OF ARBITRARY CONSTANTS, &c. 113 



■.2 



y — y 

 lim.^ol'in-x=o^('».y) = ""'-^=0 -7— = - 1 ; 



in the latter 



So in the case of the periodic series if we denote by ^ a small positive quantity 



lim.f^lim.„^„ 0(n, c + ^ = lim-f=o/(<' + = * 5 

 but we know that 



lim.„^„ lim.f^o ^ («. c ± ^) = lim.„,„ («, c) = 1 (a + 6). 



Similarly in the case of the series (14) if we denote its sum by ■^{a) = sr (p, 9), and use 

 the term limit in an extended sense, so as to understand by lim.p^„jP(p) a function of p to 

 which F(p) may be regarded as equal when p is large enough, and if we suppose to be a 

 small positive quantity, we have from (11) 



lim.e=olim.,=„ ■ar(p, 6) = lim.^^o |2c-»' fef^'da -y/^l -n-h-"'} 



° 



Hm.^„lim W(p,-e) = lim.^^o {Se-"' TV da + n/^ tt^c""'} 



''0 



''0 

 whereas equation (13) may be expressed by 



lim.^=„ lim.j^o w (p, ± 0) = lim.^„;^(^) = Sc""' fef'dp. 



*'o 



There is however this difference between the two cases, that in the case of the periodic 

 series the series whose general term is A0 (w, c) is convergent, and may be actually summed 

 to any assigned degree of accuracy, whereas the series (13), though at first convergent, is 

 ultimately divergent ; and though we know that we must stop somewhere about the least 

 term, that alone does not enable us to find the sum, except subject to an uncertainty com- 

 parable with e"*"'. Unless therefore it be possible to apply to the series (13) some transfor- 

 mation rendering it capable of summation to a degree of accuracy incomparably superior to 

 this, the equation (13) must be regarded as a mere symbolical result. We might indeed 

 define the sum of the ultimately divergent series (13) to mean the sura taken to as many 

 terms as should make the equation (13) true, and express that condition in a manner which 

 would not require the quantity taken to denote the number of terms to be integral ; but 

 Vol. X. Part I. 15 



