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 X = (fii (x, y), Y = <f}2 {x,y), 



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



1 1 1^111 . dx dy dz ^ . . 



the xy plane, denned by the equations — = -^^ = — , and it is sometimes 



Ji. jl y) 



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



"coplanar with " ^ = 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 -^ \- -pr-, and the " curl " of the vector is a vector, directed 



dx dy ' 



11 1 1 ... . oY dX 

 parallel to the z axis, of intensity ^r ■^. 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^^fx (x, y) = Cj, it is possible to find 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 



Y=:-X.'^. 



du 



9y 



The vector (X^, Yq) and the vector (R • X^), R • Yq), where R is any 

 function of xy, evidently have the same lines, and, if {X^, J^) has for 

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



