88 
the application of the theorem gives at once 
u u h=0 
ei h=M kh +n 
O= Su" Flu) - Lu Fw) =74 Zm Al (=), 
and hence 
h=M h-+-n 
De li 
Dg ") == 
h=0 mm 
From this relation we obtain taking » equal to 0, — 1, — 2, 
—M-+41 a set of M equations from which the ratios of the co- 
efficients A, can be solved. In fact, these M equations must be 
mutually independent, because they are equivalent to the ordinary 
Newron and Warne relations between the coefficients of an algebraic 
equation and the sums of similar powers of roots. 
Joining to the M equations the equation 
h=M 
lide (w) = = Arsch, 
h=o0 
we may eliminate the coefficient A, and introducing a determinate 
constant C we shall tind 
1 “w x2 a che eee . 
a on NE BERN 
OEE 
EEN 
Observing that the term «in #, (#) has the coefficient + 1, 
the constant Cis readily determined as a symmetric or as a skew 
symmetric determinant. . 
As we already remarked proposition VIII in itself suffices to cal- 
culate the coefficients A, and it is evident that in this way there might 
be deduced a second determinant also representing /’,, («). To obtain 
this second determinant we have only to replace everywhere in 
k it 
the first the symbols (=) and ( *) by u (D)p (DD), when D is the 
m mm 
greatest common measure of % and m, taking D=m for k = 0, and 
it is rather remarkable that notwithstanding the dissimilar character 
of the elements of these .determinants both represent one and the 
same polynomial. 
