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a coérdinate the sum of the omitted codrdinates, if the result be multiplied by the 
multinomial coefficient of the omitted codrdinates, as 
(axaybz) =(ax’bz)(1aly), v7 =x+y. 
9. The coefficient of a parameter of a gamma coefficient is the multinoimal 
coefficient of the corresponding preceding point. In symbols, 
(axby') =(a#i+bE,+°')(a+ly °) 
Il.- Gamma SERIES. 
10. Let there be m variables, pi, p2, °, of weights 1, 2,., and m cor- 
responding parameters, a1, a, . The gamma series of weight n is the sum 
of all terms in the variables of weight n, each multiplied by the gamma 
coefficient of its exponents and the corresponding parameters: 
(a) (ap)n = 2(qiaraza2.'") pi“ po*? , ar +2a2+°° =n. 
This series is not a function of an r’th variable and parameter for r>n, 
since the simultaneous exponent and coérdinate ar, is zero. 
By applying art. 5 to the coefficients of (ap)n, we have, 
(b) (ap)n = pi (ap) (n—-1) +..+pn—i(ap)1 +n? 
where, if r>m, p; =O. 
The last term aypy, which cannot exist if n >m, is determined by the fact 
that it is given by the coédrdinate a, = 1, and the other coordinates, zero. 
11. The difference equation 10(b) has no solution except the gamma 
series, since all values of (ap)n are determined from it by taking n= 1, 2,3, -, 
successively. It is an equation of permanent form only for 7 >m, when it 
is the general linear difference equation of n’th order with constant coefficients 
Pi, Px, , whose general solution with m arbitarty constants is therefore found 
in the form of a gamma series. The equation whose roots determine its 
solution (in the ordinary theory of linear difference equations) is, 
(a). 2? = pa” } nr ocelen +--+pm 
Symmetric functions Fn of the roots of this equation will also satisfy 
the difference equation and can therefore be expresssed as gamma series by 
certain values of the parameters. 
Since the roots of (a) are constants, the parameters will in general be 
certain functions of the roots, but we propose here to determine the sym- 
metric functions that may be expressed by gamma series with parameters 
independent of the roots: and find two sets of such functions m in each set, 
