inside a Hollow Unlimited Boundary. 
231 
Art. 300. The advantage of the present method is that with a 
given change in the boundary, we are enabled to calculate the 
resulting effects on the velocity at a distance. 
7. But another interesting extension of the preceding Article 
may be readily obtained. In Figs. 4 and 5 the idea naturally 
occurs to one. Supposing B, instead of being within the circle, 
were outside of it on the left of O', so that BED now cuts the 
circle in two points. Let us call them A 2 and A 1 . Can we get 
a solution with the line BA 2 A^D as boundary ? It will be found 
we can without very much trouble. Let angle A x 0x = and 
angle A 2 0x=a 2 . We shall again begin with the form (16) for 
equation (5). From (7) the equation of BA 2 will be 
cp sin 0 + p,0 -\-bX 
= c sin a 2 + ya 2 + bX 
/ Yn+n sin (2ft — 2)0 
(— l) n+1 x 
P‘ 
(- 1)^ +1 
( 2 n - 2 )_ 
w sin (2n — 2) a 2 
x 
••(27). 
But if, and only if, BA 2 be that stream-line which cuts OC pro- 
duced orthogonally (27) must be satisfied by 0 — it and p = OB/a, 
so we must have 
pLTT 
= c sin a 2 -f ya 2 + bX 
But by De Morgan’s Diff. Calculus, p. 608, when 7t/2 < c/> < 7r, 
2 
i sin 2 </> — q sin 4*4* + \ sin — & c * = 9 
Z T U Z Z 
Applying this to (28) it becomes 
fjLir = c sin a 2 + fia 2 + b (29), 
which gives us a 2 in terms of fi/c and b/c. 
Putting u x = 0 at B in (3), or directly from (27), we shall get 
to determine OB, 
h P 
1 +P 2 
(30), 
which is the same form as (19), but as here we must have p > 1 
we shall find that its first approximate vajue is ( p, + b)/c which 
is therefore 
>1 
(31). 
Next summing the series in (27) we shall get for the equation 
of BA 2 , 
Cp sin 0 -j- pL0 -f- 
b 
0 + tan 
= yrr . . . (31 a). 
