A TENTH MEMOIR ON QUANTICS. 
361 
693] 
It is obvious that for every covariant whatever written in the denumerate form 
(I 0 , y) & , the second coefficient is equal to the first coefficient operated upon 
by 8 ; so that the ' foregoing formulae give, in fact, the second coefficients of the 
several co variants. 
381. It is worth noticing how very much the formulae of Table No. 97 simplify 
themselves, if one of the covariants 6, c, d, e vanishes, in particular, if b vanishes. 
Suppose b — 0; writing also (although this makes but little difference) a = 1, we have 
a = 1, 
6=0, 
c — c, 
d = d, 
e = e, 
f - = — d — 4c 3 , 
9 = e 2 , 
h = 6cd + ef, 
i — ce, 
j = 9 d 2 4- ce 2 , 
k = 3 de, 
l = — 3df+ 2 c 2 e, 
m = 9 cd 2 + 3 def— c 2 e 2 , 
n = — Qcde — e 2 f, 
o = 9 d?e + ce 3 , 
p = — 9 df +12 c 2 de + ce 2 /, 
q =- 54cd 3 - 27d 2 ef+ 18c 2 de 2 + ce 3 /, 
r = 9cd 2 e + 3 de 2 /— c 2 e 3 , 
s = — 27d 3 / + 54c 2 d 2 e + 9cde 2 f— 2c 3 e 3 , 
t = - 8Id 4 /— 6d 2 e 3 + 216c 2 d 3 e + 54cd?e 2 f- 24&d& - c 2 e 4 /, 
u =-27d 5 - 18cd 3 e 2 - 4d 2 e 3 /+ c 2 de\ 
v = - 81 d 4 e/- 6d 2 e 4 -1- 216c 2 d 3 e 2 + 54cd?e/- 24c 3 de 4 - lc 2 e 5 /, 
w (not calculated). 
These values are very convenient for the verification of syzygies, &c. Take, for instance, 
the before-mentioned relation 8v = — 6dt — Gmr + nq, that is, if V = (V 0) Fi$/ y), then 
V 1 = - 6dt - 6mr + nq: calculating the three products on the right-hand side, observing