482 
MAJOR MACMAHON OR SYMMETRIC EURCTIORS 
2. Instead of taking a product of binomial linear functions of one variable, as 
above, we can, for m systems of quantities, take a product of non-homogeneous linear 
functions of m variables, and each such linear function may be taken of the form 
1 “b ^s\^\ “h d” • • • “b ^sm^m • 
As indicative of this general case it is sufficient to consider merely the case of two 
systems of quantities. Complexity of formulas is thereby avoided, but it must be 
distinctly borne in mind that all the succeeding theorems can be at once extended to 
the general case of m systems by an easy enlargement of the nomenclature and 
notation. 
I consider, then, two systems of quantities 
^ 1 ? ^'25 • • • ? 
/3i, ^0, . . . A/; 
as connected with two non-homogeneous equations, in two variables, in such wise that 
the values a^. A? of fbe variables respectively constitute one solution of the two simul¬ 
taneous equations. In order to avoid identical relations between fundamental forms, 
as well as for other reasons, which will appear, I take the number of quantities n in 
each system to be infinite. 
By analogy the fundamental relation is written 
(1 -|- a^x -b A!/) (1- “b + A.t/) • ' • (f “b + A^/) • • • 
= 1 -f + a^x^ + -f + . . . + a^^ocPif + . . . 
As shown by Schlafli this equation may be directly formed and exhibited as the 
resultant of the two given equations, and an arbitrary, linear, nondiomogeneous 
equatiori in two variables. Beyond the preliminary idea this investigation has little 
to do with the original equations or with the theory of resultants. It starts with the 
fundamental equation just written, the right-hand side of which may be put into 
the form 
1 + tayc -f + trx^a.^.X^ + ta.^^,.xij + SAiA-/ + • • • 
The most general symmetric function to be considered is 
■ • • 
which I represent symbolically by 
( Pi^h P-2'^hPz*h '• • •)• 
