443 
EXPRESSED IN TERMS OF THE ROOTS OF (px, fx. 
\ y = lr{x— hT)(X~ ki) 
~h- 
X 
1 1 
1 1 
-h- 
_^2_ 
X ^ 
I 
\K~] 
= 2 
hi 
i e h [ ^ { 
~^h,-h, \ki~kX 
{x-h){h,—k,){h^ 
^oc ~hy 
h) >,1 
{x—k^){h,~k^){h^--ki))] 
{ (^1 — + { {ki-\-k^h^—' {hJi^-^-kik^) } | 
= {x—h,)x-{-{x~h^)x—{ki+\)x+{hJi^-\-W) 
= {x^-{h,^h,)x+hA) + {x^-{ki+k,)x+kA) 
= {x‘^-\-ax-\-h)-\-{x^-\-ax-Y^) 
=/+^; 
so we find also S^ 2 ,o=<P- 
Art. (23.). The expression which is universally a conjunctive of ^and <p, con- 
tinues algebraically interpretable so long as v-\-v has any value intermediate between 
( 0 ) andm+w; when y= 0 , we must of course have ^;=0 and j'^O, and \o becomes 
the resultant of/ and (p when v-{-v—m-^n', we must also have the unique solution 
v=m and v=zn, and „ becomes necessarily fxp, which we thus see stands in a sort 
of antithetical relation to the resultant of f and <p, say (/, <p). Nor is it without interest 
to remark that/x <p=0 implies that a root of /or else of (p is zero; and (/ (p)=0 
implies that if a root of the one of the functions is zero, so also is a root of the other, 
i. e. that a root of each or of neither is zero. As i increases from 0 to w or decreases 
from m+n to m—], the number of solutions of the equation v-\-v=i in the one case, 
and the number of admissible solutions of the equation v-\-v=i in the other case, which 
is subject to the condition that v must not exceed w, continues to increase by a unit at 
each step ; there being thus w-j-1 different forms „ when and the same 
number when 1. For all values of ^ intermediate between n and (w— 1) 
(both taken exclusively) it is very remarkable that ^ will vanish, as 1 proceed to 
demonstrate. 
Art. (24.). The weight of the coefficient of the highest power of „ {v^v being 
equal to i) is and consequently, when i is greater than n, and less than 
^5 would contain fractional functions of the roots of / and (p, if there were in it a 
power x\ but „ has been proved to be always an integer function of the roots. Hence 
the coefficient of x’’ will be zero, and so more generally the first power of a? in of 
whieh the coefficient is not zero, will be x", subject to the condition (since evidently 
the weight of the several coefficients goes on increasing by units as the degree of the 
terms in x decreases by the same) that a be not less than ; let then 
ca={m—i){i—n), S-j, „ becomes of the form A:c‘""-1-B.r' &c., where A is of zero 
dimensions ; but this is impossible if i — uKn, for then A.r*""-1- &c. is a conjunctive of 
