114 
[736 
736. 
APPLICATION OF THE NEWTON-FOUPIER METHOD TO AN 
IMAGINARY ROOT OF AN EQUATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 179—185.] 
I consider only the most simple case, that of a quadric equation x- = n 2 , where 
n 2 is a given imaginary quantity, having the square roots n, and — n; starting from 
an assumed approximate (imaginary) value x = a, we have (a + Kf = n\ that is, 
o.ot. 9 7 a 2 — n* , 7 a- + n 2 
a 2 + 2 ah = ?i 2 , h — _— , and a + h = —=— ; 
2a 2 a 
that is, the successive values are 
a- + n 2 cq 2 + ?i 2 
and the question is, under what conditions do we thus approximate to one determinate 
root (selected out of the two roots at pleasure), say n, of the given equation. 
The nearness of two values is measured by the modulus of their difference; 
thus a nearer to n, than a 1 is to n, means mod. (a — n) < mod. (cq — n), and so in 
other cases; in the course of the approximation a, a ly a 2 , ... to n, any step, for 
instance a to oq, is regular if cq is nearer to n than a is, but otherwise it is 
irregular ; the approximation is regular if all the steps are regular, and if (after one 
or more irregular steps) all the subsequent steps are regular, then the approximation 
becomes regular at the step which is the first of the unbroken series of regular 
steps. 
We do by an approximation, which is ultimately regular, obtain the value n, if 
only the assumed value a is nearer to n than it is to — n; or, say, if the condition 
mod. (a — n) < mod. (a -f n) is satisfied, and the approximation is regular from the beginning