GENERATING FUNCTIONS IN THE THEORY OP NUMBERS. 
133 
Also for — 4, 
,,3 _ - 1 - o-,) = 3. 
Art, 32, In order to proceed to the general case it is necessary to make a digres¬ 
sion for the purpose of establishing certain properties of a determinant of special 
form. 
§ 4, Digression on the Tl^eoinj of Inversely Symmetrical Determinants. 
Art, 33, The determinant of special form which I have provisionally termed 
“ inversely symmetrical ” is 
1 , 
^12 
“l3 
CL 
135 
1 , 
1 
a 
13 
ai., 
In 
CL 
23 
a 
CL:, 
3« 
2.3 
‘ 1 « 
«2K 
n 
which involves different quantities a, and is such that the elements on the principal 
axis are all unity, and is inversely axi-symmetric in the sense that elements, symmet¬ 
rically placed in regard to the principal axis, liave a product equal to unity. 
Art, 34, The property of this determinant, which is of vital import to the present 
investigation, may be stated as follows :— 
“The determinant, as well as all of its co-axial minors, may be exhibited as 
functions of ^ ^ ) combinations of the ^ quantities 
a. 
2 ! --- ... ..... ^ ^ I -......... 
To establish this, first, consider the determinant itself, and put 
so that 
— O-x, x + I + I, x+ 2 • • • *^i/ — 1 , y> 2^)? 
Observe that the combinations 
^X, X + 1 - X + If 
"Yx, X + I 1- * 
yx,y {x<y-l) 
