174 



where it is desired not only to shade a diseased eye, but also to pro- 

 tect its nerves from strong light admitted by the sound eye. When 

 not only coloured light but a certain degree of darkness is required, 

 this can be readily and delicately graduated by employing shades of 

 different depths of colour. 



6. Masks of gelatine paper for protecting the eyes of travellers 

 against the glare of snow-fields and of sandy deserts. 



III. " On the Theory of Definite Integrals." By W. H. L. 

 RUSSELL, Esq., B.A. Communicated by A. CAYLEY, Esq., 

 F.R.S. Received October 30, 1854. 



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 1854, 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 



