§40] SYMBOLIC NOTATION
40. General Notations. The binary n-ic
f=aoXi n +naiXi n ~ 1 x 2 + . . . + (^ja t x l n ~ k x 2 k + . . .+a n x 2 n
is represented symbolically as a x n = (3 x n =. . . , where
a x =otiXi +«2X2, Px = PiXi~\~I3 2 X 2 , . . . ,
ai n — ao, oci n ~ 1 a 2 =ai, . . ., ai n ~ k a 2 k = a k) . . . ,
«2 n = a n ; /3i n = ao, . . . •
A product involving fewer than n or more than n factors a 1,
a 2 is not employed except, of course, as a component of a
product of n such factors.
The general binary linear transformation is denoted by
T: #1 = £1-^1+771X2, x 2 = £2X1 + 172X2, (£17)5^0
7
where (£17) = £1172 — £2171. It is an important principle of com
putation, verified for a special case at the end of § 39, that
T transforms a x n into the nth power of the linear function
(ai £1 +«2 £2)X\ + (parli -\-a 2 rj 2 )X 2 =OL£Xi-\-oi, 1 X 2 ,
which is the transform of a x by T. Further,
(1)
«É «H
ai a2
£1 ’ll
ft ft
ft ft
£2 J72
= («/3)(£t?),
where (a/3) =oci(3 2 —a 2 f3i = — (/3a). Thus
~ a,,/3j) n = (£??) n (a/3) n ,
so that (a/3) n is an invariant of a x n =p x n of index n. Since
(/3a)' n represents the same invariant, the invariant is identically
zero if n is odd.
EXERCISES
1. (a/3) 2 is the invariant 2(a 0 O2—#i 2 ) of a x 2 = p x 2 .
2. (a<3) 4 is the invariant 21 of a x i =p x i (§ 31).
3. (a/3) 2 (/37) 2 (7a) 2 is the invariant 6/ of a x i =/3 x i =y x i (§31).
4. The Jacobian of a x m and p x n is