OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
483 
Observe that the summation is in regard to the expressions obtained by permuting 
the n suffices 
1, 2, 3, . . . n. 
The weight of the function must be considered as bipartite; it consists of the two 
numbers 
Pi + i-’a + + • • • = 
</i + + ^3 + • • • = 
and I speak of the biweight %q. 
The sum Sp + Xq may be called the whole weight, or simply the weight. Asso¬ 
ciated with any number tv there will be a weight tv and a biweight corresponding to 
every composition of tv by means of two numbers, including zero as a number. By 
composition is meant partition, in which regard is paid to the order of the parts; for 
instance, 21 and 12 are different binary compositions of 3, and 30, 21, 12, 03 
constitute the system. 
3. It is necessary to introduce the notion of the partition of the bipartite number 
which denotes the biweight. 
Thus of the biweight Sp, %q the expression 
{Pi<hPi<l^Pz% • • •) 
may be termed a partition. 
The dual symbols qt-^q-^, P^Ptz^ • • • ^^e the parts of this partition; the parts 
are themselves bipartite and may be termed biparts. 
We have thus a biweight denoted by a bipartite number partitioned into a number 
of bipartite numbers termed bi parts. 
It is convenient to arrange the biparts so that the sums of the symbols which 
compose them are in descending order of magnitude from left to right. 
According to usual practice repetitions of biparts are denoted by power symbols; 
thus 
{P\(li) = {PiPlh<li)- 
4. In the notation just explained the fundamental relation is written 
(1 + a^x + l^^y) (1 + a^x + ^.qy) . . . 
= 1 + (lb) X + (01) y + (Id^) + (To bl) xy + (oT^) / 
+ (10^) x^ + (I 0® Ol) x^y -f- (10 01^) xy® + (Off)y® + . . . 
where (10 01®) denotes % and in general = (10^ 01^?). 
Observe that here the number of quantities m each system is considered to be 
infinite, and that the right hand side of the equation is taken with unit and not 
3 Q 2 
