670 PROCEEDINGS OF THE AMERICAN ACADEMY. 



K =/(«) • F(v), h v = 4> (u), i/r (v), 



it IS 



possible to find two functions, / — — -r > / — 7— , of u and v respec- 



J/(m) J iffiy) 



tively, the gradient of each of which is expressible in terms of the other. 



A solution of Laplace's Equation and any function of its conjugate are 



orthogonal functions the ratio of the gradients of which is a function of 



the second function. 



Vector Potential Functions of Plane Solenoidal Vectors. 



If u, v define a system of orthogonal curvilinear coordinates in the xy 

 plane, and if Q u , Q v , Q z are the components of a vector Q, taken in the 

 directions in which u, v, z increase most rapidly, the components of -the 

 curl of Q in these directions are 



,pft |Y«.y|, K \°(V)-'X\, 



[_9o 9z\ h v J J \_9z \ h u J 3u J 



Kh [i(x)- !(£)} 



We may denote these quantities by A"„, K v , K z , respectively. 



If Q is to be a vector potential function of a given solenoidal plane 

 vector (0, V, 0), which has the u curves for lines, we may assume that 

 the components of Q involve u and v only, and since in this case, K u = 0, 



K v = V, write Q z ■= F(u), where V= — h u . — -^ . Any vector of 



CI iff 



the form [Q u , Q v , F(u)~], where Q u , Q v , are any functions of u and v 



subject only to the condition t- y" = ^r- ( -y- 1 j, is a vector potential 



function of a solenoidal vector which has the u curves as lines, and there 

 is no vector of this latter kind which does not have as a vector potential 

 a vector of the form just given. In most cases it is simplest to make 



If, now, we ask what condition must be satisfied by the function u in 

 order that the curves of the family 11 = c may be the lines of a vector 

 the tensor of which involves u only, we learn that, since V is of the 

 form — h u • F'{u), it is necessary and sufficient that h u be a function of 

 u only. That is 



