ON FAMILIES OF CURVES WHICH ARE THE LINES 



OF CERTAIN PLANE VECTORS EITHER 



SOLENOIDAL OR LAMELLAR. 



By B. O. Peirce. 



Presented April 8, 1903. Received April 29, 1903. 



If a vector function has no component parallel to the axis of z and if 



the tensors of its components taken parallel to the axes of x and y can 



be expressed by the scalar point functions A= </>! (x, y), Y = <£ 2 fay), 



which are independent of z, every line of the vector is a curve parallel to 



dx dy dz . . 



the xy plane, defined by the equations — = -=_=-—, and it is sometimes 



convenient to call the vector itself " plane " and to say that it is 



"coplanar with " z = 0. The projection on the xy plane of any line of 



such a vector is itself a line of the vector, and a survey of the whole 



field can be obtained by studying the lines which lie in this plane. 



The "divergence" of a vector coplanar with the xy plane is the 



9X 9T 



quantity -= 1- -^r— , and the " curl " of the vector is a vector, directed 



dx dy 



• p • . d J dX ' . it . 



parallel to the z axis, of intensity -= ^— . If the divergence is zero 



in any region, the vector is said to be " solenoidal " in that region ; a 

 vector the curl of which vanishes is said to be " lamellar." 



Given any family of curves in the xy plane represented by the equa- 

 tion u EE./i (x, y) = c u it is possible to fiud an infinite number of plane 

 vectors which have the u curves as lines, by assuming in each case X at 



pleasure, and then making 



9u 



die 

 dy 



The vector (X , Y ) and the vector (R • X , R ■ Y ), where R is any 

 function of xy, evidently have the same lines, and, if (X , Y ) has for 

 lines the u curves, no other vector has the same lines unless it is of the 



