106 
MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
Co = AoJ-^ C'o-A'oJ-L ^o-^oJ- 
^'oJ 
2 
are absolute irreducible invariants proper to the rank 2. Let 
K‘'s-r|;)Co = F. 
K4.-Pa!,)v = «’ 
J dx ~ r 
a.r 
, a 
p'a^)c«=F; 
P'|)C'.= G'; 
then F, F', G, G', 0r, are eight educed invariants proper to the rank 
three. But instead of q and p, or q and 'p, we can substitute the first differential 
coefficients of any unchanging quantity, say of any one of the absoiute invariants 
Cq, C'f), &Q, and thus educe new invariants. All these, however, can be expressed 
in terms of the set of eight already retained ; for we at once have 
and therefore 
t = rF'-/F. 
F ; 
1 /oCfi d 0C(i 0 \ 
j~\dq 0.33 033 0y/ 
C'o = J (F' G - F G'), 
which proves the statement.'^ 
Hence, through the present class of eductive operators we are able to derive from 
the simultaneous invariants jDrope]’ to a rank n double the number of educed invariants 
proper to the next higher rank ; but it is not to be inferred that they are all irredu¬ 
cible, or that they form the complete system of irreducible invariants proper to 
that rank. 
29. The foregoing linear operators are not the only eductive operators ; in fact, 
each new invariant suggests a new eductive operator. For the fundamental property 
of non-variation on the part of 2 :, the differential coefficients of wdiich are combined 
into invariants, enables us to substitute for 2 ; any other unchanging quantity, such as 
an absolute invariant. Thus, for instance, if I be any absolute invariant, then 
/ffi Y 
\0y/ 0*^^ 
01 01 0T_ /^Y , 
dx dq dx dy \ dx ) dy~ 
* Similarly for functions of 2 + \z' ; tbus 
F (s -f- \z') — F -|- \ {jp -|- F') + X“-t- -f V (G -f- 
