298 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Tensor curl ^ = ^„ • /^^ . ^r- ( — | . (15) 



If P„, P„, P^ are the components of a vector P taken in the directions 

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

 are 



r 





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

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

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

 write 



n = =tM, (19) 



where 



f'=-^" •!;/(»)• (20) 



r 



9u' 



Any vector of the form [P„, P„,y(?/) /r], where P„ and P„ are any 

 functions of u and v subject only to the condition 



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 mentioued which does not have as a potential function 

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

 P^=P, = 0. 



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

 which follow are true. 



(o) If Q is to be solenoidal, r- V j h^ must be either constant or a 

 function of u only ; that is, 



F^^^^^^^, (22) 



