OF ARTS AND SCIENCES. 



221 



drawn through L, is a doublet of strength* /a— and axis making an 



angle of 180° with that of the first, at the inverse point of L with 

 respect to the circumference. 



It is to be noticed that the flow function given by (i) is the real 

 factor of the imaginary part of the function 



t^ = — /x 



z — a 



a' 



2 — _ 



a 



4. By differentiating (1) partially with respect to y and — cc, 

 respectively, we get the velocity components at the point (x, y) cor- 

 responding to the case which we have been considering. They are 



u = D^\p = ix 



-Ji^-^y-f 



r- ("-!^T 



r 



\{x-af-\-y''Y a 



' ih^f 



+f 



(2) 



v = — D^^^ = 2ixy. 



X — a 



r^ {ax — 7^) 



[{x-ay + y'Y 



"1(^-9" 



y- 



(3) 



If we make a approach zero in order to obtain the flow in a thin 

 circular plate due to a doublet at the centre, tp as given by (1) grows 

 larger without limit, but u and v approach the definite limits 



Mn 



( a;2 — ?/2 1 ) 



/-]- r- ) 



\ [_x' + 



Vq = 



r]- 



2 ixxy 



(4) 



(5) 



and these expressions solve the problem. 



This case may be treated in another way, however. In Figure 4 let 

 OL =0L' = 8 and 03f. OL = OL' .OM' = OA' =■. r^; then, if 

 there are sources of strength m a,t L and 31 and equal sinks at L' and 

 jW, one of the lines of flow due to this combination will consist in part 

 of the circumference of radius r drawn about as centre. 



* This does not as^ree with the statement marie in the first edition of 

 Basset's Treatise on Hydrodynamics, p. 56, where tiiere seems to be a typo- 

 graphical error. 



