532 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QIT ANTICS. 
(p 4 (x, y) : those ultimately so called by him are 
y)=4:{cp 3 (x, y)} + J<p t (x, y) (J= No. 19), 
<p 4 (tf, y) = 4 {<&(#, y)}+-3J<p 2 (x, y)+ y), 
where ^p,(x, y) is the cubicovariant ( — 27A 2 D+9ABC — 2B 3 , ' . . 1 J(x, y) 3 of <p{(x, y ), 
=(— 3 A, — B, — C, — 3DX^, y) 3 , ut supra. 
The co variant <p 2 (x, y) has the property that if the given quintic (a, . . .~Jx, y) 5 con- 
tains a square factor ( lx-\-myf , then <p 2 (x, y) contains the factor Ix+my: \<p 3 (x, y)\ and 
\<Pi{x, y)} are co variants not possessing the property in question, and they were for this 
reason replaced by <p 3 (x, y) and <p 4 (x, y) which possess it, viz. <p 3 (x, y) contains the factor 
lx-\-my, and <p 4 (x, y) contains (lx-\-my)\ being thus a perfect cube when the given 
quintic contains a square factor. 
288. The covariants cp^x, y ) and <p 2 (x, y) are included in my Tables, viz. we have 
<Pi(x, y)= — 3No. 16, 
,y)=— No - 23 
(observe that in No. 23 the first coefficient vanishes if <7 = 0, &=0, which is the property 
just referred to of <p 2 (x, y)) ; the other two covariants, as being of the degree 7 and 9, are 
not included in my Tables, but I have calculated the leading coefficients of these cova- 
riants respectively, viz. 
Table No. 85 gives leading coefficient (or that of x 3 ) in <p 3 (x, y), and 
Table No. 86 gives leading coefficient (or that of x 3 ) in cp 4 (x, y). 
The coefficients in question vanish for a= 0, #=0, that is, <p 3 (x, y) and <p 4 (x, y) then 
each of them contain the factory ; if the remaining coefficients of <p 4 (x, y) were calculated, 
it should then appear that for a=0, b = 0, those of x 2 y, xy 2 would also vanish, and thus 
that <p 4 (#, y) would be a mere constant multiple of y 3 . 
Table No. 85. 
a 3 ce 'f 
+ 1 
- 1 
ab 3 df 2 
+ 
64 
b'cf 2 
a 3 d' 2 f~ 
+ 15 | 
a‘bcdf 2 
- 94 
ab 3 ef 
— 
54 
Vdef - 
144 
a 3 de 2 f 
-32 
d?bce 2 f 
+ 86 
ab 2 &f- 
— 
48 
b*e 3 + 
135 
a 3 e 4 
+ 16 
a?bd 2 ef +106 
ab 2 cdef 
+ 
184 
b 3 c 2 ef + 
108 
a 2 bde 3 
- 96 
abrce 3 
— 
135 
b 3 cd 2 f + 
288 
d 2 c 3 f 2 
+ 63 
ab'd'f 
— 
272 
b 3 cde 2 - 
450 
crc'def 
-188 
ab 2 d 2 e 2 
+ 
243 
b 3 d 3 e + 
80 
a 2 c 2 e 3 
+ 32 
abc 3 ef 
— 
66 
b 2 c 3 df - 
360 
| a?cd 3 f 
+ 60 
abc~d 2 f 
+ 
212 
tfc 3 ? + 
135 
a 2 cd 2 e 2 
+ 68 
ab<?de 2 
+ 
148 
b 2 c‘d 2 e + 
360 
a 2 d i e 
- 36 
abcd 3 e 
— 
412 
b 2 cd A - 
160 
abd° 
+ 
144 
be 3 / + 
108 
ac'df 
— 
36 
bc'de — 
180 
ac 4 e 2 
— 
48 
bc 3 d 3 + 
80 
ac 3 d 2 e 
+ 
124 
ac 2 d* 
- 
48 
±32 +415 +1119 +1294 
* M. Hermite, p. 17, has erroneously written <p s (.v, y)-{- 4A<p l (x, y), instead of 4<p 3 (or, y)-\-A<p 1 (x, y) ; the 
latter expression is that which he really makes use of, and the formula in the text is correct. 
