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 



B = r(v)-- 9 ^L,^ = o,x=r(v)-^l (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 



^P + £, (4) 



r'dr dx v y 



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

 curves as lines, \_R • F(u), 0, X'F(n)'], where F is any single-valued, 

 differentiable function, is another solenoidal vector which has the same 

 lines. If two solenoidal vectors, [i? 1} 0, X{\, [i? 2 > 0, X 2 ], symmetrical 

 about the x axis have the u curves as lines, the ratio R x / R 2 , or X x j X 2 

 is a function of u only. 



Whatever u is, the vector which has the components 



W 9u „ W 9u 



or the components 



K 9x> Q > + K'Tr> (5) 



W 9v W 9v 



h v ' 9r ,u, h v ' 9x' {b) 



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

 h u , h v are the gradients of the functions u and v, — has the u 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 that 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. 



