694]
DESIDERATA AND SUGGESTIONS.
405
The diagram has a remarkable property, in virtue whereof it in fact represents a
group. It may be seen that any route leading from some one point a to itself, leads
also from every other point to itself, or say from h to b, from c to c,..., and from
l to l. We hence see that a route, applied in succession to the whole series of
initial points or letters abcdefghijkl, gives a new arrangement of these letters, wherein
no one of them occupies its original place; a route is thus, in effect, a substitution.
Moreover, we may regard as distinct routes, those which lead from a to a, to b, to
c,...,to l, respectively. We have thus 12 substitutions (the first of them, which leaves
the arrangement unaltered, being the substitution unity), and these 12 substitutions
form a group. I omit the details of the proof; it will be sufficient to give the
square obtained by means of the several routes, or substitutions, performed upon the
primitive arrangement abcdefghijkl, and the cyclical expressions of the substitutions
themselves: it will be observed that the substitutions are unity, 8 substitutions of
the order (or index) 2, and 8 substitutions of the order (or index) 3.
It may be remarked that the group of 12 is really the group of the 12 positive
substitutions upon 4 letters abed, viz. these are 1, abc, acb, abd, adb, acd, adc, bed,
bde, ab. cd, ac. bd, ad. be.
Cambridge, 16th May, 1878.
No. 3. The Newton-Fourier imaginary problem.
[From the American Journal of Mathematics, t. ii. (1879), p. 97.]
The Newtonian method as completed by Fourier, or say the Newton-Fourier
method, for the solution of a numerical equation by successive approximations, relates
to an equation f(x) = 0, with real coefficients, and to the determination of a certain
real root thereof a by means of an assumed approximate real value f satisfying
prescribed conditions: we then, from derive a nearer approximate value £ by the
formula £i = £-vKS:; and thence, in like manner, £, f 2 , f 8> ••• approximating more
/(f)
and more nearly to the required root a.
In connexion herewith, throwing aside the restrictions as to reality, we have what
I call the Newton-Fourier Imaginary Problem, as follows.
Take / (u), a given rational and integral function of u, with real or imaginary
coefficients; £, a given real or imaginary value, and from this derive £ by the formula
f , and thence £, £ 2 , each from the preceding one by the like
formula.
A given imaginary quantity x + iy may be represented by a point the coordinates
of which are (x, y)\ the roots of the equation are thus represented by given points