236 Mr Sharpe, Liquid Motion from a Single Source 



same quantity to be discharged in the same time through the 

 asymptotic cylinder, we shall get 



C 2 -b a = -2/xa (57). 



Or we may suppose V= 0, then from (52) it is easy to shew that 

 the quantity of fluid discharged through the asymptotic cone at 

 infinity in unit of time = 27r6„« (I — cos X ). 6 X being the semi- 

 angle of the asymptotic cone, therefore we must have 



2irb a (1 — cos 0j) = ^/xair (58), 



and remembering from (53) that cos d x = G. 2 /b a we get for C 2 the 

 same value as is given by (57). From (58) we see that as X 

 cannot vanish without p vanishing, which we do not suppose, the 

 bowndary cannot be -parabolic at infinity. 



11. Looking at (45) and (47) let us put for brevity a,,, for 

 a n /(/jt, — b ). Then (51) may be written, being the equation to 

 BA (Fig. 1), 



= — fia — b»a cos 6 + = Vr" sin 2 6 - (/n 



— ((i — b ) sin 



3a ri+ i2a^ + *\a n (n + l) 



b ) a cos — 



.(59). 



(53) may be written, being the equation to A D (Fig. 1), 



= - fia - b a cos + <> Vr 2 sin 2 — (/m — b ) a — 



— ((*>— b ) sin 6 



"2a 2 

 Sr 



8r 2 



nr r 



P ' 



(60). 



As the curves BA, AD meet in the same point A on the circle 

 AG A' it is evident that when r = a the two equations (59) and (60) 

 should give us the same value for 0, therefore the coefficients of 

 (p _ ft o ) in both equations must be equal. We must therefore 

 establish the truth of the following equation, 



sin 



P' 



% 2re + l p ,' 



i n(n+l) 

 This can be done thus. From (50) 

 sin . P ' 



n(n + l) i, 



Applying this to (61) it becomes 



l-cos0-sin 2 6>...(61). 



P„ sin Odd. 



/oH- 



j P 2 + 2 (2n + 1) a n P n [d0 = l-cos0- sin 2 



