MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
8. If, then, we have an invariant f of index A., such that 
F ( . . . , Z,;,, „,...) = J\/’( . . . , Zm^n, . . .), 
and we substitute for J and for all differential coefficients 7im, n, ^nd then expand, 
retaining all small cjuantities of the first order, we have the following equations 
derived from a comparison of corresponding terms. 
From the terms which are multiplied by eX 
(2m + ri) = 3X/ . . . . 
. . . (i.); 
from the terms in 6Y 
tt (2n -f m) — 3X/ .... 
n 
. . . (ii.); 
from the terms in e 
Ai/ — SSm (m + — 1) ^ • • 
. . . (in.); 
from the terms in 6 
Ao/ — Stn (m + n — 1) ■ — 0 . . 
. . . (iv.) ; 
from the terms in /S + eY 
^ 3 /- + -0 . . . . 
• • • (v.); 
and from the terms in 
« + 6»X 
A4y = 1 , = 0 .... 
. . . (vi.). 
Equations (iii.)-(vi.) determine the form of the function /; when the’ form is 
obtained, the index is derivable by inspection, and equations (i.) and (ii.) are then 
identicall}^ satisfied. 
9. Before considering these equations, characteristic of the invariants, one remark 
should be made. If the quantities e and 6 are absolutely zero so that the transforma¬ 
tions are 
X = X + aY, y = ;8X + Y, 
that is, transformations to which a binary form is subject, the terms which, in what 
precedes, give rise to equations (i.)-(iv.) do not exist, and, therefore, these equations 
do not exist; but there are terms in and a, and, therefore, equations (v.) and (vi.) 
survive, being in fact the partial differential equations determining those covariants 
which can be expressed in terms of partial differential coefficients of the form with 
reo’ard to the variables. 
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