14] 
ON LINEAR TRANSFORMATIONS 
and © be any constant derivative whatever of U, then 
is a derivative of U, and its value, neglecting a numerical factor, may be found by 
omitting in the symbol □, which corresponds to the derivative ©, the factors which 
contain any one, no matter which, of the symbolic numbers. 
If, for example, 
or 
— ^ D 210 = © = Qabcd — 4ac 3 — Fbd 3 + 3b 2 c 2 - a 2 d 2 , 
□ = 12 3 .34 2 .13.42; 
then 
d_ 
dd 
2 d d . d 
xy do + ^di-y 
reduces itself, omitting a numerical factor, to 
12 2 13 UUU = — \B 1 {U, B,(U, U)}. 
This may be compared with some formulas of M. Eisensteins (Grelle, vol. xxvn. 
[1844, pp. 105, 106]; adopting his notation, we have 
O = aa? + Sbx 2 y + Sexy 2 + dy s , 
F = 3^ Bz (O, O) = (ac — b 2 ) x 2 + (ad — be) xy + (bd — c 2 ) y 2 , 
®i = 
d> 1 = - 
STÏÏ 
d 3 0 d 2 0 dO d 3 (f> 
v 2 
d :i O 
~ a i , d . d „ d 0 d , ,, 
where D is the same as ©. Hence to the system of formulae which he has given, 
we may add the two following: 
I /dO dF dO dF\ 
II \dx dy dy dx) ’ 
2 d 2 0 dO (M> dO\ 
dx 3 dxr dy dx 2 dy dx dy dy dy 2 dx) 
the first of which explains most simply the origin of the function O x . 
It will be sufficient to indicate the reductions which may be applied to derivatives 
of the form 
C atftf y(U, V, IT) = 23°.31^. 12 v UVW, 
where U, V, W are homogeneous functions. In fact, if 
& + ijy = 
the above becomes, neglecting a numerical factor, 
(H,. 23)“. (Hi. 31)“. (B,. 12)' UVW,