950 



3. "On the Algebraic Expression of the Number of Partitions of 

 which a given number is susceptible." By Sir John F. W. Herschel, 

 Bart., K.H., F.R.S. &c. 



The object of this paper is to exemplify and extend the mode of 

 analysis explained by the author in two former communications to 

 the Royal Society " On the Development of Exponential Functions," 

 and " On Circulating Functions," to a case in the theory of numbers 

 in which they afford remarkable facilities, viz. that of the partitions 

 of which a given number is susceptible. The separation of the 

 symbols of operation from those of quantity, in the mode explained 

 in the former of those communications, allows of the expression of 

 the sums of certain series entering into this theory, under a form 

 susceptible of resolution (by a theorem here given) into two portions, 

 one of v/hich, a rational function of the independent variable or 

 number to be subdivided, expresses approximately, as a rational frac- 

 tion, the number of partitions ; the other, a periodic or circulating 

 function, expresses the fraction, less than unity, by which the other 

 portion differs from an exact integer number, and which, applied 

 with its proper sign to that former portion, reduces it to an integer. 

 In the mode of procedure here followed all subdivision into cases 

 according to the numerical constitution of the number to be subdi- 

 vided is avoided, and a uniform treatment is carried throughout. 



May 30, 1850. 

 The EARL OF ROSSE, President, in the Chair. 



Sir Benjamin C. Brodie, Bart., gave notice, that, at the next or- 

 dinary Meeting of the Society, he would propose the Right Ho- 

 nourable Lord Londesborough for election into the Society. 



Mr Weld communicated the following particulars respecting the 

 original model of the safety-lamp which was presented to the Society 

 at this meeting by Joseph Hodgson, Esq., F.R.S. 



In November 1815, Sir Humphry Davy read a paper before the 

 Royal Society ' On the Fire damp of Coal Mines, and on Methods 

 of lighting the Mines so as to prevent its Explosion.' In this commu- 

 nication he described a safe light, " which became extinguished \Y\i&n 

 introduced into very explosive mixtures of fire-damp:" but as this 

 fell short of the philosopher's wishes, he instituted a fresh series of 

 experiments, which resulted in his invention of the safety-lamp 

 described in a paper read before the Society in January 1816. 

 "The invention," he says, "consists in covering or surrounding the 

 flame of a lamp or candle by a wire sieve;" and he adds, "when a 

 lighted lamp or candle screwed into a ring soldered to a cylinder of 

 wire gauze, having no apertures except those of the gauze, is intro- 

 duced into the most explosive mixture of carburetted hydrogen and 

 air, the cylinder becomes filled with a bright flame, and this flame 

 continues to burn as long as the mixture is explosive." The model 

 in the possession of the Royal Society answers in every respect to 



