527] 
ON A THEOREM IN COVARIANTS. 
407 
expressed as a sum of terms each of which is sharp or inferior: and we have to 
prove this for the next following degree m +1, or writing for convenience p in place 
of to + 1, (say for the degree p); that is, for a symbol 
[12... mp], = pi p2 2 ... pm [12 ... to] 
= P[12...to] suppose. 
I write as before <r l , <r 2 , ...,c m for the index-sums of [12... to]: those of [12 ...mp] are 
therefore <r x + X 1} cr 2 + X 2 ,.., cr m + X OT , and (for the duads involving p) <r p = X x + X 2 ... + X m . 
If [12. . to] is sharp, then [12.. mp] is sharp, and the theorem is true. 
If a p < ^n, then [12.. . mp] is inferior in regard to p; and the theorem is true. 
The only case requiring a proof is when [12 ... m] is not sharp (being therefore 
inferior) and when <r p is > \n. And in this case if any one of the indices X OT 
is S (or say if P is sharp) then the theorem is true. 
Consider the expression 
where a x , <x 2 , .., a m are as before the index-sums for [12..to] and therefore the numbers 
7l <t x —- X [, .., 'll o’jfi 
are none of them negative. 
Assume that when [12... to] is inferior, and when \ x ... \ m have any values such 
that their sum is not greater than a given value <7^ — 1, the expression is a sum of 
terms each of which is inferior or sharp: we wish to show that when \ x + X 2 ... + \ m 
has the next succeeding value, =<r p , the case is still the same. 
For this purpose, introducing the V’s I write 
then supposing for a moment that X x is not =n — <r 1 and X 2 not = 0, the expression 
contains the factor V x ._p2, which is equal to and may be replaced by — V 2 .pl + V p . 12: 
we have thus 
Q — Q + il, 
where omitting the V’s 
Q' = jP1 ..pm [12...to], 
0=^191' p2 pS ..pm 12 [12...to]. 
Now for Cl the sum of the indices X 1} X 2 — 1, X 3 ..X m is cr p — 1, so that by hypothesis 
il is inferior or sharp: that is, the difference Q — Q' is inferior or sharp: so that to 
prove that Q is inferior or sharp, we have only to prove this of Q', where Q' is 
derived from Q by increasing by unity the index of pi, at the expense of that of p2