638 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
Table No. 98 (concluded). 
U as an invariant was divided and segregate, U = a 18 u 
12.0 
V divided and segregate is 
Y=a l % 
- 4-6 
V + 1 
bgr - 
5 
bjo — 
10 
cjjh + 
5 
3 s ~ 
12 
nq - 
9 
13.1 
14.4 
where the fractional coefficient is = 
dt — 
6 
vir — 
6 
nq + 
1 
W as an invariant was divided and segregate, W —a 27 w 
18.0 
Derivatives. — Article Nos. 382 to 384, and Tables Nos. 99 and 100. 
382. I call to mind that any two covariants a, b, the same or different, give rise to 
a set of derivatives (a, l) 1 , (a, by, (a, b) 3 , &c., or, as I propose to write them, ab 1, ab2, 
ab3, &c., viz. : 
ab\=d x a . d y b- 
ab2 = d/c 
cl, .a . dj) 
d/a . 
d/b 
_ ~a . d/b — 2 d x d y a . d x d y b-\- 
db3—d/a . d/b—3d/d y a . d, t d/b-\-3cl c d/a . d/d/)—d/a . d/b 
&c. 
or, as these are symbolically written, abl = \2a 1 b. 2 , ab2 = 1 2~aJj 2 , ab3=12 s a 1 b 2 , &c. ; 
where 12 = Ano — Ann, =— — y- the differentiations -f-, ~r applying to the a x 
dxy cly % dx^dy/ dot/ dy x J & 
and the — , y applying to the b z , but the suffixes being ultimately omitted : hence 
(I d- Q (t lj Cf 
if 0 be the index of derivation, the derivative is thus a linear function of the differ- 
ential coefficients of the order 6 of the two covariants a and b respectively : and we 
have the general property that any such derivative, if not identically vanishing, is a 
covariant. If the a and the b are one and the same covariant, then obviously every 
odd derivative is =0 ; so that in this case the only derivatives to be considered are 
the even derivatives act 2, aaA, &c. : moreover, if the index of derivation 0 exceeds the 
