PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 665 



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

 expression 



Mr. <? = (£) V(„) + V . ».(£) + 



and that 



Tensor curl <? = A. ■ A, [£ (£) - | (£)]. (2, 



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

 ponent perpendicular to these curves and Uis, everywhere ecpial to zero, 

 so that 



DiT . <? = £.vv„ ) + v .|(£), ( 3) 



Tensor curl Q = h u ■ h v ■ ■=- ( j J, (4) 



where h v is the gradient of v. 



In applying these expressions it is convenient to remember that 



n-> aiog (£) vw ^' og (£) . 



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

 true : 



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



§-A' og ^J = "^' (o) 



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 

 i/r (u, v) -+- x (") the partial derivative of which with resoect to v is 



V 2 (v) 

 ^V '* then ^= K ' ex(u) ' e*( M .«>. (6) 



* See equation (18). 



