PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
623 
and this being so, any other covariant whatever, expressed in the like standard form, 
is reduced to zero by the operator 
4 
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 ( , B 2 $yc, y) 2 , suppose : the equation to be satisfied is 
viz., we have 
x (Bpr-f^B.^y ) 
— 4 cy ( 2B 0 a:d-Bpy) 
— cc s 8B 0 — inySB, — y 3 SB 2 =0, 
B, — 8B 0 =0, 
2B 3 — ScB 0 — - SB 1 = 0, 
-406,-863=0 ; 
which (omitting, as we may do, the outside factor or) are satisfied by the foregoing 
values B 0 , B„ B 3 =Z>, e, — Sad — be. And if we assume only B„=Z>, then the first 
equation gives at once the value B, = e, the second equation then gives B 2 = — 3 ad — 3 be ; 
and the third equation is satisfied identically, viz., the equation is 
• 4co -j- S(3 g&c? + be) — 0, 
that is 
— 4(3(3 = 
— 4ce 
= 
+c8Z> 
+ c . 
e 
+6Se 
+ b. 
3/ 
+ Sa&d 
+ 3( — 
■bf+ce) 
which is right. 
Of course every invariant must be reduced to zero by the operation 8 : thus we have 
Table No. 97, 
a~g= 12 abd 
+ 4 b 2 c 
• + 1 e 3 , 
and thence 
crS(/= (12ad!+8&c)S6, = (12ad-\-8bc) e, 
+ 4& 3 .Sc + 4& 3 .3/ 
+ 12 ab . Sd +12 b( — bf+ce) 
+ 2e . Se +2e( — 6ad—10bc) 
which is =0, as it should be. 
MDCCCLXXVIII. 4 L 
ade b 2 f bee 
; + 12 + 8 
+ 12 
-12 +12 
-12 —20 
