2/1 
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 an of (ap)n, n = 1, 2, ', m, is the coefficient of pn. 
Thus to determine the possible parameters of a given symmetric function, 
Fn, we must take a, as the value of F'n for the roots of the equation x =1, 
this being what 11 (a) becomes when we put py, =1, and other p’s equal to 
zero. It remains to test the resulting equations, 
Fl =a)pi, F2=piF1+ap2, F3 =p, F2+p2F 1 +aszps, ete. 
13. The sum of the wth powers, Sy. 
By art. 12, we find a, = n, for the function s,, and the difference equations 
are Newton’s equations. Hence 
Sy = =(la “Nay) prt qt ona: +Non =n 
This is Waring’s formula for sy. 
14. The homogeneous products, ry. 
Here, an=1, giving the correct difference equations, 
T= Pi, T= pit + po, 73 = pit. + pom + ps, ete. 
Hence, ty = (1p)n, i. e. the coefficient of a term is the multinomial co- 
efficient of its exponents. Since the equations are symmetrical in 7, — p, we 
have also, pn = —(1[—7])n. These formulas seem to be new, as also those 
which follow. 
15. The homogeneous products, k at a time, wnk. 
Here a, is a binomial coefficient of the n’th power, whose value 
is zero for n<k, and 1 for n=k, and, 
Tk =(ap)n, ay =(—1)*-1dK1.n—-k.) 
16. By applying art. 9 to the coefficients of (ap)n, and substituting 
%_ =(1p)n, we have. 
(a). (ap)n =Qpiry—1 + @2pory —2 +  +OnDy 
We have therefore, 
Pity —1 P27 —2 P3™y 225} Paty =A PsT yn — Es elc. 
7 = 1 1 1 1 1 etc. 
sq="m = 1 2 3 4 5 ete. 
"m= 1 3 6 10 ete. 
"23 = 1 4 10:7 ete: 
Na = 1 5 etc. 
tae = 1 etc. 
