ON HAMILTON’S NUMBERS. 
289 
The question arises as to whether it is possible to deduce the Hamiltonian 
Differences, or to deduce the Hamiltonian Numbers, directly in a continued chain 
from one another without the use of any intermediate numbers. Mr. James 
Hammond has shown that it is possible, and has made the remarkable discovery that 
it is the Numbers of Hamilton, and not the Hypothenusal Numbers, which are 
subject to a very simple scale of relation. These being found, of course the Diffe¬ 
rences become known. This is contrary to what one would have expected. A priori, 
one would have anticipated that the determination of the Hypothenusal Numbers 
would have preceded that of their sums. 
I leave Mr. Hammond to give his own account of his mode of obtaining the 
wonderful formula of reduction, which, by a slight modification, I find, may be expressed 
as follows:—Using E; to denote the {i -f- l) th Hamiltonian Number augmented by 
unity, so that E 0 = 3, E L = 4, E 3 = 6, E 3 - 12, E d , = 48, . . .; and /3;m to signify 
the coefficient of t l in (1 -j- t) m ; then, for any value of i greater than unity, 
A)E i — /3jE i_i + /3 2 E;_2 — /TEi_ 3 + . . . + (—)* AE 0 = 0. 
tant part in the systematic treatment of the problem, first appears in my memoir on the subject in the 
100th volume of ‘ Cjrelle.’ 
It is proper also again to notice that what I call the Numbers of Hamilton (at all events those subsequent 
to the number 5) are not the smallest numbers requisite for fulfilling the condition above specified. Smaller 
numbers will serve to satisfy that condition taken alone; but when such smaller numbers are substituted 
for Hamilton’s the resolving equations will be less simple, inasmuch as they will contain a greater 
number of equations of the higher degrees than when the larger Hamiltonian numbers are employed. 
This distinction will be found fully explained in the memoir cited, and the smallest numbers substitutable 
for Hamilton’s are there actually determined for r equations of degrees extending from 1 to r for all 
values of r up to 8 inclusive. 
I have added nothing (for there is nothing to be added) to the fundamental formula of Hamilton 
expressed by the equation 
[X, [l, V, . . . tt] = 1 + [X - 1, X + fl, X + /A + r, . . . , X + fl 4- V + ... + tt], 
where, supposing the letters X, p, v, . . . w, to be i in number, [X, /t, v, . . . w] means the number of letters 
required in order that it may be possible to satisfy, according to the process employed by Hamilton (in 
conformity with a certain stipulation of Jerrard), a system of X equations of degree i, fi equations of 
degree i — 1, v equations of degree i — 2, . . . , tt equations of the degree 1, without solving any single 
equation of a degree higher than i. This formula, applied X times successively, will have the effect of 
abolishing X and causing [X, fi, v, . . . w] to depend on \jt !, v', . . . w'], where p, v', . . . tt' are connected 
with X, fi, v, . . . tt by means of the formula3 given at the commencement of the present paper, but where 
instead of the letters X, /<, v, . . . I have used the letters p, q, r, . . . 
It is presumable that the reduced Hamiltonian numbers would be found much less amenable to 
algebraical treatment than the Hamiltonian numbers proper; for numerical equalities and inequalities 
have to be taken account of, in determining them, which have no place in the determination of the latter 
numbers. Hamilton, as already stated, expressly alludes to the reduction of 11 to 10, but with that excep¬ 
tion has avoided the general question of finding the absolutely lowest number of letters required in order 
that a system of equations (expressed in terms of those letters) of given degrees may admit of being satisfied 
without the necessity arising to solve any equation of a higher degree than the highest of the given ones. 
MDCCCLXXXVII.—A. 2 P 
