10 



since $ is regular through V\ Therefore, since 



r = i«+J2/ + ki 



we have 



p j (i>nda = p j T ^ da = -p r r{q-n)da 



^ S+S' "^ S+S' " J S+S' 



But on S we have the boundary condition 



q • n = w X r ' n 



so 



I p^dr = -pi riu xr ' n)d a - p i r(q • n) da 



The first term on the right can be easily reduced. One form resulting from Gauss' theorem is, 

 (Reference 12, p. 52) 



Using this, 



a(n • b) (ic7 = - 



a(v^- b) + (b • V )a 



dr 



r(«k» X r • n)^a - {r ^ u)dT = (r^ x cj) V 



and 



Therefore, 



Then 



dt 



t 



V-{r X u) = a -{ V xr) - r{ V x «) = 



(r X « • y ) r = r X w 



pqdr = 



pqdr = -r X lapV - p \ T{q • n) da 

 L x^^j^ «^(rg . «) -Tglto- w)\p¥ -J-f ^pr(q- n) da [17] 



