348 VIII. NEWTONIAN POTENTIAL FUNCTION. 



of the normal component of any vector point -function outward from 

 any closed surface S within which the function is uniform and con- 

 tinuous, multiplied ~by the area of the surface, is equal to the mean 

 value of the divergence of the vector in the space within S multiplied 

 ty its volume. The theorem is here proved for a vector which is the 

 parameter of a scalar point -function V, but it is evident that it may 

 be proved directly whether this is the case or not by putting in 

 equation 17) for W and x successively X, Y, Z and x, y, s respectively. 

 Let us consider the geometrical nature of a vector point -function E 

 whose divergence vanishes in a certain region. In the neighborhood 

 of any point, the vector will at some points be directed toward the 

 point and at others away. We may then draw curves of such a 

 nature that at every point of any curve the tangent is in the direc- 

 tion of the vector point -function E at that point. Such curves will 

 be called tines of the vector function. Their differential equations are 



o\ dx dy dz 



~X = "Y~-~-~^' 



Suppose that such lines be drawn through 

 all points of a closed curve, they will 

 generate a tubular surface, which will be 

 called a tube of the vector function. Let us 

 now construct any two surfaces S l and 8. 2 

 cutting across the vector tube and apply 

 the divergence theorem to the portion of 

 space inclosed by the tube and the two sur- 

 faces or caps S 1 and S 2 . Since at every 

 point on the surface of the tube, E is 

 Fig. 123. tangent to the tube, the normal component 



vanishes. The only parts contributing any- 

 thing to the surface integral are accordingly the caps, and since the 

 divergence everywhere vanishes in r, we have 



37) C CE cos (En ) dS + f (*R cos (JR^) dS 2 = 0. 



S L S 2 



If we draw the normal to S 2 in the other direction, so that as 

 we move the cap along the tube the direction of the normal is 

 continuous, the above formula becomes 



38) (JE cos (En^ d8 = I I E cos (Rn^ dS 2 , 

 s t s 2 



or the surface integral of the normal component of E over any cap 

 cutting the same vector tube is constant. 



