MOTION OF THE GAS SPHERE 319 



The calculation of the position and strength of the successive images 

 has been carried out by Shiffman and Friedman (102) by essentially 

 the methods outlined above, and their results therefore permit evalua- 

 tion of the velocity potential in the fluid to any desired degree of ap- 

 proximation. The first stage of the process, restoration at the plane 

 boundary, evidently is accompHshed by mirror images behind the plane 

 of the sources added in the sphere, these images being matched by 

 corresponding ones at the inverse points in the sphere, and so on. The 

 successive stages obtained by Shiffman and Friedman in this way are 

 indicated in Table 8.4. 



With this known distribution of sources, the velocity potential at 

 any point in the fluid can be computed. For a point source, the value 

 of (p is given by <^ = M/r where r is the distance from the source, and 

 the value of (p for a line source is obtained by integrating the potential 

 d(p = M'dx/r for an element over the length of the source. The gen- 

 eral expression for (p becomes increasingly complicated as successive 

 images are included. Fortunately for simplicity of calculation, this 

 general result is not needed to determine the equation of motion of the 

 sphere. This is because the kinetic energy of the system can be deter- 

 mined from the value of ip on the boundaries only (see section 8.4). 



These calculations, described in section 8.8, must, however, be pre- 

 ceded by a determination of the system of images for a sphere moving 

 in translation in order to take account of the migration during its radial 

 pulsation. If the plane boundary were not present, this motion would 

 be described by a dipole source ix at the center of the sphere. The 

 image of this source giving tangential flow at the plane is evidently an 

 equal but oppositely directed dipole at the image point an equal dis- 

 tance behind the plane. The violation of the flow condition on the 

 sphere is rectified by adding another dipole inside the sphere at the in- 

 verse point Ci in Fig. 8.15 and of strength ^t(a/26)^, as is easily shown 

 from the stream function. Successive reflections thus lead to a series 

 of oppositely directed dipoles of decreasing strength, as indicated in 

 Table 8.4. The corresponding velocity potentials at points in the fluid 

 are then given hy ip = ix cos d/r'^, where d is the angle the radius vector 

 from the dipole to the point makes with the symmetry axis. It will be 

 noted from Table 8.4 that the distances of the successive image points 

 from the center of the spheres can be expressed by recurrence formulas 

 as 



On 



a 



Cn+\ = a— 7 ci 



26 - Cn 



"(i) 



