MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
107 
the same function of I as Ah is of z ^ is an invariant of index 2, and 
an 
0A 
0H 
01 
01 
- 2j’<i a, + r' 53 + 2 -I {'v - px) 5 ; - (?« - p>) a; 
3 ,'/ 
the same function of 2 and I as is of 2 and z, is an invariant of index 2. 
30 . The expressions for educed invariants can be applied as follows to obtain the 
expressions for the efiects of the operation of 0;^, @2, @3, 0^, (§ 20) on differential 
coefficients of the invariants with regard to the variables. 
3 3 3 
Let V be an invariant of index m, and let the operators q ^ ~ 2 ^ f > 
be denoted by S and 8' respectively. Then 
0?/ 
Vi = J8V -mVSJ, 
= JS'V - «iV8'J 
are invariants of index m + 2 ; they must satisfy the equations 
@1/= 0 = %/= %/= %/■ 
Hence 
J0SV = mV0SJ, 
J0ST = mV08'J, 
are satisfied for each of tlie operators 0, because J and V are themselves invariants. 
Now, actual substitution gives 
and therefore 
Now, since 
and 
0^SJ = 3 qJ, 
038J = - 3 pJ, 
038J = 0, 
= 0 , 
0^S'J = 3^'j ; 
02S'J = — 3 j 9 'J ; 
03S'J = 0 ; 
0^8'J = 0 ; 
0iSV = SqYm, 
= — SpYm, 
038V = 0, 
0^8V = 0, 
0^8'V == SqYm ; 
0 ^ 8 'V = — Spj'Ym ; 
038'V = 0 ; 
048'V = 0. 
0^ (8V) = 50^ 
0^8'V = q'® 
0V 
0.r 
^ dr 
it follows from the first pair of equations that 
p 2 
