168 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
Table No. 86 [= leading coefficient of >8']. 
a 4 ce/ 3 
+ 9 
a 3 be/ 3 
— 
9 
ddr’df 3 
+ 
120 
ab 4 cf 3 
576 
b s / s 
4- 
192 
a 4 d?/ 3 
+ 21 
aPbcdf 3 
- 
162 
aW/ 2 
- 
21 
ab 4 def 2 
4- 
672 
dee/ 2 
— 
1440 
a*def 2 
- 78 
a 3 bce 2 / 2 
4- 
99 
aWf 3 
+ 
486 
ab 4 e 3 / 
- 
359 
dd 4 / 2 
- 
192 
a 4 e 4 f 
+ 48 
a'bd?ef 2 
4- 
309 
orlrcdef- 
2160 
alfde/ 1 
4- 
3456 
b 5 de 2 / 
- 
1080 
a s bde 3 f 
4- 
12 
ddfce’f 
+ 
1023 
aVccPf* 
— 
864 
Jfe 4 
4- 
2025 
d } be 5 
- 
240 
a 2 b 2 d 3 / 2 
+ 
120 
ab 3 cde 2 / 
4- 
2094 
b 4 dd/ 2 
+ 
2592 
a 3 c 3 / 3 
- 
81 
a?bWf 
— 
1053 
ab s ce 4 
— 
3915 
b 4 de/ 
4- 
3546 
d 3 dde/ 2 
+ 
1026 
a 2 b 2 de 4 
+ 
1314 
ab s d 3 e/ 
4- 
528 
b 4 cd 2 e/ 
4- 
5280 
a 3 c 2 e 3 / 
- 
768 
a 2 bde/ 2 
- 
1863 
ab 3 d 2 e 3 
— 
45 
b 4 cdd 
— 
13500 
a 3 cd 3 f 2 
- 
738 
a 2 bc 2 d/ 2 
+ 
2538 
ab 2 dd/ 2 
— 
2592 
b 4 d 4 f 
— 
4800 
d 3 cd?e 2 / 
- 
564 
a 2 bc 2 de 2 / 
2340 
ab 2 dd/ 
— 
9747 
b 4 d 3 e 2 
4- 
7800 
cdcde 4 
4- 
1056 
cdbc-e 4 
+ 
672 
ab 2 dd 2 e/ 
— 
8496 
b 3 e 4 / 2 
— 
648 
a''d 4 pf 
+ 
756 
a 2 bcd 3 e/ 
4- 
2820 
ab 2 c 2 de 3 
4- 
26610 
b 3 c"’de/ 
— 
14040 
cdcPe 3 
- 
696 
aPbcdPe 3 
— 
7812 
ab 2 cdf 
4- 
8544 
b 3 dd 
+ 
3075 
a?bd?f 
- 
3024 
ab 2 cd 4 e 
- 
16650 
b 3 e 2 d/ 
4- 
9120 
a 2 bd 4 e- 
+ 
4572 
ab 2 d 5 e 
4- 
720 
b 3 c 2 d 2 e 2 
4- 
16350 
a?c 4 df 2 
- 
324 
abcf 2 
4- 
972 
dcd 4 e 
— 
19200 
d 2 de 2 / 
+ 
3888 
abc 4 def 
4- 
24048 
b 3 d ti 
4800 
a 2 dd 2 e/ 
— 
8748 
abc 4 e 3 
— 
4464 
b 2 de/ 
4- 
4860 
d 2 ddd 
— 
4800 
abdd 3 / 
— 
15984 
b 2 c 4 d 2 / 
— 
3240 
a 2 c 2 d 4 / 
+ 
4248 
abdcfte 2 
- 
30108 
b 2 c*de 2 
— 
8100 
d 2 c 2 d 3 e 2 
+ 
14520 
abdd 4 e 
4- 
35088 
ddd'e 
4- 
9000 
urcd 5 e 
— 
11448 
abed B 
— 
8640 
bdd 8 
— 
2400 
a 2 cP 
+ 
2592 
adef 
— 
7776 
ac 5 d 2 f 
adde 2 
4- 
5184 
4- 
12960 
ac 4 d 3 e 
— 
14400 
add s 
4- 
3840 
1+ 
00 
+ 
3258 
+ 
41253 
4 
124716 
+ 
68640 
[The values thus are cf> s (x, y) = 10 BJ — 1ODG ; (x, y) = <8'.] 
289. The equation in z is of the form 
21 
23 
<S X 
*' + I>*‘ + i z "' + I) z + D =0 - 
where Z> is the discriminant of the quintic and 21, 23, (5, X denote rational and 
integral functions of the coefficients (a, b, c, d, e, /). And the covariants <^>j 0*. y). 
<f). 2 (x, y), <f> 3 (x, y), (f> 4 (x, y) having the values given to them above, the actual value 
of 21 is obtained as a quadric function of the indeterminates (t, u, v, w), viz. this is 
= [A* 2 - QBBtv - D (A - 10 AB) v-] + D [- Bu 2 + 2D,uw + 9 (BD - 10 A A) w 2 ], 
where A = 25AB + 16(7, these quantities, and the quantity A r (= A 2 — 10ABD 1 + 9B-D) 
afterwards spoken of, being in the notation of the present Memoir as follows: 
A = J 
B = -K 
G = 9 L + JK 
1) = D 
A = 9(1OL-JK), 
N = 1152(18 L--JKL 
(= G), 
(=-Q), 
(=-9 U+GQ), 
(= O'),