



126 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
(IV.) Let F(z) = 0 have imaginary roots, of which two conjugate ones are 
pe, peé—*, where é = e* and. = ,/—1; and let their modulus, p, be smaller 
than the modulus of any other pair of imaginary roots, and also smaller than 
any real root. Then, as in (1) 
ea) _ e(7) = 
TC) Sa en ee) 8 Soothe ee Bie ae 2 
therefore as p increases to infinity, 
fi CS ee 
Art= Fue) Gb? +t FG) GE 

Let 
fir +0. 
then 
— pPA,_1 = #(P + Qu) (cos pO — usin pO) + 4(P — Qu) (cos pO + usin p6) 
= P cos pé + Q sin p0 
Ap _ Peosp+10+Qsinp+10 | 

i ate Het ea P cos pO + Q sin p6 = W, (say) , 
es up sin pO — sin p + 16 
“Q w? cos pO — cos p + 10’ 
and similarly 
Mp 41 Sin pp + 10 — sin p + 20 
Wp 41 COS p + 10 — cos p + 20’ 

whence reducing to a common denominator 
(Wp 41 We — 2w, cos 6 + 1) sin 8 
(w? cos pO — cos p+16)(wp+1 cos p+10 — cos p+20) 

Let us suppose the solution of this equation given by 
Wp41W, — 2w,cos9+1=0, 

or 
Le 
2 COS) = ae tie 
. ‘se Aa Lt ioe 
that is, =a ee 
and similarly 
Miae il bo 


Ayys Tp Appi * ‘ ; ; - (6) 
