J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 123 . 
27" quotients, the root is approximated to much more quickly: thus beginning 
with m = 5 and 6, we get | 
1:99", 1:95), O75 08: 
the constancy of these means being very remarkable 
The equation for the two smallest roots is — 


ae ee eee | = 6 
Ay » Apis Apis 
Ay1, Ap 5 Apu 
which may be written in the form—using the above notation— 
Carat oil = 0) 
24B, 1 
Boy ? 7 24By 41 
Boye 
Bh. , _1 
Bache go4B, 
Bp 
(a) Putting p = 7, this is 
x ,x, 1 |=0, which reduces to a + °87727 — 5°725 = 0 
1 
1°57 5 1 » 9.34 
1 
aT ae la 157 
the roots of which are see i.¢., 1999, and —2°871, nearly 2and — 3. 
(8) Putting p = 11, this quadratic equation becomes, 
a ,a, 1 |=0, te, a+ 8942 —5784=0 
1 
1°90, i 3-07 
205), alee T-90 
=) 
the roots of which are — =e ae , ue., 1999, and —2°893, nearly 2 and —3. 

It may be inferred from this example that the quadratic equation whose 
roots are the two smallest roots of the given equation, give the smallest root much 
VOL. XXVIII. PART I, 21 
