§38] 
DIFFERENTIAL OPERATORS 
59 
multiplied by the sum of the total products is a covariant 
with the leader S. For example, 
«0 2 S(q:2— as) 2 , ao 2 S(«2 — a3)(«3 — «l) 
3 3 
are the leaders of the covariants in Exs. 1, 2, § 36, of the binary 
cubic. The present result should be compared with the 
theorem in § 25. 
We may now give a new proof of the lemma in § 25 that 
dp — 2w=0 for any seminvariant S of degree d and weight 
w of the binary ^-ic. Whether 5 has the factor ao or not, 
the first term of the resulting covariant K is Sx a , where 
o = dp—2w. For, in each product in the above S, the roots 
a\, . . . , oL P occur 2w times in all. In K each root occurs d 
times. Hence we inserted dp — 2w factors x—ay in deriving K 
from 5. 
38. Differential Operators Producing Covariants. Let the 
transformation 
T: X=a£-\~Pi1, y = T£+5r?, A=a8 — 07^0 
replace/(x, y) by </>(£, 77). Then 
= =a <£+ y $f 
0£ dx di dy d£ dx dy 
d± = dfc&+dfcdy = pM.+ s$ m 
dv dx dy dv dx dy 
Solving, we get 
— A—= 
9y dv 9£ dx 
d <t> PdP 
7- 
dv dt’ 
or df=Dp, dif=Dip, if we introduce the differential operators 
Di 
As usual, write d 2 dif for d{d(dif)\. Since the result of 
operating with d on df is the same as operating with D on 
the equal function Dp of £ and 77, we have d 2 f=D 2 p. Similarly, 
'Ecrsd r di s f = 'ZctsD t D\P (r+5 — w).