ON THE LINES OF CERTAIN CLASSES OF SOLENOIDAL 



OR LAMELLAR VECTORS, SYMMETRICAL 



WITH RESPECT TO AN AXIS. 



By B. O. Peirce. 



Presented October 14, 1903. Received October 14, 1903. 



The lines of force due to a homogeneous body bounded by a surface 

 of revolution are curves, each of which lies in a plane passiug through 

 the fixed axis of symmetry of the body. The force in such a case is an 

 example of a vector "symmetrical with respect to a straight line," and it 

 is evident that the whole field of a vector of this kind may be studied by 

 examining the lines of the vector in any plane which passes through the 

 axis. If we represent by R, <E>, X the components of a vector taken 

 in the directions in which the columnar coordinates r, </>, x increase most 

 rapidly, <£ is everywhere zero if the vector is symmetrical about the x 

 axis. 



If u EE fi( x t y) — k is the equation of a family of curves of one 

 parameter in the xy plane, which are lines of a vector symmetrical with 

 respect to the x axis, we may regard fi(x, ?■) = k, </> = m as the equa- 

 tions of all the lines of the vector, and the components of the vector 

 satisfy the equations 



• = °- *-f7 + *'t = °- W 



Given any family of curves, u = k, in the xy plane, without multiple 

 points or points of intersection with each other or with the x axis, it is 

 possible to form an infinite number of vectors symmetrical about the 

 x axis which shall have the u curves which lie on one side of it, as lines ; 

 for we may choose either R or X wholly at pleasure and determine the 

 other by means of equation (1). It is evident that if two such vectors, 

 [A*!, 0, Xi], [i? 2 , 0, X 2 ], have the same lines, R x / R 2 = X x / X. 2 . 



If v = f 2 (x, y) = I represents a family of curves in the xy plane 

 orthogonal to the family u = k, so that 



9 A • ?fl + 3/» • ¥l - o , (2) 



3x 9x 9y 9y 



