VECTOR ANALYSIS. 43 



The existence of the minimum requires that 



fffu w. Sw dv = 0, 



while SCD is subject to the limitation that 



V.&o = 0, 



and that the normal component of (So> at the bounding surface vanishes. 

 To prove that the line-integral of UCD vanishes for any closed curve 

 within the space, let us imagine the curve to be surrounded by an 

 infinitely slender tube of normal section dz, which may be either 

 constant or variable. We may satisfy the equation V.&o = by 



making & = outside of the tube, and 8wdz = Sa^r within it, So, 



ds 



denoting an arbitrary infinitesimal constant, p the position- vector, and 

 ds an element of the length of the tube or closed curve. We have then 



Jjju W.SOD dv =Ju (ti.Soo dz ds =Ju oa.dp 8a = Saju w.dp = 0, 

 whence fu w.dp = Q. Q.E.D. 



We may express this result by saying that uw is the derivative of a 

 single-valued scalar function of position in space. (See No. 67.) 



If for certain parts of the surface the normal component of to is not 

 given for each point, but only the surface-integral of o> for each such 

 part, then the above reasoning will apply not only to closed curves, 

 but also to curves commencing and ending in such a part of the 

 surface. The primitive of UCD will then have a constant value in 

 each such part. 



If the space extends to infinity and there is no special condition 

 respecting the value of at infinite distances, the primitive of UOD 

 will have a constant value at infinite distances within the space or 

 within each separate continuous part of it. 



If we except those cases in which the problem has no definite 

 meaning because the data are such that the integral Juco.wdv must 

 be infinite, it is evident that a minimum must always exist, and (on 

 account of the quadratic form of the integral) that it is unique. That 

 the conditions just found are sufficient to insure this minimum, is 

 evident from the consideration that any allowable values of Sco may 

 be made up of such values as we have supposed. Therefore, there 

 will be one and only one vector function of position in space which 

 satisfies these conditions together with those enumerated at the 

 beginning of this number. 



6. In the second place, let the vector o> be subject to the conditions 

 that Vxw is given throughout the space, and that the tangential 

 component of w is given at the bounding surface. The solution is that 



