271 



which can be linearly expressed in terms of each other, and either of these 

 sets suffice to express in linear form all of the symmetric functions sought. 



12. The parameter ar? of (ap)n, n = 1, 2, ' ', m, is the coefficient of pv- 

 Thus to determine the possible parameters of a given symmetric function, 

 Fn, we must take a^ as the value of Fn for the roots of the equation x =1, 

 this being what 11 (a) becomes when we put p-q =1, and other p's equal to 

 zero. It remains to test the resulting equations, 



Fl=a,pi, F2=p 1 Fl+a 2 p2, FZ^p l F2+p 2 Fl-\-a i p i , etc. 



13. The sum of the n'th powers, s^. 



By art. 12, we find a^ = n, for the function s^, and the difference equations 

 are Newton's equations. Hence 



S^ = S(l«i ?ia rl )pi 1 --prj ""I, tti+ • • +n«7j =n 

 This is Waring's formula for s^. 



14. The homogeneous products, ir^. 



Here, av = l, giving the correct difference equations, 



TTl=pl, TT2=pnri+p 2 , Tr3=Pnr2+P2TT2+P3, etc. 



Hence, x^ = (lp)n, i. e. the coefficient of a term is the multinomial co- 

 efficient of its exponents. Since the equations are symmetrical in x, — p, we 

 have also, pv = —(l[—ir])n. These formulas seem to be new, as also those 

 which follow. 



15. The homogeneous products, k at a time, irnk. 



Here & v is a binomial coefficient of the n'th power, whose value 

 is zero for n <k, and 1 for n=k, and, 



v nk = (ap)n, a v = ( - 1) k ~ 1 (1 Kl.n - k.) 



16. By applying art. 9 to the coefficients of (ap)?i, and substituting 

 v t] = 0-P) n , we have 



(a). (ap)/i = aip 1 7r 7? _i+a 2 p27r 7? _2 + ' ' +a v Pv 

 We have therefore, 



Slj = 





V^v — » 



P2TT V — 2 



p-iT-rj — 3 



V&v- 



-4 Pi 



*q-l, 



etc. 



X 



V = 



1 





1 



1 



1 





L 1 



etc. 



X 



Vi = 



1 





2 



3 



4 





|5 



etc. 



X 



7)2 = 







1 



3 



(i 





10 



etc. 



X 



Vi = 









1 



4 





10 



etc. 



X 



Vi = 











1 





5 



etc. 



X 



Vo = 















1 



etc. 



etc. 



















