GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
129 
given quadratic equation, which are expressible as irrational functions of the co¬ 
efficients of Vg. The redundant form is 
_ 1 _ 
(1 SjXj) (1 — SgXj) (1 S 3 X 3 ) 
of a doubly infinite character. 
Also 
— (2'* — 1 — o-„) = 2, for n = 3. 
Art. 31. The case n = 4. 
The matrix being that connected with the determinant | | we have the rela¬ 
tions :— 
cqi = C%3 — ^>25 ®33 — PS) ^'44 = 
I 1 — Pl2> 1 *^^ll'^^33 I — Pl35 I I — 2^235 
I 1 — Pl-i’ I *^22^^44 I ~ P 24 ) I ^^33^^44 1 “ ^34) 
I ^11%2^'33 I — Pl235 1 ^hl^22*^^'44 I — P’l24> 
I *^'^11*^33*^44 i ~ Pl34’ I *^22*^33*^% ! — P234> 
1 *^ll*^^22*^33*^h4 i — Pl234 5 
and thence 
*^^12%l ~ 9 'i 2> ^13%1 ~ fZl3’ *^23*-'^32 “ 9'23’ 
^14*^41 — S'W) *^24*^42 = 9'24) *^34*^h3 = ^Zsi ’ 
and introducing six undetermined quantities, 
^12 — *^129'i2! ®13 — “i 32'13’ *^14 — ®‘]49'l4) *^^23 — ^23^hs> *^24 — %i5'24> ^^34 — “siTsO 
_Jl — 1 . —2. 
«21 — J *^31 — „ > *^41 — „ 5 C^32 — „ 5 ^^42 — „ s *^^3 — » 
*13 *13 *14 *23 *24 *34 
and thence by substitution in the remaining relations. 
i^i, 
*^129'i2> 
*^135'i3 
Pn 
“129'i25 
°^14*Zl4 
1 
? 
^12 
P-2, 
“23fe 
C< 
II 
1 
*13 
P2, 
“24*Z24 
— Pl24 
1 
*13 ’ 
1 
CCcyo 
Ih 
1 
5 
*14 
1 
*24 
P4 
MDCCCXCIV.—A. 
