524 
MAJOR MACMAHOR OR SYMMETRIC FUNCTIONS 
Hence the ivhole number of asyzygetic functions of differences of a given weight is 
given hy 
(1 -*)(!- (1 - ... (1 - ... 
§ 13 . Special Fundamental Ident ity of Finite Order. 
72 . By taking the fundamental identity of infinite order syzygetic relations 
between monomial symmetric functions were avoided. Whenever the fundamental 
identity is taken of a finite order > 1 certain such relations of necessity arise. 
Professor Cayley (‘Collected Papers/ vol, 2, p, 454 , and ‘Phil. Trans.,’ 1857 ) 
takes a fundamental identity equivalent to 
(1 + (1 + ttoX -|- / 3 oy) = 1 + 2 hx--\- + 2 fxy + cy®, 
and finds identically 
he — hg‘^ ~ eld fi- ‘2.fgli = 0, 
the condition that the expression to the right shall break up into two linear factors. 
I take as the fundamental identity 
(l + ape (igj) (l + ctyx + = 1 a-^pc a^py + + eiipcy -j- a^yf, 
and observe that the syzygetic relation must connect monomial symmetric functions, 
each of which is symbolised by a partition containing more than two biparts. The 
symmetric functions must he of the same biweight of the form pp since the quantities 
a must occur symmetrically with the quantities y8. Of the blweight 1, 1, there exists 
no partition containing more than two biparts. Of the biweight 2,2, we have the 
four partitions 
(2u dC), (02 id"), (TTIddT), (id-dT^), 
and if the corresponding symmetric functions can be linearly connected, so that no 
fundamental symmetric function of weight greater than 2 occurs, the linear function 
must vanisli. 
From the tables, yios? § 14 , liiweight 22, partition (10^ OH), we find 
(20 dl^) — (IT Id 01) + (02 Id^) = — 4a3yrto3 + 5 
the terms involving (:%, a^-y, and a^^ disappearing. 
