NOTES AND REFERENCES. 
603 
53—60. XI (a) and (b), XII (a) and (b): unsymmetric, united. Is referred to in the 
next mentioned paper. 
Durfee. The Tabulation of Symmetric Functions, Amer. Math. Jour. t. v. (1882), 
pp. 348, 349. XII (a) and (b); symmetric, united. 
MacMahon. Symmetric Functions of the 13 ic , Amer. Math. Jour. t. VI. (1884), 
pp. 289—300. XIII (6); symmetric. 
Cayley. Symmetric Functions of the roots for the degree 10 for the Form 
1 + bx +y—5 + ... = (1 — ax)(l — /3x)(l — <yx) ... Amer. Math. Jour. t. vn. (1885), pp. 47— 
56. II to X (b), unsymmetric. The calculation of the tables for this new form 
(MacMahon’s) of the coefficients afforded a complete verification of the (b) tables, 
showing that there was not a single error in these tables as published in the 
Philosophical Transactions. 
Durfee. Symmetric Functions of the 14 ic , Amer. Math. Jour. t. ix. (1887), 
pp. 278—292. XIY (b) symmetric, the arrangement is different from and seemingly 
better than that in the tables XII (b) and XIII (b). 
MacMahon. Properties of a Complete Table of Symmetric Functions, Amer. 
Math. Jour. t. x. (1888), pp. 42—46. 
Memoir on a New Theory of Symmetric Functions, Amer. Math. 
Jour. t. xi. (1889), pp. 1—36. (a) and (b) Tables for the weights 1 to 6 and their 
several partitions. To explain this, observe that the general idea is to ignore the 
coefficients altogether, regarding them as merely particular symmetric functions of the 
roots: thus the (b) table for the weight 4 (partition l 4 ) is in fact the table IY (b) 
giving the symmetric functions (4), (31), (2 2 ), (21 2 ), (l 4 ) in terms of (l 4 ), (I s ) (1), 
(12)2, (i)2 ? (l) 4 , that is in terms of the combinations e, bd, c 2 , b 2 c, b A of the 
coefficients, but that the other tables weight 4 to a different partition, give the values 
of symmetric functions (combinations of the foregoing) which are expressible in terms 
of other symmetric functions of the roots: for instance weight 4 (partition 21 2 ) gives 
(4), (31), (2 2 ), and (21 2 ) in terms of (21 2 ), (21) (1), (2) (l 2 ) and (2) (l) 2 . A leading 
idea in this valuable memoir is that of the “ Separations ” of a Partition. 
150. The theory is developed in an incomplete form. If to fix the ideas we 
consider a quintic equation (a, b, c, d, e, f\x, l) 5 = 0, then a single equality a = ¡3 
between the roots implies a onefold relation between the coefficients (a, b, c, d, e, f): 
this is completely and precisely expressed by means of a single equation (V = 0, 
where V is the discriminant, = a 4 / 4 + &c.). Similarly a system of two equalities 
a — /3 = y, or a = /3, y = S as the case may be, implies a twofold relation between 
the coefficients (a, b, c, d, e, f) and the question arises, to determine the order of 
this twofold relation, and to find how it can be completely and precisely expressed, 
whether by two equations A = 0, B = 0, or if need be by a larger number of equations 
A = 0, B — 0, (7 = 0, &c. between the coefficients; this is not done in the memoir, 
76—2