298 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Tensor curl Q = h u • h v . ^— ( — ] . (15) 



If P u , P v , P<f> are the components of a vector P taken in the directions 

 in which u, v, </> increase most rapidly, the components of the curl of P 

 are 



*'= h -;\H£)-U'- p A' (l7) 



and, if P is to be a vector potential function of a given solenoidal vector, 

 Q = [0, V, 0], which has the u curves as lines and is symmetrical about 

 the x axis, we may assume that P u and P v involve u and v only, and 

 write 



P* = ^, (19) 



r 



where 



r=— •!-/(«). (20) 



Any vector of the form [P u , P v ,f(u) jr~\, where P u and P v are any 

 functions of u and r subject only to the condition 



9_(P, 



3u\ h 



!)-«£)• 



is a vector potential function of a solenoidal vector, symmetrical about 

 the x axis, which has the u curves for lines ; and there is no vector of 

 the kind last mentioned which does not have as a potential function 

 a vector P of the form given. It is usually convenient to make 

 P u = P v = 0. 



It is easy to see from the foregoing equations that the statements 

 which follow are true. 



(a) If Q is to be solenoidal, r- V / h u must be either constant or a 

 function of u only ; that is, 



V= ^-F(u) } (22) 



