190 



Lord Rayleigh : Hydrodynamical Xotes. 



addition to the latter of ty = — Joor 2 , for then ^ + ^ satisfies 

 V 2 (^ + ^ ) = ; and this is perhaps the simplest method of 

 obtaining it. The results are in harmony ; but the fact is 

 not immediately apparent, inasmuch as Stokes expresses the 

 motion by means of the velocity-potential, whereas here we 

 have employed the stream-function. 



That the subtraction of ^wr 2 makes (10) an harmonic 

 function shows that the series multiplying r 2 can be summed. 

 In fact 



2 ^ sin (mrd/ot) _ cos (20 — a) 1 

 mr[jrir — -%ar) 2. cos a. z 



so that 



+, 



/„_ 1„2. 



? ,s cos (29 — a.) 

 2 COS a 



M 7r/a 



+ 8a 2 S 



sin nir6 J x 



— '-. (IV) 



nir{irir 2 — 1» 2 ) 



In considering the character of the motion denned by (11) in 

 the immediate vicinity of the origin we see that if «< \ir, 

 the term in r 2 preponderates even when n = l. When a = ^7r 

 exactly, the second term in (11) and the first term under X 

 corresponding to n = l become infinite, and the expression 

 demands transformation. We find in this case 



flco = lr 2 + — (6-Itt) cos 2#4 r 2 sin 2$(- logr- ~) 



' 17 \7r Z7T/ 



2 x r 2n su\2n0 

 + rjT* n ( n »_l) ' 



(12) 



the summation commencing at n = 3. On the middle line 

 6 = ±77, we have 



Q.,2 9 r r s „io -^ 



The following are derived from (13) : — 



r. 



-***• 



r. 



-§*-*. 



r. 



Hk*- 



00 



•ooooo 



0-4 



•14112 



08 



•13030 



01 



•02267 



0-5 



•16507 



09 



•07641 



02 



•06296 



06 



•17306 



10 



•60000 



0-3 



•10521 



0-7 



•16210 







The maximum value occurs when r='592. At the point 

 r='592, ^ = i7r ; the fluid is stationary. 



