HISTORICAL SKETCH. 
13 
A misprint may be noted in 
It should read 
= 1 . 
In 1887 Durfee published in the American Journal of Mathematics 
the fourteenthic arranged according to his second (dictionary) plan and 
checked not only as was his twelfthic, but also by MacMahon’s formula (17). 
In 1898 a formula 10 was published by Metzler which, when the value in 
terms of the a’s of say ^a\ala\a° 4 is given, makes it possible directly to write 
all those terms in say 2a? _0 at _1 a i~ 2 a\~ z , involving only the coefficients 
di, d 2 , d 3 , d 4 . 
s From the equation for ^,alalala 4 in the homogeneous form 
i a\a\a\u . 4 = d 3 d 2 d x d 0 — 3d%d% — 3d 4 d\d 0 + 4a 4 a. 2 a 0 + 7d 5 d x dl — 12a 6 tto 
is derived the equation 
V 4-0 4-1 4 
CL o rt - 
Z a 
^9 
~ 2 a\ 3 = d 4 _ 3 d 4 _ 2 d 4 . 
1&4-0 3d 4 - 3 d 4 . 
_ 4 cq _ icq _q -l - 4n 4 _ 4 n 4 _ 2 n 4 _o 
+ terms involving d 5 , d 6 , etc. 
giving in the nonhomogeneous form 
^a\alala 4 = d 4 d 3 d 2 d 4 — 3d\d\ — 3cc 4 a| + 4a 4 cc2 
+ terms involving d 5 , d G , etc. 
(19) 
During the year 1899 there appeared in the American Mathematical 
Monthly a series of articles on symmetric functions by Roe. Not content 
with Cayley’s remark as to the ease with which resultants could be calcu¬ 
lated if the appropriate symmetric functions were at hand, he carried the 
matter farther, identifying the problems of calculating all resultants and 
calculating all symmetric functions. From a consideration of the former 
problem he derives a complete set of formulas for the latter. He arranges 
them in three classes (fundamental, reduction, and normal) and expresses 
them by means of a new symbol 
(*) 
which represents the coefficient of (P) in [P]. 
His fundamental relations consist of the Cayley-Betti law of symmetry 
(10) and Metzler’s formula, (19). His reduction formulas consist of four 
which he designates as reduction (or derivation) formulas and one which 
he terms the formula for the completely reducible form. Of the four 
reduction formulas, one is Hammond’s formula, another provides for the 
calculation of [l x ], the third for the reduction of 
d% [x 4 x 2 . . . x n ] 
10 W. H. Metzler: Proceedings of the London Mathematical Society, vol. 28, 1897-8. 
