MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
95 
These are not necessarily the simplest forms obtainable, bnt every simultaneous 
solution of Aj/" = Aj/* = Ag/’ = A^/* = 0 can be expressed as a functional combination 
of M 4 , Q 5> Qg’ Q?' ^9; • • •> Qi3’ 
It will be seen that the new irreducible functions Qg, . . ., Qig are linear in the 
quantities Pg, Pjq, . . ., Pjg, and are therefore linear in the partial diflPerential 
coefficients of the fourth order. In this respect they apparently differ from Qg, Qg, 
Q^, which are the irreducible invariants of the third rank in differentiation; but, if 
we take instead of Q- an equivalent invariant 1447q"Qg + Q-^, which is 
288^q«Pg + 144VPgP8 + 
the law of successive formation of the invariants (the new) Qg, Qg, Q^ is similar to 
that for the functions Qg, Qjg, • • Qia- 
18. But, before it can be asserted that Qg, Qjq, . . . Q^g are invariants, it must be 
shown that they severally for a common value of X satisfy the equations (i'.) and (iih). 
Now the following results are easily obtained :— 
/ = 
ffi/ = 
X = 
P 5 
0 
0 
0 
3Ps 
3Pg 
1 
P 7 
6 P 7 
6 P 7 
2 
Ps 
9Ps 
GO 
3 
P 9 
0 
0 
0 
Pio 
•^Pio 
3Pio 
I 
Pll 
6P11 
6Pn 
0 
Pic 
9 Pi 2 
9 Pi 2 
3 
Pi. 
12Pi2 
I2Pi2 
4 
