OF ARBITRARY CONSTANTS, &c. Ill 



p sufficiently great, we may transform the series about where it ceases to converge into one 

 which is at first rapidly convergent, and thus a quantity which may be taken as a measure 

 of the remaining uncertainty will become incomparably smaller even than ^, much more, 

 incomparably smaller than the modulus of e'"^. But if 6 = or = tt, the above transform- 

 ation fails, since q becomes infinite. In this case if we want to calculate u closer than to 

 admit of the uncertainty to which we are liable, knowing only that we must stop somewhere 

 about the place where the series begins to diverge after having been convergent, we must 

 have recourse to the ascending series (1) or (3), or to some perfectly distinct method. The 

 usual method by which Sm^ is made to depend on Ju^dx would evidently fail, in consequence 

 of the divergence of the integral. 



9. In applying practically the transformation of the last article to the summation of 

 the series (4), it would not usually, when p was very large, be necessary to go as far as the 

 part of the series where the moduli of consecutive terms are nearly equal. It would be 

 sufficient to deduct /,' 2Z... from the logarithms of ^£^.„ Atj+j..,, where I is nearly equal to 

 the mean increment of the logarithms at that part of the series, to associate the factor/ whose 

 logarithm is I with the symbol D, and take the differences of the numbers, 



However, my object leads me to consider, not the actual summation of the series, but the 

 theoretical possibility of summation, and consequent interpretation of the equation (4). 



10. The mode of discontinuity of the constant C having been now ascertained, nothing 

 more remains except to determine that constant, which is done at once. Writing v — la for 

 a in (4) after having put for u its first expression in (3), we have 



Se"' fe-'^'da = -y/ - xCe"'-- + — - ... 

 > a 2a3 



whenee, putting a=<X) , we have C= v — 1 ir^. Hence we get for the general expression 

 for C in (4), 



C = \/^ TT^, when < fl < tt, I , ^ 



. I (10) 



C = - V - 1 TT^ when tt < 0<27r; I 



and therefore from (3) and (4) 



1 1 1.3 1.3.5 



2e 



-' f e'-da = W- 1 TT^e-V 1 + -^ + il^ + ^_^+ (11) 



the sign being + or — according as 6, the amplitude of a, is comprised within the limits 

 and TT, or tt and 27r. • 



Writing a\/— 1 for a in (11), which comes to altering the origin of by — , we find 



, r 1 , 1^1 1 1-3 1-3.5 



Jo a ^a^ 9.'a^ 2° a' 



