GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
153 
Art. 59. The important point to notice is that it is legitimate to put the unde¬ 
termined quantities equal to any integral functions of x-^, x„, . . . x ,^—a fact, for the 
general order, that becomes obvious on examination of the above processes. 
As subsequently appears, it is such integral functions that usually present them¬ 
selves in arithmetical applications. 
Art. 60. As an example of the applications to arithmetic which s^varm about the 
theory, consider the important condensed form {vide Art. 14);— 
_ 1 _ 
1 2 ^]^ 2 1 ) 1 ) 1 ) ^a^p^y 
... (^21 !■) (^33 (^33 I) . . . 1) . . . Xn_ 
and, at first, consider the form of Order 3. 
The matrix of the redundant form is easily found to be either 
1 “12^31 
1,3^31 
J_ 
^12 
1 
U3 
13 
1 
*33 
U3 
“ 23^32 
— 1 
or the similar matrix with Cg^ written for Cjg. Since 
X 
^13 — 
31 
^21^32 
^31 — 0 
we have, taking Cg^ and putting (a^g, a^g) = (I, 1) a particular redundant product 
(^1 + ^21% + ^31^3)^' (^^1 + ■'^2 + + ^2 + •'^3)^'- 
In this, the coefficient of xY^xJ'^x^^^ (which is equal to the coefficient of the same 
term in the condensed form) is arithmetically interpretable as in Art. 15. 
Art. 61. If, however, we put {vide Art. 59) 
(“12> «23; O 3 0 = (-^U -^’2 
'2 5 
''SI 
we obtain a form which may be written 
^2 
(Xi -f \,^^X^X^ -f XgoXg^a-gXga-^)^* (1 fi- Xg + XggXga'g)^^ ( , , ^ ^ 4" f fi" -Xs ) , 
MDCCCXCIV.—A. X 
