94 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
u-^, u^, Vq, Vg, ; 
'^’o = — 1^3^- 
Wr; = Vr — ; 
Vg = + Gw/, 
IV^Q = — fw^Vg, 
— GWo '«.7 + 
= '^] 2 . — l''' 2 '^' 8 - 
(iii) When these functional combinations are substituted in turn for /’in A^f, the 
equations additional to those in § 12 can be transformed to 
«1 = 
GpVg - 
1 
0 
I> 
0 
11 
5pw>io - 
+ 
3iqv-, 
iq Apcii = 
4piyji - 
- 27^22 
+ 
S^q-ro, 
iq Apy^.j = 
1 
CO 
-'Vn 
4- 
9upiVj, 
II 
CO 
-f* 
<1 
: 2'pV^^ 
+ 
I2upv^, 
and therefore, if we write 
-13 
= P 
13’ 
W 
12 
3 ~ ^12’ 
Wji _ _ W, 
— p :a_o _ p 
105 
= P95 
these equations become 
W-l^A^Pg - 
Wj^A^^Pio = 
wpA^Pig = 
wpA^Pig — 
4PlOJ 
— 3P;^i + SiqPg, 
— '^■Pi2 "P biqP,;, 
Pl3 H~ ^^^4^75 
12 iqPg. 
In addition to the former irreducible solutions, Q^, Qq, Q 5 , which were obtained 
from equations in § 13, the following irreducible solutions can be obtained ;— 
Qi 3 — '^%P]3 Ps"’ 
Qi3 = 18w/Pi3 + 3w,P8Pi 3 + 81w/Pg - 2P83, 
Qu = 72w/Pii + 24 w/P8Pi 2 + 2w,P82P,3 + I 44 U/P 5 + 108W/P/ - Pgp 
Qio = 21 Gw/Pio + 108 w/P8Pii + iS^i/PgSp^^ _|_ w^PgSP^g 
- (l08iq«P8P5 + 81w/P82Pg + 18W/P83P, + P/), 
Q, = 129 GW/P 9 4- 8G4w/P8Pio + 21Gw/P82Pi, + 24w/P83P,, + 
~ (21Gw/P82p, + 108iq4VPG + IHw/Pg^P, 4 IPg^). 
