108 MATHEMATICS: 0. E. GLENN 
there results a covariant^ 
p-2 0--1 
[K,\ = [Co + (K,) ] xi*- ' + 2 S (2) 
r=0 s=0 
where (Ki,) is the pure invariant 
(Kj,) = Cp^i + C2(/,-i) + . . . + C(o--i)(^_i). (3) 
Now if we form the product of any two binary forms, as of and 
g„ = (bo, hi, . ., hnlxi, X2Y {m ^ n), 
where w + ^ is a number of the form <t {p — 1), and construct the formulas 
analogous to (2), (3) for the result, we get 
j(P-l)<T-l 
(fmgn) = 2 ^^i^J(P-^)-i' 
i=(\ j=i 
p-2 0--I t 
[fmgn] = Mo + {ffngn)]Xi^~^ S 2 ^ ^i^(<r-s)(p-l)-r-i ^l^'^'^^i' 
r=Q s=0 i=0 
This process of constructing the concomitants (fmgn), Umgn] from the product 
fm • gn is analogous to transvection and symbolical convolution in the theory of 
algebraic concomitants. 
An example of the co variant (1) is obtained from the following semin vari- 
ant^ of a quadratic form /2: 
o P-'i- P 
o = ao ai — ai . 
This seminvariant satisfies the two necessary and sufficient conditiofis that it 
may be the leading coefficient of a co variant, viz., of 
Ci = {a^Xi + aioc^^ — {a^%\ + a\X^{a^x^ + la^XiXi + (HX'f)^~^^ (4) 
where powers of Xi are to be reduced by Fermat's theorem. 
2. Concomitant scales. — If K is any modular co variant of order {rp — v) 
X (p — 1) = m, then by application, to Ky of the latter of the two modular 
invariantive operators 
7-. p ^ , p ^ \ \ p ^ 
E = ao"^ — + a/ h ..+<-—, 
dao dai oa^ 
p ^ , p ^ 
8X1 5X2 
there is obtained a set or scale of/x = ^+ ?^+2 concomitants 
(K), cc^ [K]{\ = 0,1, . .,p- 1), c'K = 0, 1, . . . , v). (5) 
This set, which we call a ii-adic scale^ for K, is analogous to the set obtain- 
able by Convolution from a symbolical algebraic covariant 
Tl{ah)alblcl"-- , 
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