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 C, 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 AJ) as boundary ? It will be found 

 we can without very much trouble. Let angle A-fix = a x 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 + fM0 + 62 



(-!)« 



sin (2n-2)0' 



c sin a 2 + pa 2 + bX 



p™~* 



(- l) n+1 



(2n - 2) 



sin (2n — 2) a 2 



2n-2 



.(27). 



But if, and only if, BA 2 be that stream-line which cuts OC pro- 

 duced orthogonally (27) must be satisfied by 6 = it and p = OB/a, 

 so we must have 



p,7r = c sin a 2 + A* a 2 + °X 



(-1) 



n+i 



sin (2n — 2) a 2 " 

 2n-2 



.(28). 



But by De Morgan's Biff. Calculus, p. 608, when tt/2 <<f) <ir, 



- sin 2(f) — -r sin 40 + ^ sin 6</> 



&c ~^-- 

 &G -~2 2- 



Applying this to (28) it becomes 



(jltt = c sin a, + /xa 2 + b (-^ + |j (29), 



which gives us a 2 in terms of /j./c and b/c. 



Putting u x = at B in (3), or directly from (27), we shall get 

 to determine OB, 



0=c-*- 



bp 



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 value 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 6 + pud + 



6 + tan 



-i(P 



r + i 



tan# 



= /ct7r. ..(31 a). 



