359 
693] 
A TENTH MEMOIR ON QUANTICS. 
which (as is at once verified) 
are consistent with the fundamental relation 
then it is easy to verify that 
/ 2 = — a?d + a 2 bc — 4c 3 ; 
( æ |- 4c 2'S- S )^ = 0; 
and this being so, any other covariant whatever, expressed in the like standard form, 
is reduced to zero by the operator 
and we have thus the means of calculating the covariant when the leading coefficient 
is known. 
Thus, considering the covariant B, the expression of which has just been obtained, 
= (B 0 , B u B 2 \x, y) 2 , suppose: the equation to be satisfied is 
x (B X x + 2B 2 y ) 
- 4>cy ( 2B 0 x + B 1 y) 
— x 2 BB 0 — xyBB x — y 2 8B 2 = 0, 
viz. we have 
B x - BB 0 = 0, 
2B 2 - 8 cB 0 - 8B X = 0, 
— 4 cB x — BB 2 = 0; 
which (omitting, as we may do, the outside factor a 2 ) are satisfied by the foregoing 
values B 0 , B X) B 2 , = b, e, —Sad —be. And if we assume only B 0 = b, then the first 
equation gives at once the value B x — e, the second equation then gives B 2 =—Sad—Sbc; 
and the third equation is satisfied identically, viz. the equation is 
that is, 
which is right. 
— 4ce + B (3ad + be) = 0, 
— 4ce = — 4ce = 0, 
4- cBb + c . e 
+ bBe +b . 3/ 
+ 3 abd + 3 (—bf+ ce) 
Of course every invariant must be reduced to zero by the operation 8: 
have, see the Table No. 97, 
a 2 g = 
thus we 
12 abd