84 
MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
enter. If the determination of the invariants be made from the point of view that 
they are simultaneous solutions of the four equations, one method of proceeding will 
be to adopt Jacobi’s process. 
Any function f, which satisfies (iii.)-(vi.), must also satisfy each of the equations 
[A„ Aj = 0 
for r, 6’ = 1, 2, 3, 4. Forming these, it is easy to show that 
[A„ A,] = X S ■( 
{d(AJ) 3(A,/) 
1 i{^'- 
^'^m. n 
a (A./) a (A,/-) 1 
dz.,.' { 
= {m — n) %,n,, 
vf 
dz„, 
after substitution and collection. Since this does not vanish in virtue of any one of 
the given equations, we must have a new equation 
^ 5 /= 
0/ 
’’/u, n 
(vii.). 
to be associated with the rest. But this is the only additional equation ; for 
^ ^1] ~ '^2/ — b 5 [^35 '^3] ~ ~ b ; 
[A 3 , Aj=0; [A„Aj-0; [A^, A J == 2A J'= 0 : 
[A 3 , Aa]^ -2A3/=0; [A^, A,] = 0 ; [A^, A J = 0. 
It is easy to verify thaty* = satisfies (vii.). Every invariant will be a simultaneous 
solution of (iii.)-(vii.). 
It may be noticed that the equation (vii.) can be otherwise obtained; it arises by 
equating the left-hand sides of equations (i.) and (ii.) to one another, for, on re-arrange¬ 
ment of this, we have 
(m — n) z„,^„ = 0. 
Hence, equation (vii.) may be considered as replacing either (i.) or (ii.); and any 
function, which is a simultaneous solution of (iii.)-(vi.) and for the same value of X 
satisfies (i.) and (ii.), will also satisfy (vii,). 
One deduction as to the character of the invariants can at once be made from the 
form of the equations. 
