124 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
sooner than the simple equation for the smallest root. For example, in the 
above p=7 gives, by the simple equation, 1:57 as the root, whereas the 
corresponding quadratic gives 1999. 
(IIL) The quantities w,, , w,,; may be reduced to determinants also. Thus 
Usps =Ay Apyi—Aps Apis 

= b, by vats bp Ap b, b, ee Dp41 Ap+1| = b, by owe Das Ap—1 b, b, a eis Dp+2 Ap+2 
DD ists Opt Ogg 1 6,4... 065 Ap Bch wipas Ops L Oy. sO eee 
by b by bb 
i 1 ae! Wal L- a& 
pA pt2 p pr+3 
(the underwritten letters indicating the order of the determinant above). 
Let the p- and (p+ 2)- determinants be raised to the (p+1) and (p+ 3) orders 
respectively, and separate the determinant of the (p+3) order into two, the 
constituents in whose first vertical lines are respectively (read downwards) 
0,0; 0-0. 2 and: 07 1 0:0. oe Tints 


Uon+3—= b, Ones. Ge Up 1b, by. +. Op+2 Ope | aw 1b, b,...b Ap b, by ++ Opie Ap+2 
1 b, mele bp -1 Op—-1 0B, Oy ++. Epa ptt | 06 Dates Op ag Ap—1 Op 6... . Upeg Ap+1 
b ay b, Gy by a, b a 
Let ies ) | Laer | 
gt p+3 pti pt+3 
— 1b, Re oe bp Ap 0 b, eee Op+2 O42 
0 6, by... Op Gpa || 15, ... Gages 
shat Sac aels nataecastnts ol. .30ph ee 
0, ity || wanna eae 
a O,, ae 
ites sl 
fag pte 
or adding the determinants in the first line, and putting 
Cie OWA. Mian sr ESM, Piatra Os)a Coa 
1 b, alo 8 bps Ap—2 if! b, Op—2 Ap—2 
SS aig Re ee 3 Es OE 
Lae Dre Tih Cah Ses as ok RC MCMICL seer cheery 
