314 A THIRD MEMOIR UPON QUANTICS. [144 
then 
a = 5 Xp 2 , b = 4>pq\ + p 2 p, c = (2 q 2 + pr) A + 2pqp, 
f = 5 r 2 p, e = r 2 \ + 4gr/a, d = 2^rA + (2g 2 4- pr) p; 
and these values give 
P = K (6q 2 — pr), P'= K (10q 2 — lopr), 
M = K. 10pq, M' = K. 10qr, 
lY = K.5p 2 , N' =K.5r 2 , 
where the value of K is 
8 (pp 2 — 2qp\ + r\ 2 ) 2 (pr — (fy. 
The table No. 29 is the invariant of the twelfth degree of the quintic, given in 
its simplest form, i.e. in a form not containing any power higher than the fourth of 
the leading coefficient a: this invariant was first calculated by M. Faa de Bruno. 
No. 29. [See U. No. 29, p. 294.] 
The tables Nos. 30 to 35 relate to a sextic. No. 30 is the sextic itself; 
No. 31 the quadrinvariant; Nos. 32 and 33 the quadricovariants (the latter of them 
the Hessian); No. 34 is the quartinvariant or catalecticant; and No. 35 is the 
sextinvariant in its best form, i.e. a form not containing any power higher than the 
second of the leading coefficient a. 
No. 30. 
( 
a +- 
No. 31. 
ag + 
1 
bf - 
6 
ce + 
15 
d 2 - 
10 
b+ 6 
c + 15 
d+ 20 
e + 15 f+ 6 
9+ 1 ; 5*. V/ 
No. 32. 
±16 
ae + 1 
of + 2 
ag + 1 
bg + 2 
eg + 1 
bd - 4 
be — 6 
ce — 9 
cf - 6 
df - 4 
c 2 + 3 
cd + 4 
d 2 + 8 
de + 4 
e 2 + 3 
±4 
±6 
_j- Q 
±6 
±4 
5», yY 
No. 33. 
ac + 1 
ad + 4 
ae + 6 
of + 4 
ag + 1 
bg + 4 
eg + 6 
dg + 4 
e? + 1 
b 2 - 1 
be — 4 
bd + 4 
be + 16 
bf + 14 
cf + 16 
df + 4 
<f -4 
/ 2 -l 
c 2 -10 
cd — 20 
ce + 5 
d 2 -20 
de — 20 
e 2 -10 
±4 ±10 ±20 ±20 ±20 ±10 ±4 
±1 
±1