PEIRCE. — ON CERTAIN CLASSES OP VECTORS. 301 



so that (32) would be satisfied. If the equation of the same family of 

 straight liues had been written in the form 



/ r - r \ 



v = tan M ) 



\x — x J 



we should have had 



h v = l/u, 



so that v would have satisfied an equatiou of the form (33). 



(h) If Q is lamellar, and if O is a scalar potential function of Q, Q 

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



, „ d 2 Cl r , . dQ, ,• 



*-"s + x »*' < 35 > 



(i) If the tensor of Q has the same value at all points of the x r plane 

 Q is lamellar if, and only if, h v is constant or a function of v alone. 



(j) If the tensor of a vector, Q, which has the u curves as lines, is a 

 function of u only, its divergence is 



(k) If the tensor of Q is expressible in terms of v, the tensor of its 

 curl is 



- *■•*■• r -k{Q- (37) 



If, for instance, the u curves are a family of straight lines, the tensor of 

 the curl of such a vector must be zero. 



(I) If Q is to be solenoidal as well as lamellar, equations (13) and (15) 

 yield Lame's well-known condition that L (y)/h* must be expressible in 

 terms of v alone, so that 



9 (Liyy 



(m) A vector symmetrical about the x axis and directed everywhere 

 parallel to it, is solenoidal only when its tensor has the same value 

 throughout every one of its lines. 



