PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 665 



tion or otherwise, that the divergence of Q is given by the well known 

 expression 



and that 



Tensor curl Q =:z riu ' '',■ 



9(1 V /?,, J 9v \h„ 



(2) 



If the lines of Q coincide with the u curves, the Vector has no com- 

 ponent perpendicular to these curves and U'ls, everywhere equal to zero, 

 so that 



Di,.g = £.V^.) + *.-!({), (8) 



Tensor curl Q = h„ ' ^^v ' ^lj\ (4) 



where h„ is the gradient of v. 



In applying these expressions it is convenient to remember that 



5l0gf ^ j ^oy ^ 5i0,L' 





K' 9u ' /^/ 9v 



It is easy to see from (3) and (4) that the statements which follow are 

 true: 



(a) If Fis to be solenoidal,* we must have 



The second member of this equation is expressible as a function of u and 

 V ; if it be integrated with respect to v while u is considered constant, 

 and if the arbitrary function ;( (u) be added to the result, we shall get 

 \p (u, v) -\- X (u) the partial derivative of which with resoect to v is 



- ^^ ; then V= h,, ■ exW • e-^-O'-f). (6) 



* See equation (18). 



