122 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
or, of 
Gi ee age) ee , 72 1a 
A, ? Agyt, Ap +. 
nit BS ? Ay 41 
Take the same quadratic as above for an example; the above determinant 
quadratic is the same as the given equation ; for we have 
a —5e +6 = 0 
A, —5Aj;41 + GA,+2 = 0 
A,-1— 5A, + GAS g = 0) 
whence the determinant quadratic in question. 
This quadratic does not simplify, however, so easily in other cases. 
Consider the expansion—quite general—of a fraction having 
a+ 5xn°—2a—24, tie., (vx—2)(a + 3)(u + 4) 
for denominator, and any other integral rational function of not more than 
three dimensions in the numerator. For example, expand the fraction 
24. + 62 + 3a? gm 
ELD =o Pea ee +Bapat.-. 

The values of B, beginning from x = 1, are 
16 1174.10? 23701.10° 
75 1016.10° 29893.10' 
29: 1547.10* 34699.10° 
938: 1587.10° 
5924: 2125.10° 
From these arises the series of quotients, = , which give the following 
values, beginning with n = 3, 
W=8- 4, B56) eee Oo) 0, a es 
a = 74, 88, 121, 277, 157, 284,179, 215, 1:90, 207. 
nm+1 

in which latter series the continued approximation to the number 2 is manifest. 
It is noticeable that these coefficients are alternately greater and less than 
the root to which they approach. If the means are taken of the (27—1)th and 

