J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 125 
MLM, .. MM, 
Unprs =| dy 0s bs 4. ~ Dp Opin Boxe %+e 
De We Oa a, « Ose De Dear Ces 
LOS os OE ORE Oa iG, 

The minors of the constituents of the first horizontal lines of d, and d, are 
the same ; and d, and d, may be written 
d, = 0,M, + 0M, +...+2% Mpita M, 
d, = 0M, + 6M, +... + 025M, 4 + a, 1M, 
where M,, M,, ... M,, are the minors of the Ist, 2d,... p” constituents of 
the first horizontal row of d, . 
In 13, let there be added to the 1st vertical row, the next p—1 vertical 
rows, each multiplied by the minor written above it with its sign changed—then 

Wes =O Uz Os..'- - Oper Opts Epia 
GeO AOy geeky” Osea Caen 
GaN be hye Op Oy Op 
1 b, b, a 
1 OG; 
RnR) ree oy a set i 1 
prt3 
1.€., Urp+3 18 the same as — A,.., with the constituents a,,a,.... 1,0, 0 written 
down its first column instead of b,,1,... 0. 
In like manner it may be shown that w.,,, is the same as —A,,., with 
constituents d,41, Gp, Uo-..d, 1, 0, written down its first column instead 
mor, Lb, ... 0 
The quadratic equation 
2 nes 
Usp44 0 — Urp ys ®@ + Urpig = O 
may therefore be written in the following form :— 

Apdo, . Dp+2%pr2 | 2 — | Ay Dab... Opsatpye | % + | Up :b3... Opsidpys | = 0 
Oy 0,0, ... bp 1Gpit Op—10,b,.. » Dp41%p44 Apetples +» Op OX 
Beeld... 0p Gp Ope olOy a. Op » Oy (Gite 5) COERR Re) ome 
1 reyalO yrs Orel ne o OO Me hee a Oy 
0 alee wk OMG ore -nisco el oe (One ee putin Mage 
pts pt3 pt2 
These determinant forms of the coefficients have already been made known. 
