Motion of a Viscous Incompressible Fluid. 



331 



discontinuity. When x is real — the case in which Stokes 

 was mainly interested, or a pure imaginary, the calculations 

 are o£ course simplified. 



If we take as S, and S 2 the two series in (8), the real 

 and imaginary parts of each are readily separated. Thus if 



$i = s 1 + it 1 , S 2 = S 2 + ^ 2 , 

 we have on introduction of irj 



■h 



1 



9y 



9y 2 



2.3.5.6^ 2.3.5.6.8.9.11.12 

 9t? 3 9V 



h 2.3 + 2.3.5.6.8.9 



.sv 



v 



3.4 



93^10 



3.4.6.7.9.10 



t 2 =v— . 



9V 



■W3 



9 4 V 



3.4.6.7 T 3.4.6.7.9.10.12.13 



(10) 



(11) 

 (12) 

 (13) 

 (14) 

 le t], t%. 



in which it will be seen that s lt s 2 are even in 77, wh 

 are odd. 



When v < 2 these ascending series are suitable. When 

 77> 2, it is better to use the descending series, but for this 

 purpose it is necessary to know the connexion between the 

 constants A, B and C, D. For x — iri these are (Stokes) 



Thus for the first series Si (A = l, B = in (8)) 



log D = T5820516, C=D «**/«; . . 

 and for S 2 (A = 0, B = l) 



log D' = 1-4012366, -C' = D'<r*»/ 6 , . . 

 so that if the two functions in (9) be called 2i and 2 2 , 



S^CSi + DSs, S 2 =0% + D'S 2 . . . 



(15) 



(16) 



(17) 

 (18) 



These values may be confirmed by a comparison of results 

 calculated first from the ascending series and secondly from 

 the descending series when 17 = 2. Much of the necessary 



