360 Professor Sir William Rowan Hamilton on 



the disappearance of the image must rather be ascribed to a 

 peculiar action of the oxygen of the atmosphere, on which 

 subject I recently made a preliminary communication to Sir 

 David Brewster and Professor Magnus, and intend shortly to 

 publish the details in these Annalen. Dr. Draper is so con- 

 vinced of the radiation of these images, that he pretends to 

 preserve them by covering them, and in this scientific man- 

 ner has discovered specific light analogous to specific heat ! 

 Konigsberg, 30th April, 1843. 



XLV. On an Expression for the Numbers of Bernoulli, by 

 means of a Definite Integral ; and on some connected Pro- 

 cesses of Summation and Integration. By Sir William 

 Rowan Hamilton, LL.D., P.R.I.A., Member of several 

 Scientific Societies at Home and Abroad, Andrew^ Professor 

 of Astronomy in the University of Dublin, and Royal Astro- 

 nomer of Ireland. 



THE following analysis, extracted from a paper which has 

 been in part communicated to the Royal Irish Academy, 

 but has not yet been printed, may interest some readers of the 

 Philosophical Magazine. 



1. Let us consider the function of two real variables, m and 

 n, represented by the definite integral 



P °° L /sin x\ m 

 y =/ dx\ I coswx; . . (1.) 



in which we shall suppose that m is greater than zero; and 

 which gives evidently the general relation 



Jm, — n "m, n* 

 By changing m to m + 1 ; integrating first the factor 

 x~ m ~ d x, and observing that x~ m sin x m ^" 1 cos n x 

 vanishes both when x = 0, and when x = oo ; and then putting 



the differential coefficient -, — (sin x mJt cos n x) under the 



dx ' 



form 



| sin x m {(m + 1 + n) cos (n x + x)-+ (m+l—ri) cos (nx — x) }; 



we are conducted to the following equation, in finite and par- 

 tial differences, 



2 m w . , ■" = (m + 1 4- n) y . . +(m + l—n)y •, (2.) 



and if we suppose that the difference between the two va- 

 riables on which y depends is an even integer number, this 

 equation takes the form 



