40 VECTOR ANALYSIS. 



where r denotes the distance from a fixed origin, then throughout the 

 space 



and in each continuous portion of the same 



u constant. 



For, if anywhere in the space in question Vu has a value different 

 from zero, let it have such a value at a point P, and let u be there 

 equal to b. Imagine a spherical surface about the above-mentioned 

 origin as center, enclosing the point P, and with a radius r. Con- 

 sider that portion of the space to which the theorem relates which is 

 within the sphere and in which u<b. The surface integral of Vu 

 for this space is equal to zero in virtue of the general condition 

 V.Vu = 0. That part of the integral (if any) which relates to a 

 portion of the spherical surface has a value numerically not greater 



than 47rr 2 (-r- ), where (-?-] denotes the greatest numerical value of 



du vfr*' v*v 



-7- in the portion of the spherical surface considered. Hence, the 



value of this part of the surface-integral may be made less (numeri- 

 cally) than any assignable quantity by giving to r a sufficiently great 

 value. Hence, the other part of the surface-integral (viz., that relating 

 to the surface in which u = b, and to the boundary of the space to 

 which the theorem relates) may be given a value differing from zero 

 by less than any assignable quantity. But no part of the integral 

 relating to this surface can be negative. Therefore no part can be 

 positive, and the supposition relative to the point P is untenable. 

 This proposition also may be generalized by substituting V. [t Vu] = 



for V.Vu = 0, and r 2 = for . 



or dr 



82. If throughout any continuous space (or in all space) 



then throughout the same space 



t = u+ const. 



The truth of this and the three following theorems will be apparent if 

 we consider the difference t u. 



83. If throughout any continuous space (or in all space) 



and in any finite part of that space, or in any finite surface in or 

 bounding it, 



then throughout the whole space 



Vt Vu, and t = u+ const. 



