164 THEOEY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



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

 Green. 



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

 tinuous everywhere outside of a certain closed surface, 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 S 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 



(6) 



and a harmonic function is determined everywhere by its values 

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

 the surface S. 



Since 



I 



r 1 or 

 dn r' 2 dn 



If x dr .dr , , dr} cos(nr) 

 = - jcos (nx) -z- + cos (ny) 4- cos (nz) g^ f = - - , 



we may write (6) 



p 4ir JJ V^ 2 r dn) 



Consequently, we may produce at all points outside of a closed 

 surface 8 the same field of force as is produced by any distribution 

 of masses lying inside of S, whose potential is F", if we distribute 

 over the surface S a surface distribution of surface-density, 



In the general expression (5), the surface integral representing 

 the potential due to the masses tuithin S, the volume integral 



AF, 

 - dr 



