652 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
Moreover, as regards the covariants AA2, AA4, &c., we take what are properly the 
half-values, 
he. of AA2 = A 0 A. 2 — A 4 3 (instead of A 0 A 3 — 2A 1 A 1 + A 2 A 0 ), 
,, „ AA4=A 0 A 4 ,— 4A 1 A s -f-3A 3 2 (instead of A 0 A t — dA^g+GAoAo — 4A 3 A L — A 4 A 0 ), 
&c., 
and similarly 
he. of BB2 = B 0 B 3 - (PO 3 
„ „ CC2=C 0 .*C 9 -(iC 1 ) s 
&c. 
Any one of these leading coefficients, for instance l.c. of AC2, is equal to the 
corresponding covariant derivative, multiplied, it may he, by a power of a • the index 
of this power being at once found by comparing the cleg-orders, these in fact differing 
by a multiple of 1.5 the deg-order of a. Thus 
act 2, A 0 A 3 — A 4 3 cleg-orders are 2.6, 2.6 or aa2=A 0 A 3 — A 4 3 , 
act A, A 0 A 4 — 4 A , A., + 3 A A, cleg-orders are 2 ‘2, 4T2 or ««4 = (A () A t — 4 A, A ;; + 3 AT) ; 
we have in fact 
A 0 A,— A 1 3 =l.c — 0 3 =c 
A 0 A 4 — 4A 1 A 3 + 3A 2 3 = 1 . (cdb — 3c 2 ) — 4 . 0 ./ + 3 . c 3 , = a 3 & 
and aa2=c, 
and aai = b. 
As another instance, and for the purpose of showing how the calculation is actually 
effected, consider the derivative ch2, which is to be calculated from the leading 
coefficient of OH2, ==C 0 .pL — 2 .^C 1 .p^ 4 ~T 5 'C' a .H 0 : this is 
= c {h/dg — 2abd — eh) 
~2 • — Z) 
+ (iad>~c 2 )h 
= column next written clown ; but this column contains congregate terms which have to 
be replaced by their segregate values (see Table No. 96, deg-order 8T6) ; and we thus 
obtain 
cvj 
crb s 
erbh 
a~cg 
abed 
W 
cVi 
i cm, 
+W C 9 
+i 
~2 abed 
o 
-w 
a Ah 
+ 3 
+ 2 
o 
Li c n 
+ fl 
i 
3 
i 
3 
4~t 
i 
3 
— 1 
— 2 
+ 2 
i _i mil L 
3 3 i~ 3 O 6 
0 
0 
0 
