GENERATING FUNCTIONS IN THE THEORY OE NUMBERS. 
113 
and thence we easily pass to the true generating function 
_ 1 _ 
I — S.r'jiCg — 2 SA’jiTj.rg — 3 — ... — (?i — 1) . . . x^ 
In the paper many examples are given. 
Frequently the redundant and condensed generating functions are differently 
interpretable; we then obtain an arithmetical correspondence, two cases of which 
presented themselves in the “ Memoir on the Compositions of Numbers.” 
A more important method of obtaining arithmetical correspondences is developed 
in the researches which follow the statement and proof of the theorem. 
The general form of V„ is such that the equation 
= 0 
gives each quantity Xg as a homographic function of the remaining n — 1 c|uantities, 
and it is interesting to enquire whether, assuming the coefficients of V,, arbitrarily, 
it is possible to j)ass to a corresponding redundant generating function. 
I find that the coefficients of V„ must satisfy 
2 " — + 71 — 2 
conditions, and, assuming the satisfaction of these conditions, a redundant form can 
be constructed which involves 
71—1 
undetermined quantities. In fact, when a redundant form exists at all, it is 
necessarily of a (71 — 1 )-tuply infinite character. 
We are now able to pass from any particular redundant generating function to an 
equivalent generating function which involves n — 1 undetermined quantities. 
Assuming these quantities at pleasure, we obtain a, number of different algebraic 
products, each of which may have its own meaning in arithmetic, and thus the number 
of arithmetical correspondences obtainable is subject to no finite limit. 
This portion of the theory is given at length in the paper, with illustrative examples. 
Incidentally interesting results are obtained in the fields of special and general 
determinant theory. The special determinant, which presents itself for examination, 
provisionally termed “inversely symmetric,” is such that the constituents symmetrl- 
cally placed in respect to the princij^al axis have, each pair, a product unity, whilst 
the constituents on the principal axis itself are all of them equal to unity. The 
determinant possesses many elegant properties which are of importance to the prin¬ 
cipal investigation of the paper. The theorems concerning the general determinant 
are connecte'd entirely with the co-axial minors. 
I find that the general determinant of even order, greater than two, is expressible 
SIDCCCXCIV.—A. Q 
