
J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS, 121 
IL. Let Uop 4.4 = ern — Ae Ne 3 
H/o(r)\?_1 ar). 9) 1 
=(¢o mere + 2 wiry FG) GaP FE 
[| 


g(r)? 1 (7) - 9(s) 1 if y] 
F(’) 2p + 4 7 LEG F(s) ppt gp ts T pts gpl 

II 
g(r). p(s) (r—s)(s—7) 7] . 
= LFOrO oar]: 
when p increases indefinitely, supposing 7 and s to be the two smallest roots 
of F(z). =0, 
— (7). g(s) (r—s)(s—7) 
alc Yp+4 = Bry EG) Geers : 

Again, let — W453 = Ap1A;i2— ApAy +1 
Bs Borate a eyo) (A 1 
=2 [¢ CW pea + (7). F(s) Ps? 8 + pet Fe) 
g(r) PI g(r). G(s) 1 1 y] 


— F(r) peta IMO Fs) pe tlopr2 + p+2,pti 

= g(r). 9(s) (r—s)(s—r)(s+7) 
ae Gnee Fs) (rs)? +3 ] 
when p increases indefinitely, supposing 7 and s to be the two smallest roots of 
ee es 
2p+3 —_ F(r). F(s) F(s) (7s)? +2 
Hence, remembering that p is to be increased indefinitely 
Uap +3 
—— as 
Uap +a 
EE Tes 
ae f} 
Wop +4 
therefore 7 and s are the roots of the quadratic equation 
2 
Usp 440° — Usp 432 + Uy ig = 0, 
that is, of 
(Az41— A,A, 4 0)a°— (AJA, 41 — p—1Apy2)@ + (Aj—A, _A,4.1) wn 
