461 



silicate to carbonate, and then back again to silicate, the only limit to 

 the process would be the satisfying of the mutual affinities of the 

 silica and the basic oxyds present. 



III. " On the Comparison of Transcendents, with certain appli- 

 cations to the Theory of Definite Integrals." By GEORGE 

 BOOLE, Esq., Professor of Mathematics in Queen's College, 

 Cork. Communicated by Professor W. F. DONKIN, F.R.S. 

 Received March 16, 1857. 



(Abstract.) 



The following objects are contemplated in this paper : 



1 st. The demonstration of a fundamental theorem for the sum- 

 mation of integrals whose limits are determined by the roots of an 

 algebraic equation. 



2ndly. The application of that theorem to the comparison of 

 algebraical transcendents. 



3rdly. Its application to the comparison of functional transcendents, 

 i. e. of transcendents in the differential expression of which an arbi- 

 trary functional sign is involved. 



4thly. Certain extensions of the theory of definite integrals both 

 single and multiple, founded upon the results of the application last 

 mentioned. 



In the expression of the fundamental theorem for the summation 

 of integrals, the author introduces a symbol, 9, similar in its defini- 

 tion to the symbol employed by Cauchy in the Calculus of Residues, 

 but involving an additional element. The interpretation of this 

 symbol is not arbitrary, but is suggested by the results of the inves- 

 tigation by which the theorem of summation is obtained. All the 

 general theorems demonstrated in the memoir either involve this 

 symbol in their expression, or are immediate consequences of theorems 

 into the expression of which it enters. 



The author directly applies his theorem of summation both to the 

 solution of particular problems in the comparison of the algebraical 

 transcendants, and to the deduction of general theorems. Of the 



2 M 2 



