§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