602 
NOTES AND REFERENCES. 
147. Upon looking at any one of the Tables, for instance VIII (a), it will be 
noticed (1) that the partition symbols in the outside top line and left-hand column re 
spectively are differently arranged, (2) that the numbers of each pair of equal numbers 
(see the Memoir) are not symmetrically situate, and (3) that the table is what may be 
called a half-square ; viz. the squares above (or, in the case of a (b) table, those below) 
the sinister diagonal are all vacant ; the squares in the sinister diagonal itself are all 
occupied by units (+1 or — 1 as the case may be). It is possible (and that in many 
ways) to give the same arrangement to the partition-symbols in the outside line 
and column respectively, and at the same time to retain the half-square form of the 
table : or (what is far more important) we may with Faà di Bruno, give the same 
arrangement to the partition-symbols, and at the same time make the table sym 
metrical, viz. cause the two numbers of each pair of equal numbers to be sym 
metrically situate in regard to the dexter diagonal of the square—but we cannot at 
the same time retain accurately the half-square form of the table. The general 
principle is that in the outside column (or line) the partition-symbols which are 
conjugate to each other have symmetrical positions, while the self-conjugate symbols 
are collected at the middle of the column (or line); there is then in regard to these 
self-conjugate symbols a sort of dislocation of the sinister diagonal, the units which 
belong to them being transferred to the dexter diagonal, and in the sinister diagonal 
replaced by zeros, for instance at the crossing of the two diagonals we may have 
10 l 
instead of 
0 1 l 
Again as remarked by Fiedler, the two corresponding tables (a) and (b) may be 
united into a single table ; the sinister diagonal is the same for each of them, and 
if we then insert into the (b) table below the sinister diagonal the numbers of the 
(a) table, we have a table which is to be read according to the lines for the numbers 
above and in the sinister diagonal ; and according to the columns for the numbers 
in and below the same diagonal. This may be called a United table: it may be 
unsymmetric, or be rearranged so as to be made symmetric. 
The tables have been rearranged as above, and extended to the order 14 : I give 
the following references. 
Fiedler. Elemente der Neueren Geometrie &c. (1862), pp. 73 et seq. (II. to X, 
(a) and (b) united, unsymmetric). 
Faà di Bruno. Sur les Fonctions Symétriques, Coviptes Rendus, t. 76 (1873), pp. 
163—168 (II to VIII, (b), symmetric, there is some error in VIII, inasmuch as it 
is presented without the dislocation of the sinister diagonal). 
Théorie des Fonctions Binaires, 8vo. Turin &c. 1876. II to XI (b) sym 
metric. 
Durfee. Tables of the Symmetric Functions of the Twelfthic, Amer. Math. Jour. 
t. v. (1882), pp. 45—60. XII (a) and (b) unsymmetric. 
Behorovsky. Tafeln der symmetrischen Functionen der Wurzeln und der Coeffi- 
cienten-Combinationen vom Gewichte eilf und zwolf. Wien, Denks. t. 26 (1883), pp. 
A Table thus arranged may be called Symmetric.