128] GREEN'S FORMULAE. 371 



vanishes. Accordingly the infinite sphere contributes nothing. For 

 the small sphere the case is different. The first integral 



- / / VdG) 



*J *J 



becomes, as the radius s of the sphere diminishes, 



4) -^//* 



The second part 



however, since -^ is finite in the sphere, vanishes with s. Hence 



there remain on the left side of the equation only 4:7tV P and the 

 integral over 8. We obtain therefore 



the normal being drawn outward from 8. This formula is due to 

 Green. 



Therefore we see that any function which is uniform and con- 

 tinuous everywhere outside of a certain closed surface, which vanishes 

 at infinity to the first order, and whose parameter vanishes at infinity 

 to the second order, is determined at every point of space considered 

 if we know at every point of that space the value of the second 

 differential parameter, and in addition the values on the surface 8 of 

 the function and its vector parameter resolved in the direction of 

 the outer normal. 



In particular, if V is harmonic in all the space considered, we have 



K\ V 



b) . VP - 



' 



and a harmonic function is determined everywhere by its values and 

 those of its normal component of parameter at all points of the 

 surface 8. 

 Since 



r 



== - i | cos (nx) || + cos (ny) ~ r y + cos (ne) f z } = 



cos(nr) 



a f 



24 



