45 
19 16-1 7. J Operators applied to Solution of Equations. 
act convergently. These give 
1379 
”658~ 
/ 49 y/1379\ 2 _ 1 / 1 37 9 \ 3 
+ \658 ) \ 658 / 658 \ 658 / 
0 /49y/1379\ 3 5 / 49 V 1 V1379y / 1 y/1379\ 5 
+ \658/ V 658 ) ',658 658 M 658 ) ' 658/ V 658 / 
+ 5 
+ 14 
49 \ 3 / 1379 y 
658/ V 65 8 / 
/ 49 y/1379\ 
',658/ \ 658 / 
5 
- etc. 
1 37 9\ 5 
658 / 
+ etc. 
+ etc. 
We develop, of course, in the direction of the important terms. The rate 
of convergency for these latter begins at T5 and increases gradually to 
•6, and by the aid of the logarithms of the three coefficients we may 
calculate the root with fair ease. The alternative method is to subtract 
the integral part of the root — this is seen from the opening term alone 
to be 2 — as in Horner’s process. 
The reduced equation is y 3 — 43 y 2 + 474y — 251 = 0. 
251 
The new operand suggests ’5 as the first figure of the new root, and 
now instead of subtracting • 5 we divide the root by ’5, division being an 
accelerated form of subtraction, and this latter being the kernel in 
Horner’s unerring process. 
With t/ = *5(1+0) we have 
•125(1 +0) 3 - 10-75(1 + 0) 2 + 237(1 +6)- 251 =0, 
or 
0 3 - 830 2 + 17270 -197 = 0. 
197 
The operand here gives or T as the opening figure in 0. Put 0 
therefore = *1(1 + 0^, so that we have 
•001(1 + Oi) 3 - -83(1 + dj) 2 + 172-7(1 + 0 X ) - 197 = 0, 
or 
•001^ 3 - -8270J 2 + 171-O430J - 25429 = 0, 
from which we have 6 1 in the form 
25 Jl 29 j827_ / 2M29 y = ^ 47018 
171-043 171-043V171 043/ 
And x = 2 + *5[1 +4(1447018)], or 2-5573509 correct to five decimal places. 
To get a still closer approximation, we must, of course, include more 
terms in the calculation of 6 V or put 0 1 = T(l + 0 2 ). We are merely 
