hope to bring before the Royal Society in 
the near future.—Added August 25 , 1893 , P. A. M.] 
