286 
PROFESSOR SYLVESTER AND MR, J. HAMMOND 
the 5 th degree, upon which by Being’s method the general equation of that degree 
can be made to depend, has necessarily imaginary coefficients except in the case 
where four of the roots of the original equation are imaginary, and also pointed out a 
method of obtaining the absolute minimum degree M of an equation from which any 
given number of specified terms can be taken away subject to the condition of not 
having to solve any equation of a degree higher than M. # The numbers furnished by 
Hamilton’s method, it is to be observed, are not minima unless a more stringent 
condition than this is substituted, viz., that the system of equations which have to 
lie resolved in order to take away the proposed terms shall be the simplest possible, 
i.e., of the lowest possible weight and not merely of the lowest order; in the memoir 
in ‘ Crelle,’ above referred to, he has explained in what sense the words weight and 
order are here employed. He has given the name of Hamilton’s Numbers to these 
relative minima (minima, i.e., in regard to weight) for the case where the terms to be 
taken away from the equation occupy consecutive places in it, beginning with the 
second. 
Mr. James Hammond has quite recently discovered by the method of generating 
functions a very simple formula of reduction, or scale of relation, whereby any one of 
these numbers may be expressed in terms of those that precede it: his investigation, 
which constitutes its most valuable portion, will be found in the second section of this 
paper. The principal results obtained by its senior author, consequential in great 
measure to Mr. Hammond’s remarkable and unexpected discovery, refer to the proof 
of a theorem left undemonstrated in the memoir in ‘ Crelle ’ above referred to, and 
the establishment of certain other asymptotic laws to which Hamilton’s Numbers 
and their differences are subject, by a mixed kind of reasoning, in the main apodictic, 
but in part also founded on observation.t It thus became necessary to calculate out 
the 10th Hamiltonian Number, which contains 43 places of figures. The highest 
number calculated by Hamilton (the 6th) was the number 923, which comes third in 
order after 5 (the Bring Number), 11 and 47 being the two intervening numbers. It 
is to be hoped that some one will be found willing to undertake the labour (consider¬ 
able, but not overwhelming) of calculating some further numbers in the scale. 
The theory has been “a plant of slow growth.” The Lund Thesis of December, 
* For’ instance, an equation of not lower than the 905th degree may be transformed into another of 
that degree, in which the 2nd, 3rd, 4<th, 5th, 6th, 7th, terms are all wanting, by means of the successive 
solution of a ramificatory system of equations, of no one of which the degree exceeds 6, whereas by the 
Jerrard-Hamiltonian method this transformation could not be effected for the general equation of degree 
lower than the 6th Hamiltonian Number, viz., 923. So for the analogous removal of 5 consecutive 
terms the inferior limit of degree of the equation to be transformed would be 47 by the one method, but 
44 (the lowest possible) by the other. In the case of 4 consecutive terms Hamilton could not avoid 
being aware that 11, the 4th number which I have named after him, might be replaced by 10, as the 
lowest possible inferior limit of the equation to be transformed. 
f In the 3rd section, communicated to the Society after the 1st and 2nd had gone to press, the 
empirical element is entirely eliminated, and the results l'educed to apodictic certainty. 
