PROFESSOR CAYLEY’S NINTH MEMOIR ON QI7 ANTICS. 
21 
The development is 
-\-ax- 
+a%x i +l) 
-\-ct 3 (x 6 +x 2 ) 
+ a 4 (x s +x 4 + 1) 
1 
(b) 
+« 2 (- 4 +!) 
which is 
where 
=A(x)- 
; A 
(1 — «, 2? 2 )(1 — « 2 ) ’ 
and, since 4 A (-) contains only negative powers, the required number is 
00 \0C J 
= coefF, a? of in 
( 1 — ax 1 ) ( 1 — a 2 ) 
indicating that the covariants are powers and products of (ax 1 and a 2 ), the quadric itself, 
and the discriminant. Compare Second Memoir, No. 49, according to which, writing 
therein a for x, the number of asyzygetic covariants is 
= coefF. a 9 in vtt z\ • 
» (1 — «)(1 — a ) 
334. For the cubic (a, b, c, d$jx, y f the number of asyzygetic covariants a 9 x' x is 
1 — X 
coefF. in 
(1 — a) (l — ax)( 1 — aa? 2 )(l —ax 3 ) 
or transforming as before, this is 
1 - 
= coefF. a'hf in 
(1 — ax 6 ) (1 — ax) ( 1 — ax~ *) (1 — ax~ 6 ) 
the function is here 
A W-? A (;)• 
where 
. . 1 — a 6 x G 
W (1 — a^ 3 )(l— a 2 o.’ 2 )(l — d 6 x 3 )( 1 — a 4 ) 
(that this is so may be easily verified) ; and since the second term contains only negative 
