Mr. G. Boole on the Comparison of Transcendents. 551 
The author directly applies his theorem of summation both to the 
solution of particular problems in the comparison of the algebraical 
transcendents, and to the deduction of general theorems. Of the 
latter the most interesting, but not the most general, is a finite ex- 
pression for the value of the sum 
sfoun dx, 
where @ and y denote any rational functions of 7; the equation by 
which the limits of the integrals are determined being of the form 
"= x, in which x is also a rational function of 2. 
The forms of 9, W, and x are quite unrestricted, except by the 
condition of rationality. Previous known theorems of the same 
class, such as Abel’s, suppose W a polynomial and specify the form 
of g. In the author’s result, the rational functions ¢, p, 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 z, 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, 3, 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 
emt ne ac lenb sash Ca 
(" ser(= @—d, @w—A, x) 
in which ¢ (2) is a rational function of x, and in which a, a... @, are 
positive, and A,, A... A, are real, the number of those constants 
being immaterial, may be reduced to the evaluation of a definite 
integral of the form 
V(x) f(r) de, 
in which ¥(v) is a rational function of v of the same order of com- 
plexity as the function g(x). Two limited cases of this theorem are 
referred to as already known,—one due to Cauchy, the other pub- 
lished by the author some years ago. 
The remainder of the paper is occupied with applications of the 
above general theorem of definite integration. Of the Notes by 
which the paper is aecompanied, the first discusses the connexion 
between the author’s symbol and Cauchy’s, and contains twa 
theorems, one exhibiting the general solution of linear differential 
equations with constant coefficients, the other the general integral of 
rational fractions. Both these theorems involve in their expression 
the symbol ©. The second Note is devoted to the interpretation of 
some theorems for the evaluation of multiple integrals, investigated 
in the closing section of the paper. 
