192] VECTOR POTENTIAL. 513 



Where the vorticity vanishes the space contributes nothing to the 

 integrals, so that the latter may be taken over all the vortices. Thus 

 we see that Q, which determines the solenoidal part of q, is deter- 

 mined by curl q. Consequently both parts of q are completely 

 determined and the theorem is proved. If q is solenoidal div. q 

 vanishes, cp = 0, and q = curl Q. Accordingly every solenoidal vector 

 may be represented as a curl. If q is irrotational curl q = and 

 = 0, so that every irrotational vector is lamellar, as we saw in 31. 

 The vector Q, whose components are formed as potential func- 



tions for densities > ^ > respectively, is called the vector potential 

 of the vector - We may thus abbreviate our results in the vector 



_ Tt 



equations, 



71) q = vector parameter <p -\- curl Q, 



Let us verify that Q as determined is solenoidal. We shall 

 distinguish the point of integration by accents, so that 



r 2 - (x - xj + (y - yj + (, - zj. 

 Differentiating we have 



) S - i/// 1 6) * - - ///' A 



In like manner 



,, 9U, 9V ,8W 

 75 ) g^ + 37 + -07 



- ^ [ ff(g cos (*) + r,' cos (y) + {' cos (*)) ^ 



WEBSTER, Dynamics. 



