MOTION OF THE GAS SPHERE 325 



For a point outside the sphere at 6n, we have the same integral except 

 for replacing Cn by 6 n and changing the limits: 



j/'^«-f^j;;;";^-™G") 



The integral for a source inside the sphere is thus independent of its 

 position, and hence line sources contribute to the same extent as a point 

 source of equal strength. As remarked in section 8.8, each of the nega- 

 tive line sources introduced in the sphere is exactly cancelled by a point 

 source and their net contribution is zero, leaving only the original source 

 at the center. Putting m = a^ gives ^ira^ for this term. This can- 

 cellation does not occur for the images behind the plane and the expres- 

 sions for the line sinks and sources in this region can be obtained by 

 direct integration and addition. Using the notation of Shiffman and 

 Friedman (102), the result can be expressed 



(8.46) 



where the Z)„ and dn are defined by the recurrence formulas 



(8.47) 



The first term comes from the original point source, the first term in 

 brackets from its image in the plane; the DnS are from successive point 

 sources behind the plane, and the logarithms from the corresponding 

 negative line sources. 



The second and third integrals in Eq. (8.45) involving the dipole 

 potential (pz require a knowledge of the potential at the sphere directed 

 along it. For a dipole ^ located a distance s from the center of the 

 sphere, the velocity potential on the sphere is 



fjL cos 6 cos d 



^2 _|_ g2 _ 2as cos d 



where 6 is the polar angle of the radius a with the axis of symmetry. 

 The direct integration of this function does not lead to simple results 



