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