564 Mb. stokes, ON THE CRITICAL VALUES OF 



("^ F{a:, K)dx, may be numerically less than a given constant e which may be as small as we 

 •'I 

 please, X increases beyond all limit when h vanishes. 



The reasoning of the preceding article leads to the following theorems. 



If F = r F{x, h) dx, if Fo be the limit of V when ^ = 0, and if F(x, 0) =f{x) ; then, if 



'^a p CD 



the convergency of the integral V do not become infinitely slow when h vanishes, / /(.r) dx must 

 be convergent, and its value must be Fj. But in the contrary case either F must become infinite 

 when h vanishes, or the integral /" /(«) dx must be divergent, or if it be convergent its value 



•'o 



must differ from F„. 



When the integral { f{x) dx is convergent, if we denote its value by U, we shall have in all 

 cases 



F(x, h) dx approaches when h is first made to vanish and then 



X to become infinite. 



The same remarks which have been made with reference to the convergency of series such as 

 (15) or (30) for values of a* near critical values will apply to the convergency of integrals such 

 as (50), (59) or (65). 



The question of the convergency or divergency of an integral might arise, not from one of the 

 limits of integration being eo , but from the circumstance that the quantity under the integral sign 

 becomes infinite within the limits of integration. The reasoning of the preceding article will 

 apply, with no material alteration, to this case also. 



41. It may not be uninteresting to consider the bearing of the reasoning contained in this 

 Section and a method frequently given of determining the values of two definite integrals, more 

 especially as the values assigned to the integrals have recently been called into question, on account 

 of their discontinuity. 



Consider first the integral 



r " sin ax 

 u= / dx, (71). 



J, X 



where a is supposed positive. Consider also the integral 



/•» . sin a a? 



V = / 6"*' dx. 



J, X 



It is easy to prove that the integral v is convergent, and that its convergency does not become 

 infinitely slow when h vanishes. Consequently the integral u is also convergent, (as might also be 

 proved directly in the same way as in the case of v,) and its value is the limit of ti for h = 0. 

 But we have 



dv r" k. • , " 



— =- / e~ sinaxdx = — — ; 



dh J„ a- + h- 



,A 

 whence v = C - tan - ; 



a 



