MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
87 
In order to obtain the most general simultaneous solution of Ag/= 0, A^/ = 0, 
0, Aj._/=: 0 , it will be sufficient to obtain the irreducible functional combina¬ 
tions of til, Vg, Vg, Wrj, which satisfy ^ 4 ^/ = 0. Now it is easy enough to shoAV 
that 
— 0 ; 
and that 
A^^i ■— p, 
WlA^Vg = - 3Vg, 
U^A^Vg = 3 pVg — ‘ZWr., 
= 22nVrj — Vg, 
= 'pv^ + 6w/. 
If, then, we write 
these equations become 
^i^A^Pg = , 
— Pg’ 
Wl^A^Pg = - 2 P 7 , 
iq^A^Pg = _ 3Pg ; 
and therefore, bearing in mind that A^'^^^ = 0, we have 
A,(Pg2+ 12 i^/P7)= 0, 
A,(P83 + IS^q^P^Pg + 54^^/Pg) = 0, 
A4(P8P6-P7' + 2VP5)= 0. 
Hence the most general simultaneous solution of A^f = 0, A.^f = 0, Agf = 0, A^f = 0 
can be expressed as a functional combination of u^, and 
Q? — Pg^ H“ i2tf/P^, 
Qe = Ps® + ISw/P^Pg + 54u^‘^Pg, 
Q 5 = PsPe - P/ + 2<P5. 
13. Before considering the question as to whether these functions satisfy (i.) and 
(ii.), and, therefore, also (vii.), it is desirable to modify their expressions. 
We have already had the quantity iq; it is the same as Aq, so that we write 
= Aq ; 
and it will be convenient to write 
tig = A^ = (zgQ, Z. 21 , Zj_ 2 , ^osX^on ^]o)^’ 
