743] 
143 
743. 
ON THE NEWTON-FOURIER IMAGINARY PROBLEM. 
[From the Proceedings of the Cambridge Philosophical Society, vol. hi. (1880), 
pp. 231, 232.] 
The Newtonian process of approximation to the root of a numerical equation 
f(u) = 0, consists in deriving from an assumed approximate root £ a new value 
£ = £ > which should be a closer approximation to the root sought for: taking 
the coefficients of f(u) to be real, and also the root sought for, and the assumed 
value to be each of them real, Fourier investigated the conditions under which 
is in fact a closer approximation. But the question may be looked at in a more 
general manner: f may be any real or imaginary value, and we have to inquire in 
what cases the series of derived values 
/(J) 
/'(*)’ 
ft = ft- 
/(ft) 
/'(ft)”’’ 
converge to a root, real or imaginary, of the equation f(u) = 0. Representing as usual 
the imaginary value f-, = x + iy, by means of the point whose coordinates are x, y, 
and in like manner £ lt =x 1 + iy i , &c., then we have a problem relating to an infinite 
plane; the roots of the equation are represented by points A, B, 0,...; the value 
£ is represented by an arbitrary point P ; and from this by a determinate geometrical 
construction we obtain the point P 1} and thence in like manner the points P 2 , P 3 ,... 
which represent the values £>, f 3 ,... respectively. And the problem is to divide 
the plane into regions, such that, starting with a point Pi anywhere in one region, 
we arrive ultimately at the root A ; anywhere in another region we arrive ultimately 
at the root P; and so on for the several roots of the equation. The division into 
regions is made without difficulty in the case of a quadric equation, but in the next 
succeeding case, that of a cubic equation, it is anything but obvious what the division 
is : and the author had not succeeded in finding it.