406 ON A THEOREM IN COVARIANTS [527 
degree by a number which is greater than \n — 1: in fact, suppose it inferior in 
regard to 1, then the order is 
(n — ctj) + (n - cr 2 ) + .. + (n — cr m ), 
where each term after the first is at least = 1, that is, the order is at least 
= n — (T 1 +m — 1; hence order — degree is at least = n — cr x — 1; viz. being less than 
\n, this is greater than ^n— 1. 
Conversely, if for any symbol order-degree is —1, then the symbol is not 
inferior. 
A symbol [12... to] is sharp when any index is otherwise it is flat; viz. 
this is so when each index is < \n. The symbol is sharp as to any particular duad 
or duads when the index or indices thereof is or are each of them > \n. 
The subsidiary theorem is now as follows: “ A symbol is inferior or sharp: or it 
can be expressed as a sum of symbols each of which is inferior or sharp ”—or what 
is the same thing, the only symbols which need to be considered are those which 
are either inferior or sharp. 
Thus for the degree 1 the symbol is [1] (which is simply unity) <t x = 0, and the 
symbol is inferior. 
—fc 
For the degree 2 the symbol is [2], = 12 ; if k<^n the symbol is inferior, if 
k then it is sharp. 
• n . •. * £ ~y 
A proof is first required for the degree 3, here [123] = 12 13 23 (/3 + 7, y + a, 
a + /3 each = or < n) which may very well be neither inferior nor sharp; for instance, if 
2 2 2 
n = 5, we have 12 13 23, where each index being = 2, the symbol is not sharp; and 
each index-sum being = 4 the symbol is not inferior. But writing the symbol in the 
2 2 2 
form V 1 V 2 V 3 12 13 23, then by means of the relation 
Vj. 23 + V 2 .31 + V 3 .12 =0, 
(or, what is the same thing, Vj. 23 = V 2 .13 — V 3 .12), the symbol becomes 
V 2 V 3 12 2 13 2 23 2 (V 2 . 13- V 3 . 12), 
= V 2 3 V 3 12" 13 3 23 — V 2 V 3 2 13 3 12* 23, 
where each term, as containing an index 3, is sharp. To complete the reduction, 
observe that calling the expression 21 — 33, then in the term 21 interchanging the 
numbers 2 and 3 we obtain 21 = — 33, and thence 21 — 33 = 221; so that the whole is 
2 V 2 2 V 3 12 13 23, viz. it is a multiple of 12 13 23. 
I prove the general case, substantially in the manner used by Dr Clebsch, as 
follows. We assume that the theorem is proved up to a particular degree m: that 
is, we assume that every symbol belonging to a degree not exceeding m can be