§20] 
ANNIHILATOR OF INVARIANTS 
35 
By § 16, a function I{ao, . . . , a P ) is invariant with respect 
to every transformation S t if and only if it is isobaric. 
Finally, the function must be invariant with respect to 
every T„; under this transformation let 
Differentiating partially with respect to n, we get 
»=0 \t / [ dn 
since r] = y is free of n, while %=x—nr\. The total coefficient 
of £ P- V is 
(P s 
|M'_| 
(.* ) 
\j/ 
' 0w 
\J-V 
(P~— 0, 
the second term being absent if j — 0. But 
Hence ] 
~=Mi-1 ü = i,...,ÿ). 
(2) 9/Mo,..,2l g ) =j4 _3/ ^ d/ +3 ^ _3/ + + ^ 3/ 
dn dAi dA 2 0H 3 0H* 
Now /(do, . . . , a p ) is invariant with respect to every 
transformation T„, of determinant unity, if and only if 
/(Ho, . . . , A p )=I(a 0 , . . . , a p ), 
identically in n and the a’s. This relation evidently implies 
dI(A 0 , . . . , A p ) _q 
0W 
Conversely, the latter implies that /(Ho, . . . , A P ) has the 
same value for all values of n and hence its value is that given 
by n = 0, viz., 7(ao, • • . , a p ). Hence I has the desired property 
if and only if the right member of (2) is zero identically in 
n and the a’s. But this is the case if and only if