693]
A TENTH MEMOIR ON QUANTICS.
391
384. The Canonical form (using the divided expressions, Table No. 98) is peculiarly
convenient for the calculation of the derivatives. Some attention is required in regard
to the numerical determination : it will be observed that A is given in the standard form
(A 0 , A 1} A 2 , A s , A 4 , Arfax, y) 5 , while the other covariants are given in the denumerate
forms B = (B 0 , B 2 , B 2 \x, y) 2 &c. : these must be converted into the other form
B = (B 0 , \B U B^x, y) 2 , (7= (Co, t V(7 4 , $C s , G 6 $x, y) 6 , &c., the numerical
coefficients being of course the reciprocals of the binomial coefficients. We thus have,
for instance, the leading coefficients,
but
l.c. of AG2 = A 0 .JgC a -2. A^^ + At.Co,
„ „ BC2 = B 0 . J '- E a 2 -2.±B 1 .±C 1 + B 2 .C 0 .
Moreover, as regards the covariants A A 2, AA4<, &c., we take what are properly the
half-values,
l.c. of AA2 = A 0 A 2 — A 2 2 (instead of A n A 2 — 2A l A l + A 2 A () ),
„ „ AA4> = A 0 A 4 — 4>A 1 A 3 + 3A 2 2 (instead of A 0 A 4 — 6A 2 A 2 — 4iA 3 A 1 — A 4 A 0 ),
& c.,
and similarly
l.c. of BB2=B 0 B. 2 -(^B 1 ) 2 ,
„ „ CG2 = C 0 .J 5 G 2 -(%G 1 ) 2 ,
&c.
Any one of these leading coefficients, for instance l.c. of A(72, is equal to the
corresponding covariant derivative, multiplied, it may be, by a power of a: the index
of this power being at once found by comparing the deg-orders, these in fact differing
by a multiple of 1.5 the deg-order of a. Thus
aa2, A 0 A 2 — A-?, deg-orders are 2.6, 2.6 : or aa2 = A 0 A 2 — Af,
(Wi4, A n A 4 — 4AiA 3 + 3A 2 2 , deg-orders are 2.2, 4.12: or aa4 = (A 0 A 4 — 4>A 1 A 3 + 3A 2 2 ) ;
we have in fact
A 0 A 2 — A 2 2 = 1. c — 0 2 = c : and aa2 = c,
A 0 A 4 — 4AiA 3 + 3A 2 2 = 1. (a 2 b — 3c 2 ) -4.0 ./+ 3. c 2 , = a 2 b : and aa4> = 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 GH2, = G„. $H 2 — 2. ^G 2 . \H 2 + y$G 2 . H 0 : this is
= c {^a 2 g — 2 abd — ch)
- 2. y{^be -1)
+ (ia 2 b - c 2 ) h
