46 
Proceedings of the Royal Society of Edinburgh. [Sess. 
showing how best to accelerate the convergency. The series for 0 above 
is already rapidly convergent. 
When we turn to the other integral-indexed operands in this equation, 
658 
49 
we find they both have a slightly divergent quadratic operator 
D 65 g . Di 1 . D 49 2 , for 4x658 is slightly >(49) 2 . But if we try the operands 
with fractional indices we shall find that these also have divergent 
operators. Now, whenever two or more operands are missing we may 
be sure of the presence of nearly equal roots. The series for the operands 
658 
49 
and 
49 
themselves show this. 
The former, beginning at 13, increases 
rapidly, and the latter, beginning at — , decreases rapidly, and they have 
nearly become equal before there is any trace of divergency. Though 
the rate of convergency be rapid at first also, it later becomes very slow, 
approaching asymptotically its maximum value, which is slightly over 1. 
This slow convergence is another unmistakable symptom of proximity of 
roots. Once we are sure of nearly equal roots we may find a first approxi- 
mation to their value in different ways, by following the series belonging 
to the operands which would give the roots in normal circumstances, or 
by finding the H.C.F. of the original function and its first derived. In 
this case we simply subtract the value of the root already found from 49, 
the sum of all three, and divide by 2. This gives 232 as the approximate 
value of the nearly equal roots, and the equation with its roots reduced 
by 23 is 
y 3 + 20 y 2 - 9 y + 1 = 0. 
There is now another point to be mentioned. In the above process of 
dividing the root by *5, T, etc., the ultimate goal was to render negligible 
the coefficient of every power of the unknown above the first, leaving the 
solution finally in the form aO = b. But till a pair of equal roots are 
separated we must solve the quadratic in 0, the coefficient of 6 2 refusing to 
become negligible. 
In the present case 20y 2 — 9y + 1= 0 gives y = '2 or *25. 
Putting y = ’2 (1 +6), therefore, we have 
•008(1 + Of + *8(1 + Of - 1*8(1 + 0) + 1 = 0 or 0 3 + 1036> 2 - 22^ +1=0, 
and the solution of this new quadratic 103d 2 — 226 + 1 =0 gives 6 = '0656 
or T479 and y = -21312 or *2295. The roots are now separated, and 
if one of them, say the first, be wanted more accurately, we must put 
0 = -06(1 +d x ) and solve for a single root 0 V 
