PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 667 



must be either constant or expressible in terras of v. If V is not lamellar, 

 h v must in this case involve u. 



(g) If Fis lamellar and if O is a scalar potential function of V, €1 

 must be expressible in terms of v and the divergence of V is equal to 



£■"«+£•" w> 



(Ji) If the tensor of J 7 has the same value for all values of a; and y, V 

 is lamellar if, and only if, h v is constant or expressible in terms of v ; it 

 is solenoidal if, and only if, 



"•<•>=■$£ («) 



(i) Whatever u is, the vector which has the components 



x _/(«) _ 3v T = fW . ?! 

 K 9x h v 9y' 



and the vector which has the components 



h v 9x h v By ' 



have the u curves for lines. The tensor of the first is a function of u 

 only, that of the second a function of v only. 



(j) If a solenoidal vector has the u lines for curves, its curl must be 

 of the form </>(«) ' ^ 2 (») + 4>' ( u ) ' ^« 2 > where </> is arbitrary. If, for 

 instance, the u curves are concentric circumferences, the curl of the 

 vector must be expressible as a function of the distance from the centre. 



(k) If the tensor of a vector V which has the u curves for lines is a 



function of u only, its divergence is of the form V ( — ; — - — -r-^ ]. If 



J ' ° V h v c)o J 



the u curves are concentric circumferences, Fmust be solenoidal. 



(I) If the tensor of V is expressible in terms of v, the tensor of its 



h oy> 



curl is — V— • -=r^. If the u curves are straight lines emanating from 

 h v Sic 



a point, the curl is zero and the divergence a function of the distance 



from the point. The velocity in the case of a steady squirt * motion of a 



gas illustrates this. 



* Minclrin, Uniplanar Kinematics, 178, Examples 21 and 22. 



