114 PROFESSOR STOKES, ON THE DISCONTINUITY 



then equation (13) would become a mere truism. However I shall not pursue this subject 

 further, as these singular values of divergent series appear to be merely matters of 

 curiosity. 



12. In order still further to illustrate the subject, before going on to the actual 



application of the principles here establisWd, let us consider the function defined by the 



equation 



i.l , 1.1.3, 



M = l + la? - — ,v^ + -ai'- (15) 



^ 2.4 2.4.6 ^ ' 



Suppose that we have to deal with such values only of the imaginary variable a; as have 

 their moduli less than unity. For such values the series (15) is convergent, and the equation 



(15) assigns a determinate and unique value to u. Now we happen to know that the series 

 is the development of (l + a?)'. But this function admits of one or other of the following 

 developments according to descending powers of x : — 



,,,1.1.1.1.3, 



^ 2.4 2.4.0 ^ ' 



u = — X^ — %X 2 + UO * X~^ ■>r (17) 



^ 2.4 2.4.6 



Let X = p (cos Q Jrs/ -\ sin 0), and let x^ denote that square root of x which has \Q for 

 its amplitude. Although the series (l6), (17) are divergent when o < 1, they may in general, 

 for a given value of Q, be employed in actual numerical calculation, by subjecting them to 

 the transformation of Art. 8, provided p do not differ too much from 1. The greater be the 

 accuracy required, Q being given, the less must p differ from 1 if we would employ the series 



(16) or (17) in place of (15). It remains to be found which of these series must be taken. 



If Q lie between (2i - l) tt + a and (2i + 1) tt - a, where i is any positive or negative 

 integer or zero, and a a small positive quantity which in the end may be made as small as 

 we please, either series (l6) or (17) may by the method of Art. 8 be converted into another, 

 which is at first sufficiently convergent to give u with a sufficient degree of accuracy by 

 employing a finite number ohly 6f terms. If m terms be summed directly, and in 

 the formula of Art. 8 the n**" difference be the last which yields significant figures, the 

 number of terms actually employed in some way or other in the summation will be wi + « + l . 

 And in this case we cannot pass from one to the other of the two series (l6), (17) without 

 rendering u discontinuous. But when Q passes through an odd multiple of tt we may have 

 to pass from one of the two smes to the other. Now when Q is increased by 27r the Series 

 (l6) or (17) changes sign, whereas (15) remains unchanged. Therefore in calculating w for tWo 

 values of 9 differing by Stt we must employ the two series (16) and (17), one in each case. 

 Hence We must employ one of the series from = — tt to = tt, the other from = tt to 

 B a= 3rr, and so on ; and therefore if we knew which series to take for some one value 

 of 8r everything Would be determined. 



Now when '|0 it 1 the Series '(15) becomes identical Vith (1 6) when Q has the particular 

 Value 0. Hencfe (T6) ^U 'not (17) gives the true value of « wheal - tt < Q <-k. 



