Compressible Fluid movingin Incompressible Liquid. 141 



Since products of small terms are neglected, those con- 

 ditions become 



"d^fr _ ~d^' ~dcj) _ ~d(f)' 

 "dr ~" 'dr ' "dr " ~dr ' 



Substituting the values of ty, yfr', (f>, <f>' from (10), (11) 

 and equating coefficients of cos n0 and sin nd, we get 



A„= 



(13) 



K -d 



K 8 



LTrna 



^ n ~ 2irna tit*' 1 ' 



F - 1 da B 



lima at 



Hence outside the vortex, we have 



(14) 



da 



6 = — —- -flogr + S-T^-facosnO + PnSiniiO) (-) A 

 2ir at ° 'lima v7 1 



i/r= D + ^log r - 2 g^( a » cos w0 + A. si » w#) f ? J 



Suppose there are two vortices of strengths re, k ; let their 

 cross sections at time t be a and a' respectively, the distance 

 between their centres being c. Let the radii of their 

 cross sections be given by 



R=a + 'Z(* l n cosn0 + P n smn0), [ 



R' = b + t(* n ' cos nff'+Pn' sin n0y\' ' ' (15 ^ 



Let (f>, -v/r be the functions at an external point for the 

 first and </>', y\r' for the second. If (r, 6) and (V, 6') be the 

 coordinates of a point referred to the centres of the two 

 vortices, </> and -\jr will be given by (14), and 



da' 



> 



We shall fix our attention on the second vortex. If 3ft be 

 the radial velocity of a point on it and I>S the velocity per- 

 pendicular to the radius vector, both relative to the centre 



>(!0) 



