462 



latter the most interesting, but not the most general, is a finite ex- 

 pression for the value of the sum 



where <f> and vj/ denote any rational functions of x ; the equation by 

 which the limits of the integrals are determined being of the form 



m 



i|/ n =X, in which ^ is also a rational function of x. 



The forms of y, \jj, and ^ are quite unrestricted, except by the 

 condition of rationality. Previous known theorems of the same 

 class, such as Abel's, suppose ^ a polynomial and specify the form 

 of 0. In the author's result, the rational functions <, $, and x are 

 not decomposed. In a subsequent part of the paper, after inves- 

 tigating a general theorem applicable to the summation of all 

 transcendents which are irrational from containing under the sign of 

 integration any function which can be expressed as a root of an 

 equation whose coefficients are rational functions of x, he explains, 

 by means of it, the cause of the peculiarity above noticed. 



In the section on functional transcendents, a remarkable case 

 presents itself in which the several integrals under the sign of sum- 

 mation, S, close up, if the expression may be allowed, into a single 

 integral taken between the limits of negative and positive infinity. 

 The result is an exceedingly general theorem of definite integration, 

 by means of which it is demonstrated, that the evaluation of any 

 definite integral of the form 



C* 



I 



J- 



in which fy (x) is a rational function of x, and in which a l a 2 . . # are 

 positive, and \ v X 2 . . X n are real, the number of those constants 

 being immaterial, may be reduced to the evaluation of a definite 

 integral of the form 



r^(v}f(v)dv, 



in which \//(t>) is a rational function of v of the same order of com- 

 plexity as the function <j)(x). Two limited cases of this theorem are 

 referred to as already known, one due to Canchy, the other pub- 

 lished by the author some years ago. 



The remainder of the paper is occupied with applications of the 



