230 Royal Society. 



movements of their appropriate segments, and, thus far, to maintain 

 the belief in their essential independence. At the same time, the ob- 

 servations on the diffused influence of shocks accord with those of 

 the coordinate similar movements of all the segments, in proving 

 their close mutual relations and mutual influence. 



Nov. 23, 1854.— Thomas Bell, Esq., V.P., in the Chair. 



" On the Theory of Definite Integrals." By \V. H. L. Russell, 

 Esq., B.A. 



I propose in the following paper to investigate some new methods 

 for summing various kinds of series, including almost all of the more 

 important which are met with in analysis, by means of definite 

 integrals, and to apply the same to the evaluation of a large number 

 of definite integrals. In a paper which appeared in the Cambridge 

 and Dublin Mathematical Journal for May 1S54, I applied certain 

 of these series to the integration of linear differential equations by 

 means of definite integrals. Now Professor Boole has shown, in an 

 admirable Memoir which appeared in the Philosophical Transactions 

 for the year 1844, that the methods which he has invented for the 

 integration of linear differential equations in finite terms, lead to the 

 summation of numerous series of an exactly similar nature, whence 

 it follows that the combination of his methods of summation with 

 mine, leads to the evaluation of a large number of definite integrals, 

 as will be shown in this paper. It is hence evident that the discovery 

 of other modes of summing these series by means of definite integrals 

 must in all cases lead to the evaluation of new groups of definite 

 integrals, as will also be shown in the following pages. I then point 

 out that these investigations are equivalent to finding all the more 

 important definite integrals whose values can be obtained in finite 

 terms by the solution of linear differential equations with variable 

 coefficients. Again, there are certain algebraical equations which 

 can be solved at once by Lagrange's series, and by common alge- 

 braical processes ; the summation of the former by means of definite 

 integrals affords us a new class of results, which I next consider. 

 A continental mathematician, M. Smaasen, has given, in a recent 

 volume of Crelle's Journal, certain methods of combining series 

 together which give us the means of reducing various multiple inte- 

 grals to single ones. The series hitherto considered are what have 

 been denominated "factorial series;" but, lastly, I proceed to show 

 that analogous processes extend to series of a very complicated 

 nature and of an entirely different form, and for that purpose sum 

 by means of definite integrals certain series, whose values are obtained 

 in finite terms in the " Exercices des Mathematiques " by means 

 of the Residual Calculus. The total result will be the evaluation 

 of an enormous number of definite integrals on an entirely new type, 

 and the application of definite integrals to the summation of many 

 intricate series. 



Dec. 7, 1854.— Colonel Sabine, V.P., in the Chair. 



The following communications were read : — 



" On the Attraction of the Himalaya Mountains, and of the elevated 



