inside a Hollow Unlimited Boundary. 
225 
Let the quantities m and a m be so chosen that 
S ( a m sin mO) = 0 (5), 
for all values of 0 that lie between certain fixed limits, let us say 
from 0 to 7 t/ 2. We might here for 7 t/ 2 substitute any aliquot 
part of 7 r, but we begin with 7 t/ 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 0 = 0 to 0 = 7 t/ 2, the velocities and pressures on each 
side of the arc AC 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, we 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 6 + u y sin 6) with regard to 0, or 
u x sin 0 — u y cos 0 with regard to r comes out 
For points inside the circle, or for r< a 
n r m+i 
sin (m -f 1) 0 
cr sin 0 + yad + S 
a m (m + 1) 
+ S 
ca sm a + 
a m a 
m + 1 
sin (m -f 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 0 + ya0 + S 
a 
jm-\ ( m _ 1) 
sin ( m — 1) 0 
ca sin a + paa. 
+ S 
a m a . . \ 
sm (m — 1) a 
m — 1 
( 7 ). 
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 1 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 0 = 7 r, and we shall get from (6) 
/xa7r = ca sin a x + ya 'x x + S 
a m a 
m + 1 
sin (m + l)a! 
( 8 ). 
This equation determines or the angle AOx. If we put 
a = otj in (6) it becomes the equation to ABA' . Substituting in 
