308 
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS. 
[932 
deg. 5, 
deg. 6, 
class F, 
b 5 G.F. = 
= îc° -5, 
EF, 
bc i 
of — 4.5, 
DF, 
b 2 c 3 
æ 8 - 3.5, 
CF, 
o 3 
to 
x 7 -2.5, 
DEF, 
bed 3 
x 12 — 3.4.5, 
GEF, 
bc 2 d 2 
x 11 — 2.4.5, 
CDF, 
b 2 cd 2 
« 10 - 2.3.5, 
GDEF, 
bede 2 
a; 14 — 2.3.4.5 
'ght terms 
G. F.= 
x 5 2.3.4.5, 
before. 
class G, 
¥ G. F. = 
x 6 — 6, 
FG, 
be 5 
x n — 5.6, 
EG, 
6 2 c 4 
¿c 10 — 4.6, 
EG, 
b 3 c 3 
x 3 -3.6, 
CG, 
¥c 2 
a? -2.6, 
EFG, 
bed 4 
¿c 15 — 4.5.6, 
DFG, 
bc 2 d 3 
x li - 3.5.6, 
CFG, 
bc 3 d 2 
x 13 — 2.5.6, 
DEG, 
b 2 cd 3 
æ 13 — 3.4.6, 
CEG, 
b 2 c 2 d 2 
æ 12 — 2.4.6, 
CDG, 
b 3 cd 2 
x 11 - 2.3.6, 
DEFG, 
bede 3 
£c 18 -f- 3.4.5.6, 
CEFG, 
bcd 2 e 2 
i» 17 -i- 2.4.5.6, 
CDFG, 
bc 2 de 2 
a; 16 — 2.3.5.6, 
CDEG, 
b 2 cde 2 
x 15 - 2.3.4.6, 
CDEFG, 
bedef 2 
a ,2 ° — 2.3.4.5. 
and for the sum of the sixteen terms 
G.F. = x 3 + 2.3.4.5.6, 
which may be verified as before. 
Reducible Seminvariants—Perpétuants. Art. Nos. 60 to 64, 
60. Seminvariants of the degrees 2 and 3 are irreducible—or say they are 
perpétuants. Hence by what precedes, as regards perpétuants 
for degree 2, G. F. = a? — 2 ; 
for degree 3, G. F. = x 3 — 2.3. 
For the degree 4 (if as before b, c, d denote discrete letters), then the finals are 
b 4 , be 3 , b 2 c 2 and bed 2 . For a final b 4 —b 2 .b 2 or b 2 c 2 = b 2 .c 2 , we have evidently a product 
of two quadric seminvariants ending in b 2 and b 2 , or in b 2 and c 2 , with the same 
final term as the quartic seminvariant ; so that, considering the quartic seminvariants