1178 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



o 



curl I In = — e/i«grad/> X grad "0, ^ (13) 



o 



curl II = —e(jin + Mp) grad p X grad V, (14) 



and 



whence 

 Theorem 2: 



curl |L curl IL curl !! 



Mp Mn At' -r ^p 



o c o 



That is, curl ||p , curl ||n , and curl j| are constant multiples of one another, 

 and 



o o o 



Theorem 3: | |p , | |n , and 1 1 are irrotational if and only if 



grad /> = {p = pit)) 



or grad'U = fU = V{t)) 



or V = V{p, t). 



The following interesting relations can be obtained from (8) and (9) 

 (they are really consequences of Theorem 1) : 



o o 



curl Up = grad In /> X ||p (15 



and 



o o 



curl Ijn = grad In w X ||n. (16) 

 Now from (3) — (5) we find 



lip X fin = e^nHpkT{n + p) grad p X grad V (17a) 



= innHpkT(n + p) grad {n + p) X grad V (17b) 



= iennUpkT grad (n + pY X grad V (17c) 



= WnUpkT curl [(« + pY grad V] (I7d) 



and 



and 



L^ _ II- = grad \nv - - (n + p)\ (18) 



Mj)« line \_ e J 



Jl? + Ik ^ -{« + /,) grad U (19) 



