122, 123] DERIVATIVES WITHIN MASS. 359 



small sphere about it. Applying Green's theorem to the rest of the 

 space K U j we have to add to the surface -integral the integral over 

 the surface of the small sphere. 



Since cos (nx) <^ 1, this is not greater than g m I I = 4:iteQ m , 



which vanishes with . Hence the infinite element of the integrand 

 contributes nothing to the integral. 



In the same way that ^ was proved finite, it may be proved 



dV 3V" 



continuous. Dividing it into two parts -5 and -~ > of which the 



ox ox 



second is continuous , we may make, as shown, -~ as small as we 



please by making the sphere at P small enough. At a neighbor- 

 ing point P! draw a small sphere, and let the corresponding parts 



, 3V' , 3V" mi , 3V' 



be -^ and -5-* Then we can make ~^- as small as we please, 



ox ox ox 



dV 3V' 

 and hence also the difference -~ --- -^ L - Hence by taking P and 



3V 

 P! near enough together, we can make the increment of ^- as small 



o x 



3V 

 as we please, or ^ is continuous, and accordingly the second derivatives 



are finite. 



123. Poisson's Equation. By Gauss's theorem [ 118, 43)], 

 we have 



when r is drawn from 0, a point within 8. Multiplying by m, a 

 mass concentrated at 0, and calling F=> 



68) ~ cos (>r) dS = - 



The integral 



n} ds > 



where n is the internal normal, is the surface integral of the outivard 

 normal component of the parameter yP, or the inward component 

 of the force. 



The surface integral of the normal component of force in the 

 inward direction through S is called the flux of force into 8, and 

 we see that it is equal to 4#y times the element of mass within S. 

 Masses without contribute nothing to the integral. Every mass dm 



