MR. A. R FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
75 
Second, as an inference from the equations = 0, = 0, in Elliott’s theory, it 
follows that all pure reciprocants are invariants of the binary quantics (r, s, rjf, 
[a, b, c, rjY, . .. —but all invariants are not reciprocants—and that there are no 
covariants among these reciprocants. From the equations =0, Ag = 0, in the 
present theory it follows that all the functional invariants are algebraical covariants 
of the binary quantics (r, 5, iXc, d\q,— pf,. . . —but not all alge¬ 
braical covariants are functional invariants; and, from the other equations, that no 
algebraical invariants of these quantics are functional invariants. In particular, 
rt — 5^ is a reciprocant, but not a functional invariant; gh’ — 2pqs -j- pH is a functional 
invariant, but not a reciprocant.] 
Isolated Transformations. 
1. We may briefly consider functions which are invariantive for merely isolated 
changes of the independent variables, that is, for changes which are effected by one 
relation between x and X only, and one relation between y and Y only. For such 
transformations we have 
P = ^3 
dx 
dX 
S = s 
dx dy 
dXdY 
so that s ~ pq is an absolute invariant. Again, 
0 _ dx d 0 _ dy 0 
^~dYdy' 
so that ^|p> 0/003 and Ijq 0/0y are absolute invariantive operators, which, when applied 
to absolute invariants, will produce absolute invariants. We therefore have the 
series 
P<1 ’ 
M(s)’ 
J- 
qj bx q by] pq ’ 
w(-)’ 
q by \pq! 
qby p bxj pq 
and so on. The operators 1/p bjbx and Ijq bjby may be applied, any number of times 
in any order, to the absolute invariant s/pq (or any other invariant which is absolute), 
and the result will be an absolute invariant. 
These invariants possess their property for any general isolated transformations of x 
and y; but, if special isolated transformations are efiected on x and on y, e.g., the 
homographic transformations of the form 
aX+h a'Y + h' 
cX. + d cY + d 
L 2 
