270 



a coordinate the sum of the omitted coordinates, if the result be multiplied by tfa 



multinomial coefficient of the omitted coordinates, as 



(axaybz) = (ax'bz)(lxly), x' =.»: + //. 



9. The coefficient of a parameter of a gamma coefficient is tin multinoimal 

 coefficient of the corresponding preceding point. In symbols, 



(axhy ' " ) = (offi +bEi + ") (lx + 1 ?/ " ) 



II. Gamma Series. 



10. Let there be m variables, pi, p 2l , of weights 1, 2, , and m cor- 

 responding parameters, Oi, a 2 , ' ' . The gamma scries 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(aiaia 2 a 2 ' ) Pi l pi 2 , ai+2« 2 + ' ' =n. 

 This series is not a function of an r'th variable and parameter for r > n, 

 since the simultaneous exponent and coordinate ar, is zero. 

 By applying art. 5 to the coefficients of (ap)n, we have, 



(b) (ap)n=pi(ap) (n-l) + . . +pv~ i(ap)l+a v p v 

 where, if r>m, p r =0. 



The last term a v p v , which cannot exist if n > m, is determined by the fact 

 that it is given by the coordinate a v = 1,, and the other coordinates, zero. 



11. The difference equation 10(6) 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 n>m, when it 

 is the general linear difference equation of n'th order with constant coefficients 

 Pi, pi, , 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, 



/s m m — 1 . m — 2. , 



(a), x = p x x +p 2 x +--+Pm 



Symmetric functions Fn of the roots of this equation Avill 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 sjan- 

 metric functions that may be expressed by gamma series ivith parameters 

 independent of the roots; and find two sets of such functions m in each set, 



