110 PROFESSOR STOKES, ON THE DISCONTINUITY 



equation (2), the number of terms comprised in this about will increase with i, the order of the 

 term of minimum modulus. If we suppose that we are uncertain to the extent of n terms, the 

 sum of the moduli of these n nearly equal terms will be 



nearly. It seems as if « must increase with i, but not so fast as i. If we suppose that it is of 

 the form ki or kp, the sum of the n terms will be a quantity of the order e~^\ But even 

 if n increased as any power p of i, however great, still the sura of the n terms would be a quan- 

 tity of the order p'"'~^e'''', which when p was infinitely increased would become infinitely small 

 in comparison with the modulus e-p"«o*28 ^f ^^^ ^^^.^ multiplied by C in (4), provided 6 had 

 any given value differing from zero or a multiple of tt. Hence if B have any value lying 

 between a and v — a, or else between tt + a and Stt — a, where a is a small positive quantity 

 •which in the end may be made as small as we please, the quantity C in (4) cannot pass from 

 one of its values to another without rendering the function u discontinuous, which it is not. 

 But when = or = tt, the term Ce""' becomes merged in the vagueness with which, in this 

 case, the divergent series defines the function. Hence we arrive in a way quite different from 

 that of Art. 5 at the conclusions enunciated in Art. 6, 



8. Nor is this all. When the terms of a regular series are alternately positive and 

 negative, the series may be converted by the formulae of finite differences into others which 

 converge rapidly. In the present case the terms are not simply positive and negative alternately, 



except when 6 is an odd multiple of — , but the same methods will apply with the proper 



modification. Suppose that we sum the series (4) directly as far as terms of the order i - 1 

 inclusive. Omitting the common factor e-<"+'>* -i, which may be restored in the end, we have 

 for the rest of the series 



If we denote by X) or 1 + A the operation of passing from Hi to /x^^^, and separate symbols 

 of operation, this becomes 



(1 + e-'^~'D + e-">^~'I)' + ...)^i, 



or {l-(l + A)e-^''^^|-Vi. 



Now 1 -e-'*^-'= 1 - COS 20 + \/- 1 sin20 = 2sin 0e ^ , 

 which reduces the expression to 



(2 sin0)-^e('"^^''^{l - (2 sin0)-»e"^^^*)^^A}-V„ 

 or, putting q for (2 sin 0)"S to 



Now if p be very large, and m belong to the part of the series where the moduli of con- 

 secutive terms are nearly equal, the successive differences Afji<, A^^l^,... will decrease with great 

 rapidity. Hence if 6 have any given value different from zero or a multiple of tt, by taking 



