1887.] 



On Hamilton s Numbers. 



471 



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 be 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 conse- 

 quential in great measure to Mr. Hammond's remarkable and unex- 

 pected discovery, refer to the proof of a theorem left undemonstrated 

 in the memoir in 1 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. 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 (considerable but not overwhelming) of calcu- 

 lating some further numbers in the scale, in order to establish or 

 disprove conclusively the presumptive law of the asymptotic branch 

 of the series connecting any two consecutive semi- differences y x , ijx+i of 

 the Hamiltonian Numbers, viz. : — - 



yx+i — yJ = y22, r = c r y x ^> 



The theory has been "a plant of slow growth." The Lund Thesis 

 of December, 1786 (a matter of a couple of pages), Hamilton's 

 Report of 1836, with the tract of Mr. Jerrard therein referred to, and 

 the memoir in ' Crelle ' of December, 1886, constitute as far as the 

 senior author of this paper is aware, the complete bibliography of the 

 subject up to the present date. 



