OF AN INFINITE ANGLE. 265 



singular anomalies. It has been asserted to be equal to — , a result which is manifestly sym- 

 bolically erroneous, seeing that it does not change sign with a, a property which the expression 

 to be integrated shews must belong to the true integral. Such an objection as this would be held 

 to be fatal to a result in other branches of analysis, and I am at loss to conceive why it has 

 not been allowed the same force in this. It is true a proof has been offered that the integral ought 

 to be independent of a ; but if any thing can be inferred from that proof it is that the 

 integral ought to be indefinite in every case. The proof alluded to is as follows : 



„. Sinaa? Sin a* ,^ , Sin « 

 since ; — d.v = d(ax) = dx, 



Jrf Sino.r /•» Sin» /•< 



I dx = I dz = I 



Sin a; 



dx : 



whence it is stated that the value of the integral is in every case the same as when a = I : yet 

 as I have said before, this inference is evidently erroneous when - o is written for a. The 

 probability is that the true integral is such a function of a as is constant for ordinary values of a, 

 and changes sign with a ; I say ordinary values, because it is easy to shew that the transformation 

 fails as a approaches zero. For since the equation ax = z must be respected, by means of 

 which the transformation is effected, this shews that were a to become indefinitely small, z would 

 not be CO when x approached co ; but in that case the limits for z would be and y {y being 

 an arbitrary finite quantity). Consequently as a approaches towards zero, the integral approaches 

 towards an indeterminate form as its limit. 



The value of the integral when a = 0, would therefore seem to be isolated : and cannot be 

 inferred from the above transformation. Expressed in a series the required expression for the 

 integral is 



^'l.2.3 ^1.2.3.4.5 



which confirms the preceding reasoning in the case when a approaches zero. 



27- The next case which I shall consider is / dx, which has been stated to be 



•'o « + *■ 



equal to — e"'" when h is positive, and to — e'" when b is negative. As in the preceding case, 



Ad lid 



SO here, the symbolical inaccuracy of the integral brought forward is sufficiently indicated by 

 the acknowledged necessity of empirically changing the form of it. As the erroneous principle 

 by which this result is obtained has found its way into a great number of other integrals which, 

 as well as this, are vitiated and rendered erroneous by it, I shall endeavour fully to expose it. 



28. Denoting the required integral by P, we find 



,2o "D /-T- ; ; Sin(6.0) Sin (6 . eo ) 

 dj,P - a- P = - /, Cos bxdx = ^ . 



In the usual process, the last member of this equation is assumed to be zero: and with regard 

 to the first term of it that assumption may be allowed ; but the last term of it, it has been the 

 object of this paper to prove, is indeterminate. It is also to be remarked, that this term forbids us 

 to make b approach towards zero, because when b is indefinitely small the right-hand member 

 approximates to eo . Yet regardless of these cautions the right-liand member has been put equal 

 to zero, and the value of /•* has been then found by integration to be 



P = Ce"' + C'e-"\ 

 Vol. VIII. Paut III. M m 



