36 Lord Bayleigh on the Incidence of Aerial 



When the density of the obstacle (<r r ) is the same as that 

 of the surrounding medium, (47) constitutes the complete 

 solution. Otherwise the difference of densities causes an 

 interference with the flow of fluid, giving rise to a disturbance 

 of order 1 in spherical harmonics. This disturbance is inde- 

 pendent of that already considered, and the flow in the 

 neighbourhood of the obstacle may be calculated as if the 

 fluid were incompressible. We thus fall back upon the 

 problem considered in the earlier part of this paper, and 

 the results will be applicable as soon as we have established 

 the correspondence between density and conductivity. 



In the present problem, if ^ denote the whole velocity- 

 potential, the conditions to be satisfied at any part of the 

 surface of the obstacle are the continuity of d^/dn and of <r^, 

 the latter of which represents the pressure. Thus, if we 

 regard a% as the variable, the conditions are the continuity 

 of {(T'x) and of er _1 c?(<7%)/V/«. In the conductivity problem 

 the conditions to be satisfied by the potential (%') are the 

 continuity of x' an d of fi dx'/dn. 



In an expression relating only to the external region 

 where cr is constant, it makes no difference whether we are 

 dealing with a^ or with % ; and accordingly there is corre- 

 spondence between the two problems provided that we suppose 

 the ratio of /*'s in the one problem to be the reciprocal of the 

 ratio of the a's in the other. 



We may now proceed to the calculation of the disturbance 

 due to an obstacle, based upon the assumption that there is a 

 region over which r is large compared with the linear dimen- 

 sion of T, but small in comparison with X. Within this region 

 ty is given by (8) if the motion be referred to certain principal 

 axes determined by the nature and form of the obstacle, the 

 quantities u, v, w being the components of flow in the primary 

 waves. By (41), (42), this is to be identified with 



p -ikr / 1 \ 



when r is small in comparison with X ; so that 



o _ ik ( Aj ux + B 2 vy + C 3 wz) 



^i • . . 



At a great distance from T, (52) reduces to 



, _ ik(A x ux + B 2 vy + C 3 wz)e ~ ikr 



(53) 



(54) 



