





132 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS, 
Similarly it may be found that 
rs+st+ir 
Uap = ( ) ae) 
ts=(  ) aa 
Thus the determinant cubic becomes 
ve—(rt+s + ta’ + (rs + st + ir)a —rst = 0 
VII. Let now 
Cl Cg OO A Gt tO 
be an equation whose roots are the m least roots of F(z) = 0: and let 7 be the 
least root, then 

g(r) _ or) g(r) gC) 
Cn Fre toma Ri pgptt +--+ + Fret + Ppp = 0. 
Now Hoe is what A, , becomes when p is infinite. The above equation 
may therefore be written in the form 
Cm pe ae Cm—-1 A, + 2-5 SMT Cy Aes a € seis 
te Crp Copsey tea Onente ite lol tial Al eae wie e CpOoen— tO 
where a,_; . . . 44m, ultimately vanish when p is infinite. The number of terms 
in the second line of this equation is finite, and if « be the greatest of the 
quantities a, 1.... 44m, this equation cannot have a greater error than when 
, 
(Givi Crap at Yenc ate 
is written for its second line. But this, being the product of a finite quantity 
and a’ which is ultimately zero, ultimately vanishes, and the quantities ¢,.. . 426 
are connected by the equation 
0 
Cp i TF CnirAy oF eyes: t CGE mae = GING opens == 0 O 
Similarly, the following equations are true 
Cee + pe + eke satay ate + Bee Vy os + Cay scans = Gi. 
CmAip—s ar ¢ WEA aF oe @ @ + Cdr Bek + ClAS es = 0 ’ 
&e. 
