VECTOR ANALYSIS. 41 



84. If throughout a certain space (which need not be continuous, 

 and which may extend to infinity) 



and in all the bounding surfaces 



t=u, 



and at infinite distances within the space (if such there are) 



t = u, 

 then throughout the space 



t = u. 



85. If throughout a certain space (which need not be continuous, 

 and which may extend to infinity) 



and in all the bounding surfaces the normal components of Vt and Vu 

 are equal, and at infinite distances within the space (if such there are) 



r 2 ^ ;T/ = O' where r denotes the distance from some fixed origin, 



then throughout the space 



V^ 



and in each continuous part of which the space consists 



t u = constant. 

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



Vxr = Vxft> and V.T = V.o>, 



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

 bounding it, 



T = ft), 



then throughout the whole space 



For, since VX(T a>) = 0, we may set VW = T o>, making the space 

 acyclic (if necessary) by diaphragms. Then in the whole space u is 

 single-valued and V.Vu = 0, and in a part of the space, or in a surface 

 in or bounding it, Vu = 0. Hence throughout the space Vu = r o> = 0. 



87. If throughout an aperiphractic* space contained within finite 

 boundaries but not necessarily continuous 



VXT = VXO> and V.r = V.w, 



and in all the bounding surfaces the tangential components of T and 

 ft> are equal, then throughout the space 



T = ft). 



It is evidently sufficient to prove this proposition for a continuous 

 space. Setting VU T o>, we have V.Vi& = for the whole space, 



* If a space encloses within itself another space, it is called periphractic, otherwise 

 aperiphractic. 



