508 MAJOR MACMAHON ON SYMAIETRIC FUNCTIONS 
Theorem. 
“If 
idh<lTIMP . . .)2 = • • • + pC C ••• + ••• > 
the cofactor symmetric function P is unaltered when the partitions P-Mp . ■ .), 
{r-^sp . . .), are interchanged.” 
The function P presents itself in the first place as a linear function of separations of 
the partition of the 6 product to which it is attached. The theorem supplies linear 
functions of separations of any two partitions [pPlP • • •)’ • • •) respec¬ 
tively, of the same biweight, which are equal to one another. 
46. To make the matter clear, form a table of biweight 21 as follows :— 
{ 21 );. 
(20 01)jj 
(n io), 
which is to be read by rows.'"' 
Each term in a column is a separation of the partition of the h product at the head 
of the column. 
The separations in each line of terms as written possess the same specifications, and 
also the same numerical coefficients. In the right hand column the partition 
separated is a fundamental symmetric function, and hence each separate therein 
apjiearing is so also. Each block of separations in the right hand column is the 
expression by means of fundamental symmetric functions of the monomial symmetric 
function of the same elements whose partition appears to the left of the same line. 
The terms of the first three columns may be regarded as being formed according to 
the same law as the right hand column, and therefore according to a law defined by 
^21 ^ 20^01 ^ 11^10 
(21) 
(20 01) 
- m (^) 
(11 10) 
-(ii)(To) 
(10® 01) 
- (To®) ( 01 ) 
— (To oT) (To) 
+ (To)® ( 01 ) 
- (21) 
— (^bl) 
- (IT Ib) 
— (ib^bl) 
- M (01) 
- (To®) (oT) 
+ (ii)(To) 
-h (Tb bl) (Ib) 
-(21) 
— (20 bl) 
-(IT TO) 
— (Ib- bl) ' 
+ (20) (01) 
+ (To®) (n) 
1 
(^) 
(2b bl) 
(nio) 
(Ib^ bl) 
* Eacli term in the left hand column is equal to the aggregate of terms in any block in the same row. 
