527] 
ON A THEOREM IN COVARIANTS. 
405 
We may omit the operand UfJ 2 ... TJ m , and consider only the symbol 
k (12 12 IF ...) or V »-*.... V(12 13* IF. 
which, under either of the two forms, I represent for shortness by [12... m] : observe 
that this is considered as a symbol involving the m symbolic numbers 1, 2, 3... m, 
even although in particular cases one or more of these numbers may be wanting from 
the actual expression of the symbol: thus [123] may denote 12 , but the operand to 
be supplied thereto is always TJJJ 2 U Z . 
A sum of symbols is not in general equal to a single symbol: but a single 
symbol can be expressed in a variety of ways as a sum of symbols: the most simple 
transformation-formulae relate to three or four symbolic numbers; viz. for three such 
numbers, say 1, 2, 3, we have 
Vj. 23 + V 2 .31 + V,. 12 = 0, 
showing that in a symbol, which written with the V’s involves V z .23, this may be 
replaced by its value V 2 .13 —V 3 .12; and so in other cases. 
For the four numbers 1, 2, 3, 4 we have a group of the like formulae 
- V 2 .34 + V 3 .24- V 4 .23 = 0, 
Vj.34 . - V 3 . 14- V 4 .31 = 0, 
-V 1 .24+V 2 .14 . — V 4 .12 = 0, 
V 1 .23+V 2 .23+V 3 .31 . = 0, 
leading to 
23.14 + 31.24 + 12.34 = 0, 
which is a form not involving the V’s and consequently is applicable to the trans 
formation of invariant-symbols where the numbers 
n — ctj, n — cr 2 , ..., n — a m 
are all = 0. 
I establish the following definitions: 
A symbol [12... m] is proximate when each index-sum is <n\ otherwise it is 
ultimate; viz. this is the case when any one or more of the index-sums is or are 
= n. We may say that the symbol is ultimate as to 1 if (r x = n\ and that it is 
ultimate as to 1, 2 if <j 1 and <x 2 are each = n : and so in other cases. 
A proximate symbol which has any one index-sum thereof < is said to be 
inferior: thus if cr 1 < the symbol is inferior in regard to 1; and so if <r z and a 2 
are each < \n, it is inferior in regard to 1 and 2: and the like in other cases. 
Observe that if a symbol is inferior then in the covariant the order exceeds the