GENERATIN'G FUNCTIONS IN THE THEORY OF NUMBERS. 
115 
Without specifying at present the aritlimetical meaning of the generating function, 
I will call the portion above-written the “ redundant form,” and the essential portion, 
to which reference has been made, the “ condensed form,” 
Art. 4. As typical of the general case, put n = 3. 
It will be shown that the condensed form is 1/N, where 
N = 1 — — CgSgXg 
+ I 1 “h 1 I s^s^XiXg -j- ] bgCg [ s^SgX^a^g — j a^^^Cg [ 
The notation is that in use in the Theory of Determinants, the coefficients of N 
being the several co-axial minors of the determinant ] \ ; this determinant is 
the content of the matrix which occurs in the definition of the linear qualities X^, 
^2) Xg. 
Art. 5. In determinant form N 
1 — a^s^x^, 
C^SgSlg, 
may be written 
CioS^X-^ 
CO 
CO 
CO 
1 
1 — Cg.9gXg 
and also in the important symbolic form 
I (l ( 1 1>.2^^X:^{\ CgL^giTg) | , 
wherein, after multiplication, the a, h, c products are to be written in determinanr 
brackets. Such symbolic multiplication will be denoted by external determinant 
brackets as shown. 
Art. 6. We have now 
N 
(1 - vX)(i -soX„)(i - vW) 
(1 - SiXi)(I - ssXo)!! - S3X3) 
(1 — 3^X| -I- g^X^ - rtjSjrq) (1 — SgXg + ssXj — h^SoX\) (I — S3X3 -|- S3X3 — c.s.^.r. 
(I — SjX^) (1 SjX,) (1 S3X3) 
1 I ^1 (Xj I $2 (Xo , S3 (X3 , S0S3 I (Xg ^2^2) 1X3 
^ + “5 TA?-k — + “5 -TY-T ' 
I - .,X, 
1 SgXg 
So 
(1 ^ 2 X 0 ) (1 S 3 X 3 ) 
■ 4r% I 1 X 3 (Xi I gj^Sg I (X] (Xg | 
(1 S 3 X 3 ) (1 SjX;^) (1 ^iXj (1 ^ 5 X 0 ) 
Q 2 
5 
