188 Lord Rayleigli : Hydrodynamical Notes. 



the stream-function i/r, satisfying V 2 ^ = 0, maybe expressed 

 by a series of terms 



ir/a • /i / 27r/« • c» /i , nirla • /\ , 



r sin 7TC7/a, ?' ' &m27ra/cc, . . . r sm?nru/cc, 



where n is an integer, making ^ = when = or 6 = a. 

 In the immediate vicinity of the origin the first term pre- 

 dominates. For example, if the angle be a right angle, 



ty = r 2 sin 20 — 2xy, (1) 



if we introduce rectangular coordinates. 



The possibility of irrotational motion depends upon the 

 fixed boundary not being closed. If a<vr, the motion near 

 the origin is finite ; but if a > 77, the velocities deduced from 

 yjr become infinite. 



If there be rotation, motion may take place even though 

 the boundary be closed. For example, the circuit may be 

 completed by the arc of the circle r=l. In the case which 

 it is proposed to consider the rotation co is uniform, and the 

 motion may be regarded as steady. The stream-function 

 then satisfies the general equation 



V-> = ^Y/^r 2 + ^/r/?/ 2 = 2a), .... (2) 



or in polar coordinates 



*±+l%.+ \*±^ (3) 



dr 2 r dr r 2 ad 2 ' 



"When the angle is a right angle, it might perhaps be 

 expected that there should be a simple expression for \jr in 

 powers of x and y, analogous to (1) and applicable to the 

 immediate vicinity of the origin ; but we may easily satisfy 

 ourselves that no such expression exists *. In order to 

 express the motion we must find solutions of (3) subject to 

 the conditions that i/r = when = and when = a. 



For this purpose we assume, as we may do, that 



ty — SR ;l sin nirOIcc, (4) 



where n is integral and R w a function of r only ; and in 

 deducing V 2 ^ we may perform the differentiations with 

 respect to 6 (as well as with respect to r) under the sign of 

 summation, since i/r = at the limits. Thus 



Vt = Z 7T + --J ^-t\ t sin . . (o) 



* In strictness the satisfaction of (2) at the origin is inconsistent with 

 the evanescence of 4> on the rectangular axes. 



