526 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
ttj, ag,. . . and the other only with the single system /Sg, A) • • • 
therefore, suppress altogether the zero elements in the biparts and then proceed to 
form the tables for the several partitions (unipart) of the weight which I have 
already set forth in the ‘ American Journal of Mathematics ’ (vol. 11). 
C)f the remaining biweights 31 , 22 , 13 , it is merely necessary to calculate the two 
former, since the tables for the hiweight 13 are obviously immediately obtainable from 
those of the biweight 31 by interchanging the elements of each bipart :— e.g., by 
writing (jy) for pq. 
There are seven partitions of biweight 31 , viz. :— 
(Td^di), (iTTb'^), (^To 01 ), (^TI), (mTo), (30 dl), ( 3 l), 
and nine of the biweight 22, viz. :—■ 
(Td^dl^), (TTIddT), (IT®), (02Td^), (Mdd), (Idl'd), (22). 
( 2 d dl^), (21 dl), 
Of these the table of (20 01^) gives also the table of (02 10^) by transposing the 
elements of the biparts, and similarly the table of (21 01) gives that of(l210). We 
have thus 28 tables; but of these, the four corresponding to the partitions ( 31 ) and 
(22) are mere identities, so that the number is reduced to 24 . The earlier tables 
which are necessary are those of the partitions (10 01), (20 01), (11 10), (10'^ 01). 
Tliese are now given. Each table is read according to the lines. 
Biweight 11. 
Partition (10 d I). 
(10 01) (To) (OT) 
(H) 
— 1 
1 
1 
{10 dl) 
1 
(2d dl) 
(20) (Of) 
(21) 
— 1 
i 
1 
(2ddi) 
1 
1 
(id 01) 
Bl WEIGHT 21 , 
± 1 
(20 01 ) 
(20)(oT) 
(H) 
(10 01) 
1 ! 
1 
1 
(2l) 
(dddl) 
1 
1 
1 
