362 



Lord Rayleigh on ih 



When r = l, 







i£ = -22B n cosn0, 

 dr 



.... (26) 



and is to be made to vanish for all values of 6 except in the 

 neighbourhood of = 0. If we suppose that the integral of 

 d^jdr with respect to 6 over the whole region where d^fr/dr 

 is sensible, is 2, we find 



B o = ~2^' Btt== "^' (27) 



the second equation applying to all values of n other than 0. 

 Hence 



'' _7r.^=_. 1(1 -r 2 ) + (l-r 2 )2;V tt cos7i0, . . (28) 



or in finite terms, 



— • f=-Kl-O + (l-^) 1 _ 2rC0S , + } . - • • (29) 



The equation may also be written 

 In (29), 



2 -^ i-LoX^ • • • (30) 



1— rcosfl _AM _ cos PAO 

 l-2rcos0 + r* AP 2 AP ' 



which is a solution of V 2 ^ = 0. When multiplied by r 2 , or 

 by (1— r 2 ), it remains a solution of V 4 ^ = 0. 



In (30) we may write a? for r cos #, and if the point under 

 consideration lie upon the axis, as' 2 = r 2 . Hence on the axis, 



-27r.^=(l+ < r) 2 , (31) 



-J± = (l +X ), (32) 



equations which may be applied at all points except near as = 1. 

 It appears from (32) that the velocity transverse to the axis 

 increases continuously from #=— 1 to the neighbourhood of 

 #=+1. 



The lines of flow are readily constructed from (30), which 

 we may write in the form 



1 —OP 2 



AP =;M^' (33) 



showing how P may be determined by the intersection of 

 circles struck from and A. A few of the lines of flow are 



