PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 667 



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

 /?„ must in this case involve u. 



(g) If Fis lamellar and if f2 is a scalar potential function of F^ O 

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



(h) If the tensor of V has the same value for all values of x and y, V 

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

 is solenoidal if, and only if, 



2^^{v) = ^. (8) 



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



and the vector which has the components 



^'~ h„ ' 9x' ''- K ' 9y' 



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 (^ (a) • V^ {it) + <^' {u) ' h^, where 4> 's 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. 



(h) If the tensor of a vector F which has the u curves for lines is a 



function of u only, its divergence is of the form V ( —7 yrj j. If 



the « curves are concentric circumferences, V must be solenoidal. 



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



curl is — V — ' ~. If the « curves are straight lines emanating from 

 h„ du 



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



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



gas illustrates this. 



* Miiichiii, Uiiiplanar Kinematics, 178, Examples 21 and 22. 



