296 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and if F (y) is any single-valued, differentiable function of v, a vector 

 the components of which are 



jj=j.'(„).Si^,4, = 0,Xs#»-i^ (3) 



has the u curves as lines, and it is lamellar, for 



9R._9X 



9x 9r 



Of all the vectors symmetrical about the x axis which have the given u 

 curves as lines, an infinite number are lamellar. 



Since the divergence of a vector symmetrical about the x axis is 

 equal to 



^ + f/' (*) 



vcJr c)x 



it is evident that if [i?, 0, X] is a solenoidal vector which has the u 

 curves as lines, \_Ii ' F(u), 0, X'F'(u)^, where F is any single-valued, 

 differentiable function, is another solenoidal vector which has the same 

 lines. If two solenoidal vectors, [Ri, 0, Xi], [i?o, 0, X,], symmetrical 

 about the x axis have the u curves as lines, the ratio ^i / B^} or -^i / X2 

 is a function of u only. 



Whatever u is, the vector which has the components 



or the components 



W^9u W^9u 



K 9x' ^' + K 9i-' ^^^ 



W 9v W 9v^ 



h„ 9r^ '' h„ 9x* ^ 



— where IT is a single-valued function of the space coordinates and 

 h^, k„ are the gradients of the functions u and v, — has the zi curves as 

 lines. The tensor of this vector involves u alone or v alone, according as 

 W is expressible in terms of u alone or in terms of v alone. 



It is to be remembered that the field of a physical vector may be 

 a restricted region in space, so tliat a family of u curves which have 

 double points, or points of intersection with each other, or with the x axis, 

 may still be lines of a vector symmetrical with respect to the x axis if 

 the field of the vector is free from such singular points. 



