824 MOTION OF THE GAS SPHERE 



where (pa and (pz are the velocity potentials for unit velocity of trans- 

 lation and pulsation. 



A. Rigid surface. The normal derivatives d(p/dn must satisfy 

 boundary conditions at the sphere and a rigid plane as tabulated below 

 (see section 8.8) 



When these conditions are applied to Eq. (8.44), the integrals over the 

 plane vanish, leaving 



(8.45) 



JT _ P£^ 



wj n ''" + '^ (f ) n "'" + ^^ a ""^^^ h 



the integrals to be evaluated over the sphere, 

 tion, use has been made of the fact that 



In obtaining this equa- 



Hi- 



d<Pz 

 dn 



dn ) 



for any functions ipa^ ipz satisfying Laplace's equation. The calculation 

 of the total kinetic energy thus requires only a knowledge of the mean 

 values of (pa, ^Pz and ipz cos d^ over the sphere. Considering first the po- 

 tential (Pa of radial motion, which is represented by combinations of 

 point sources, we can easily show that the integral for a source of unit 

 strength at any point inside the sphere is simply 47ra and for a source 

 outside the sphere it is ^iro^/hn, where 6n is the distance of the source 

 from the center. 



These results are simple consequences of mean value theorems 

 (Gauss'" theorem) for potential functions and can also be demonstrated 

 by direct integration. Let the source be at a distance Cn from the 

 center and on the axis {6 = 0). Then the distance r to the ring ele- 

 ment of length 27ra sin d dd and wddth add is given by r"^ = a^ -\- c r? 

 — 2aCn cos and the desired integral for (pa = m/r is 



I I (Pads = m I 



27ra2 sin ^(/(9 



irma 



Cn 



/: 



a + Cn 



d{r^) 



= 4:Tnna 



