■ 
J 
326 
A THIRD MEMOIR UPON QU AN TICS. 
[144 
or else 
^d x U, 
%d y U, 
R U 
hd x U, 
d x 2 HU, 
d x d y HU, 
dfd z HU 
5 
d y d x HU, 
dy 2 HU, 
dyd z HU 
\d z U, 
d z d x HU, 
d z dyHU, 
d 2 HU 
or else, what I believe is more simple, a function ©„Z7, which is a linear function of 
the last-mentioned two functions. 
The relations between %U, © / C7, ©„?7 are 
- ©, U + №U=T. U 2 — 24$. U .HU, 
©„Î7 +2<èU = T. U 2 — 10$. U.HU 
I have not worked out the developed expressions. 
No. G5. 
The cubicontravariant is PU = 
bel — 1 
ad — 1 
abl — 1 
ack + 1 
abi + 1 
bej + 1 
abg + 1 
bch + 1 
acf + 1 
abc — 1 
bgi + 1 
agi + 1 
afk + 1 
a fg- 2 
a/ 2 - 1 
bg 2 - 1 
aft. + 1 
bgl + 1 
ai 2 — 1 
afi + 1 
cfk + 1 
chj + 1 
bhj + 1 
ail + 1 
bqh — 2 
cfh - 2 
aik - 2 
bij — 2 
chi + 1 
hi +1 
Pg -1 
g 2 h - 1 
fh 2 - 1 
di 2 - 1 
bjl +1 
ckl + 1 
bf -1 
ck 2 — 1 
cjk - 2 
chk + 1 
fil + 1 
gji +1 
ckl + 1 
ff +2 
fhl + 3 
fgi +3 
ÀJ -1 
/V + 2 
fgj - 1 
fgh + 3 
l 2 k -1 
ij 2 -1 
jk? -1 
fid + 3 
fjk - 1 
fv -i 
ghk — 1 
fgk -1 
g 2 k + 2 
3 -4 
f 
g.P -1 
gk 2 + 2 
gki — 1 
h 2 i +2 
fhi - 1 
ghi — 1 
gkl — 4 
hij - 1 
hki — 1 
hi 2 +2 
hi 2 -2 
fl 2 -2 
gl 2 -2 
hil — 4 
j 
jP -2 
W -2 
il 2 -2 
jkl + 3 
ikl + 3 
ijl + 3 
ijk + 3 
P +4 
J 
=t 3 
=*=3 
±3 
±7 
±7 
±7 
±7 
±7 
±7 
±13