OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
533 
§ 15. Property of the Coefficients in the Tables. 
76. There is in regard to the coefficients a very simple and important property 
which does not come into view with the unipartite theory so long as the tables are 
restricted to the j)articular cases in which the separable partitions are composed 
entirely of units. The property appears the instant we consider a separable partition 
composed of parts which are 7iot all similar. The law is the same whether the 
symmetric functions are unipartite, bipartite, or in general 7)i-partite. It depends 
upon the possibility of grouping the various separations in a particular manner. To 
make this clear suppose we are presented with a separable partition (10^ Offi). The 
nine separations may be written down in four groups, as follows :— 
Group 1. 
{Tof (blf 
(n) (To) <oT) 
(uf 
Group 2. 
(To dl^) (10) 
(dT2) (Toy 
Grou]) 3. 
(rd- dl) (dT) 
(ud®) (df)^ 
Grorip 4. 
(Id" dl^) 
(id'^) (dl'^) 
In Group 2 it will be seen that the parts (10-) of the separable partition occur in 
the separation (10)®, while the parts (01®) occur in the separation (01®), so that the 
expression {(10)®, (01®)] may be taken as defining a certain separation property of the 
separations of the group. The group in question may be denoted by Gr {(10)®, (Ol®)] 
and on the same principle the Groups 1, 3, and 4 may be denoted by 
Gr {(Id)®, (dl)®}, Gr {(Id®), (Ol)®}, Gr {(Id®), (Ol®)} 
respectively. In the separable partition (10® Ol®) the parts (10) and (01) occur each 
twice, and a group results from every combination of a partition of 2 with a partition 
of 2. If Pg denote the number of partitions of 2, the number of groups will be 
Pg® = 4. In general if the different parts of the separable partition occur a, b c, .. . 
times the number of groups of separations is P^P^Pc .... 
77, The leading property that has been adverted to is that in the expression of a 
single-bipart function by means of separations of a partition composed of dissimilar 
parts, the algebraic sum of the coefficients in each group of separations is zero. A 
corollary at once follows which will be given in its proper place. 
78, From the identity of Art. 24, viz. : — 
^ + • • • + + • • • = (1 -j- -f- . . . -h Ct./ -f- . , .) 
is derived a series of relations which ex])ress the quantities c in terms of the 
