120 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
Comparing the two expansions of on , 
SP eNO s Dp., deol OM ee lps 
Lo (Diy Aik oe Oped (comas iGo) a) 
b , G 
a i 95 me 
If p be indefinitely increased, that term on the right hand side of the 
above equation will alone be comparable with the expression on the left hand 
side, which contains the smallest value of 7, independent of its sign. In this 
case 
L( ea A,) = 1 bh, bs , POS b, 5) ay = g(r) 
eae ape 1 ’ Dis Pre ve ae B—1 Eg Ca) > 
oi Op se We 
by ik mat alt Pe 
considering now 7 to be the smallest root of F(z) = 0. 
From this 

or, if it be remembered, that p = o, 
ACH Aja =0; ° ° . . (4) 
As an example, consider the quadratic x —52+6=F(«#)=0, and let 
$(z) = F’(2) = 2a — 5; then will 
—5 + 2% Deeilo 35 OT ga 210 mn 193... 2089 xf 
=f eS yee ee ———— —————— — 
6—5at+a 6 6 6 64 69 68 Sete 
The series of values given by 7 ~*~ fer _ as p increases from 0 to 6 is as follows 
pe OP 51 42> 5 3. See Oe Or 
r = 231, 223, 217, 2:12, 208, 2°06, 2-04, 
in which the approximation of 7 to the smaller root 2 is evident. 


