8 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
Cayley used the term conjugate partition, due to Ferrers, the meaning 
of which may be shown by the following example exhibiting its calcula¬ 
tion. To get the partition conjugate to 7, 3, 2 2 a row 
of seven dots is written; under these, beginning at the 
left, a row of three, under these two rows of two dots 
each; then the dots in the columns are counted and . . 
found to be 4, 4, 2, 1, 1, 1, 1 respectively, and accord- 4 ; 4 ) 2, 1, 1, 1, 1 
ingly 4 2 , 2, l 4 is called the conjugate partition of 7, 3, 2 2 . 
In general the partition conjugate to x u x 2 , . . . x n is 
l* 1 -* 2 , 2 K2 ~ K3 , ■ • • (ft— ft*n 
Cayley’s theorem in regard to the sinister diagonal elements may now 
be given the form, 
coefficient of (P) in [Q] is ( — 1)“ (9) 
if Q is the partition conjugate to P. 
Cayley noticed a symmetry in the constituents of the tables, observing 
that 
coefficient of (P) in [P] is same as the coefficient of (P) in [P] (10) 
In the following year the Italian Betti 6 independently observed this 
symmetry and showed the necessity for it, since which time it has been 
known as the Cayley-Betti law of symmetry. 
Cayley reduced the number of coefficients to be calculated in a table 
by observing that a k would not occur in any symmetric function involving 
less than k numbers in its partition: for example, in the fifthic a 4 occurs in 
[1 3 2] and [l 5 ] only. He observed that it is possible to keep the sinister 
diagonal elements ( — 1)™ by arranging the partitions representing the sym¬ 
metric functions in the same order as those representing the coefficients, 
giving the table a triangular form. He saw also that the table could be 
made symmetrical by arranging the self-conjugate partitions (such as 4 3 3) 
at the middle, in one order for the functions and in the reverse for the 
coefficients; but that the table could not at once possess both properties. 
He expressed his preference for the latter method. 
Cayley called attention to the ease with which resultants could be 
calculated if the appropriate symmetric function tables were at hand. 
He corrected the score of mistakes in the Hirsch tables, making them 
reliable for computation. They may be found in Salmon’s “ Modern 
Higher Algebra.” 
Faa Di Bruno, who a little later took up the work, had a formula 
giving multipliers for the constituents in a row or column such that the 
sum of the products (except in the case of the row or column with the 
9 M. Betti: Annales de Tortolini, Rome, 1858. 
