670 PROCEEDINGS OF THE AMERICAN ACADEMY. 



K=f(u).F{v), h, = <f>(u), ^(v), ■ 



it is possible to find two functions, / -— - , / —^ , of u and v respec- 



J/(«) J ^(v) ^ 



lively, the gradient of each of which is expressible in terras 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 op Plane Solenoidal Vectors. 



If M, V define a system of orthogonal curvilinear coordinates in the xi/ 

 plane, and if Q,„ Q^, Q^ 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 



We may denote these quantities by ir„, K^, K^, 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^ = 0, 



K„ = F, write Q^ = F(u), where V= — h^ . . Any vector of 



the form [Q^, Q^, F{ii)'], where ^„, Q„, are any functions of u and v 

 subject only to the condition ^( "t-^' ) — o~ ( T^ )' ^^ ^ 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 u ■= 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^ ' F'(u), it is necessary and sufficient that A„ be a function of 

 u only. That is 



