
J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 131 



_ gree) _ (6-H t=)(5+ )__ g@PoO -NE-JC4+9)__ 9(9.9O? C—DE-NE+t__ 
F(r)F(s)F' (4) (rst)P+1 FO PR) s?et Gao Macs wea ; 
(e999 s=)C—s)st | o(s)Pe() (s—H(t—s)st 9(5).6OP (s—(t—s)st | ge, 
TFOFORO PACH TP GEE SFT TPO KOP eee 




+ Similar terms in each with s, 4, 7; #, 7, s; written in turn for 7, s, ¢, 
in the above expressions. 
a(r)as)e(t) (s—t(t¢—s){r%st— (s+ drst + (st)} +¢—N— At J+G—Hls—”)t } 
Puls ue ie een cree ST Te 
_ or) 9(8) 90 S-DE-9)(t-")r-D7—5)6—7) r 
=FOFOFO (rst)? +2 + &e. SS ee 
This is the greatest term in the value of w,, , provided the terms in 
BPO) | Be, 
FGF Q ~ 

be not greater. The coefficient of the term in 
gs) - o(2) 
Fo). FO 
1S 
(s—t)(t¢—-s){s?4P +2 — (s4t)stP +? 4 stP+3} 
s2P + 3842p + 3 

which vanishes: similarly for the other terms of this form. Thus the only 
terms in w;,, which remain are of the form given in (9’): and when p becomes 
infinite 
pe Lae CHCl S Oe) 6=7) 
» = Fo). FG). FO (rst) PF ke PaaS) 

The term w,,_, may be calculated in the same way. For it is thus found 
that 


t —t)(t— —t)(t—s)(s+t —t)(t—s) ; 
Usp—1 = Cee Sage S oe Se +terms in t, ies and thy :,) 
TaN 
+terms in OKO) &c., whose coefficients vanish ; 
F(s)[F') 
therefore when 7, s, ¢, are the three smallest roots, and p is infinite, this 
becomes 
EOE OO) =D) GS Est) 
et = FOFOF(O (rahe 
VOL. XXVIII. PART L pia Bi 
