560 Mr. stokes, on THE CRITICAL VALUES OF 



The equation J{6l) is applicable to the case in which \/r(/3) is an arbitrary function, and a, 

 Q, &c., are given. If however \//(/3) should be given, we may find (p^(l3) or »//-^(/3) by the same 

 rule as before. 



In the case in which •vi/(/3) is given, we may find the values of a, Q, &c., without being able to 

 evaluate the integral (59). For this purpose it is sufficient to expand a|/-(/3) according to negative 

 powers of /3, and compare the expansion with that furnished by equation (6l). 



35. The same remarks as to the cases in which we are at liberty to put x for a apply to (60) 

 as to (51), with one exception. In the case in which /(.r) approaches zero as its limit, and is at 

 last always decreasing numerically, or at least never increasing, as x increases, while //(a?) dw is 

 divergent at the limit co , it has been observed that 0(/3) remains finite when j3 vanishes. This 

 however is not the case with ■^'(/S), at least in general. I say in general, because, although 



/ /(«) dx increases indefinitely with its superior limit, we are not entitled at once to conclude from 

 thence that / cos /3 .r/(.i') d.» becomes infinite when /3 vanishes, as will appear in Section III. It may 



X-" coi (ix d,v, where 1 > re > 0, that if f(x) = F(a:) + CaT", 



where F(,r) is such that fF(x)dx is convergent at the limit co , x/'(/3) becomes infinite when /3 

 vanishes; and the same would be true if there were any finite number of terms of the form C.r"". 

 There is no occasion however to enquire whether >(/ (/3) altvays becomes infinite : the point to consider 

 is whether the integral {pf)) is always convergent at the limit zero. 



In considering this question, we may evidently begin the integration relative to as' at any 

 value x„ that we please. Suppose first that we integrate from x' = x^ to x = X, and let '5r(j8) be 

 the result, so that 



(;3) = - f'^ fix') cos j3x'dx'. 



TV J. 



TT 



Let •23-, (/3) be the indefinite integral of •ijr (/3) rf/3 : then, c being a positive quantity, we get from 

 the above equation 



73", (/3) - ■ZB-re) = - f^f(x') \sinlix' - sincaj'} -^ . 



Now put X= CO . Then since / f(x') ^dx is a convergent integral, and its convergency 



remains finite (Art. 39.) when fi vanishes, as may be proved without much difficulty, its value 

 cannot become infinite, and therefore S7-,(/3) does not become infinite when (i vanishes. Now 



/■w((i) cos fix d^ = -sr0) cos fix + xfnr^i^)sin^xd^, (62), 



when X is positive ; and when x = 0, 



/•sr(^)rf(,3)=7!r,(i3): 

 hence in either case f-sr(li) cos/3.rd/3is convergent at the limit zero. Now the quantity by 

 which •23- (/3) differs from \j/(/3) evidently cannot render (59) divergent, and therefore in the case 

 considered the integral (59) is convergent at the limit zero. 



By treating f sr^ji) e-"^ cos (ix dji in the manner in which /w(/3) cos/3a'dj3 is treated in 



(62), it may be shown that the convergency of the former integral remains finite when h vanishes. 

 Hence, not only is the integral (59) convergent, but its value is the limit to which the integral 

 similar to (53) tends when h vanishes. 



