§42] 
SYMBOLIC NOTATION 
67 
Covariants as Functions of Two Symbolic Types, §§ 42-45 
42. Any Covariant is a Polynomial in the a x , (a/3). This 
fundamental theorem, due to Clebsch, justifies the symbolic 
notation. It shows that any covariant can be expressed in 
a simple notation which reveals at sight the covariant property. 
While a similar result was accomplished by expressing 
covariants in terms of the roots (§36), manipulations with 
symmetric functions of the roots are usually far more complex 
than those with our symbolic expressions. 
The nature of the proof will be clearer if first made for 
a special case. The binary quadratic a x 2 has the invariant 
K = aoa2 — ai 2 
of index 2. Under transformation T of § 40, a x 2 becomes 
(« £ Xi+a,X 2 ) 2 =^4oA’i 2 + . . ., +o=«£ 2 , A 1 =a^a v , A 2 =a 2 . 
Hence A 0 A 2 -Ai 2 equals 
<*£ 2 0, 2 —affifOnfir, = (£v) 2 K. 
We operate on each member twice with 
(1) v= _& 
9£i9*?2 9&941’ 
and prove that we get 6(a0) 2 = 12K, so that K is expressed 
in the desired symbolic form. We have 
i£v) = £142 — ¿2171, 
+ (?l) 2 = 2(£,) ii, ++0) 2 = 2(iu) +2i)2 (1, 
dv2 dkioV2 
-~-{^v) 2 = —2(^77) ¿2, ^ — (^) 2 = -2(^??)+2r7i^2, 
9171 9 £29*71 
F(^) 2 = 6(^), F 2 (^) 2 = 12, 
since F(^rj) =2, by inspection. Next 
(2) F« £ j8, = F(ai £1 | of2¿2)(0m +02772) =«i02 £*20i = («0)•