Flow of Viscous Liquids. 359 



These series may be summed. In the first place, 2>" sin nd 

 is the real part of — i2 (»'***)", or of 



•ire 



Thus 



tr n 



l-re ie 

 r sin 



sm no = 



(17) 



l-2rcos0 + r 2, ' ' ' 



Again, Sn" 1 ^" sin nd is the real part of — i%n~ l {re x6 ) n , or of 



Hog (l — re ie ); so that 



Sw-y sin h0 = tan- 1 , rsm6> (18) 



1 — rcosa v 



Thus, as the expression for -^ in finite terms, we have 



7T . t|t = 



(1— 7' 2 )rsin0 , OJ _ _ x r sin fl 



1 — 27' cos + 



H- 2 tan' 



1—7' COS 0' 



(19) 



In (19) the separate parts admit of simple geometrical 

 interpretation. The second represents simply twice the angle 



Fig, 1. 



PAO, fig. 1, which is known to constitute a solution of 

 V 2 -v|r = 0. In the first term, 



rsinfl _PM _ sin FAQ 



l^rcosfl + r 2 AP 2 AP J 



which is also obviously a solution of V 2/ ^ = 0. The remaining 

 part of (19) is not a solution of \/ 2 yjr = 0; but it satisfies 

 V 4 ^ = 0, as being derived from a solution of V 2 ^ = by 

 multiplication with r 2 . 



On the foundation of (19) we may build up by simple 

 integration the general expression for yjr subject to the con- 

 ditions that d^jr/dr vanishes over the whole circumference, 

 and that dty/d0 has any prescribed values consistent with the 

 recurrence of -xfr. 



