inside a Hollow Unlimited Boundary. 



225 



Let the quantities m and a m be so chosen that 



S (a m sin mO) = (5), 



for all values of that lie between certain fixed limits, let us say 

 from to 7r/2. We might here for 7r/2 substitute any aliquot 

 part of 7r, but we begin with ir/2 as the simplest supposition. 

 We shall see presently that condition (5) can be satisfied in 

 an infinite number of ways. Supposing it so satisfied, we see 

 that when r=a, that is for points on the circumference of the 

 circle from = to 6 = tt/2, the velocities and pressures on each 

 side of the arc AG A' are continuous. The general shape of 

 the stream lines will be like Oead. If there is any stream line 

 such as ABA' which cuts xO produced orthogonally, Ave shall take 

 it for a rigid boundary. We shall directly shew that there is 

 such an one. The equation to the stream lines, which is got 

 either by integrating r (u x cos + u y sin 0) with regard to 0, or 

 u x sin — u y cos with regard to r comes out 



For points inside the circle, or for r<a 



sin (to + 1) 



cr sin + fia0 + S 



_a m (m + l) 

 + S 



= ca sin a + fiau 



Oj m 0j 



m + 1 



sin (m + 1) a 



(6), 



where a and a are the polar coordinates of the point a where the 

 particular stream line we are considering cuts the circle r = a, 

 so that a may be called the parameter of the system of stream 

 lines. 



For points outside the circle, or for r > a 

 cr sin + fxaO + S 



-rsm(m — 1)0 = ca sin a 4- aaa 



(to — 1) 



+ S 



a m a 



TO — 1 



sin (to — l)a 



o\ 



These stream lines are of course continuous and touch each 

 other at the point a. If there be such a stream line as ABA' and 

 if for this particular stream line a = a x then farther supposing the 

 constants so chosen that the point B is within the circle r = a (and 

 we shall presently see that they can be), (6) must be satisfied by 

 r= OB and = tt, and we shall get from (6) 



fxair = ca sin a x + pai^ + S 



a m a 



■m 



— - sin (to + l)oti 



•(8). 



This equation determines a x or the angle AOx. If we put 

 a = a x in (6) it becomes the equation to ABA' . Substituting in 



