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LV. RemarJcs on a Paper by the Rev. Brice Bronwin, On the 

 Solution of a particular Differential Equation. By George 

 Boole, Esq. 



To the Editors of the Philosophical Magazine atid Journal. 

 Gentlemen, 



THE current Number (April) of the Philosophical Ma- 

 gazine contains a paper by the Rev. Brice Bronwin, On 

 the Solution of a particular Differential Equation, upon which 

 I beg leave to offer a few remarks. Mr. Bronwin is pleased to 

 consider his researches as supplementary to an investigation of 

 my own, which was published in the Cambridge Mathematical 

 Journal, New Series, Jan. IS*?, and he is led by them to dis- 

 pute the accuracy of certain conclusions at which I had arrived. 

 My immediate design in the present communication, is to show 

 that these conclusions are, on the contrary, perfectly lawful ; 

 but in doing this, I must ask permission to make such accom- 

 panying observations as may render it unnecessary for me to 

 trouble you, under any circumstances, with another letter on 

 the subject. 



It will not ,be irrelevant to premise, that the original design 

 of my investigation was to integrate the equation of Laplace's 

 functions in such a form as would permit the calculation of 

 their actual values. This object I had previously attempted 

 by a method which I have given in the Philosophical Trans- 

 actions for 1 844, part 2, On a General Method in Analysis. 

 That method is founded on the proposition, that every linear 

 differential equation and every linear equation in finite differ- 

 ences, whose coefficients are rational functions of x\ can be 

 reduced to the form 



/o'^W«+/iWp"+/2Wp'"..=x, . . _(i.) 



TT and p being operative functions, which satisfy the conditions 

 7rp2^ = p(,r+ 1 )tl, /(7r)p'« =/(?»)p'"- 



It may be mentioned, that the integrable forms of a differ- 

 ential equation of this description can be determined in various 

 cases, and especially in this, viz. when the equation has but 

 two terms in its first member. Now the equation of Laplace's 

 functions, treated according to the above method, was found 

 to have three terms in its first member; nor was it included 

 in those other forms of the same kind, the integrability of 

 which I had at that time ascertained. 



It occurred to me, therefore, to devise an analogous form 

 ol' the differential equation, in which the operating symbol tt 

 should be replacetl by two symbols tt,,,, x,„ and 1 was led to 

 consider the equation 



■jr„7r.,u + (jpu = 0, (2.) 



