
J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 129 | 
Let F(z) = (@ —1r)*(z) then we may write i += = f\e), and 


g(@) f) f@) (s) 1 
Fa) ~@—7ta—r +2(40-—) 
=....4+{(p+)58- ea = Oo g 


If 7 be the smallest root, then when p = » 
Ap +1 p 
pt+il 


Y =r We may say. 
Again 
lrg = NG — Ape A, 5 
= (450ee 4 ee) + ee eee (p+ 2)f(7) 2 Se. 
yp +3 
'- the terms neglected ultimately vanishing, when p is infinite, compared with 
those retained : that is, ultimately 
(p? + 2p + 1 —p? — 2p) (SM)? 




Urp4+2 = = 2p + 4 
LO 
pep + 4 
Also 
. Ug ta Net pa Ny 2 
ultimately 
_ {p+ 1) (p + 2)— wp + 3)} Lal 
p2P +9 
— 2F@)|? 
pp +e d 

therefore the determinant quadratic becomes 
2 .f (r)\” f (r)\? 
Ee 2 T0"2  OE aa 
or 
“v—I¢we+r—O, 
i.¢., the determinant quadratic is true also for equal least roots. 
(VI.) In the case in which F(z) = 0 has real and unequal roots, it has been 
proved that the equation 
0 
II 

oe 
