PROFESSOR CAYLEY’S TEETH MEMOIR ON QUANTICS. 
655 
On the form of the Numerical Generating Functions : the N.G.F. of a Sextic. 
Article Nos. 385, 386. 
385. It is to be remarked that the R.G.F. is derived not from the fraction in its 
least terms, which is algebraically the most simple form of the N.G.F., but from a 
form which contains common factors in the numerator and denominator : thus for the 
quadric, the cubic, and the quartic, writing clown the two forms (identical in the case 
of the quadric) these are — 
Quadric 
N.G.F. 
1 
1 — ax l 1 —a 2 
Cubic 
N.G.F. 
1 — ax-\-a 2 x 2 
1 — a 4 . 1—ax : 3 . 1 —ax 
Quartic 
N.G.F, 
1 — ax 2 + a 2 x 4 
1 —a 2 . 1 — a?. 1—ax 4 . 1—ax 2 
1 —a G x & 
l— a 4 . 1—ax?. l—c?x 2 . l—a s x 3 . 
1 —a 6 x 12 
l— a 2 . 1 —a 3 . 1—ax 4 . 1 —a 2 x 2 . l—a?x 6 . 
For the quin tic the two forms are, N.G.F. = 
( 1 
-a 6 
+ a 12 )® 0 
+(-l 
+ a 4 
+ 2 ft 6 
— ft 12 ) ft* 1 
+( 
+ ft 2 
— ft® 
+ a 10 )* 2 
+(-l 
+ a 4 
+ a , 6 
+ ft 8 
-a 10 
— a 12 ) ft* 3 
+ ( + 1 
+ a 2 
— eft 
-ft 6 
- « 8 
+ ft 12 )ft 2 * 4 
+( 
—a 2 
+a 
-a 10 
)ft 3 * 6 
+(+l 
-2ft 6 
-ft 8 
+ ft 12 ) a 2 * 6 
+(-l 
+ ft 6 
— a 12 ) ft 3 * 7 
divided by 
1 — a 4 . 1 — a 6 . I— a 8 . 1 — ax 5 . 1 — ax 3 . 1—ax 
4 P 
MDCCCLXXVIii. 
