672 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If for r and t in (25) we substitute their values as obtained from (20) 

 and (21), we shall get the equation 



q' 2 r< — 2 p' q' s' + p' 2 t' = 0. (26) 



and this is (8) in expanded form. 



If equation (18) or its equivalent (26) is satisfied, it is evident that 

 by choosing F(ii) at pleasure we may find an infinite number of solenoidal 

 vectors which have the u curves as lines and have tensors which involve 

 u only. 



A comparison of equations (9) and (18) shows that the condition that 

 the u curves be possible lines of a set of solenoidal vectors the tensors of 

 which involve u only, is the condition that the v curves be possible lines 

 of a set of lamellar vectors the tensors of which involve u only. 



If Q is a vector potential function of a solenoidal vector which has the 

 u curves for lines, and a tensor expressible in terms of v, — h u F'(u) is a 

 function of v, and h u must be expressible as the product of a function of 

 u and a function of v, that is, 



*.=/(«)"*(*)■ (27) 



If for u in this differential equation we substitute w, defined by the 



/' du . 



, we get the simpler equation 



h w = <j>(v) or **= = 0. (28) 



It is to be noticed that to has the same lines as u, and that (27) and 

 (28) define the same curves; the equations (11) and (28) are evidently 

 equivalent. 



If u is such that a solenoidal vector, V, can be found which has the 



u curves for lines and a tensor expressible in terms of v, its x and y 



( V 9o V 9v\ 1Jf 

 components are I -r- . -s-, -=- • yr- ] • 11 we denote these components 

 \ h v dx h v 9yJ 



by Xj Y, every other solenoidal vector which has the same lines has 



components of the form X'if/(u), T'\p(u), and the vector is not a 



function of v alone unless the factor if/ (u~) degenerates into a constant, 



and the vector is a simple multiple of V. 



A comparison of (10) and (27) shows that if the u curves are possible 



lines of a solenoidal vector the tensor of which is expressible in terms of 



