PEIRCE. — ON CERTAIN CLASSES OP VECTORS. 297 



If a set of ortliofjonal curvilinear coordinates in the plane <^ = be 

 defined by the functions, u =fi(x, r), v =/,(a:, r), and if 



U= ^(x, r), V= r](x, r), $ = 0, (7) 



represent the magnitudes, at the point (x, r) of the components, taken 

 in the directions in which u, v, (f> increase most rapidly, of a vector Q, 

 symmetrical with respect to the x axis, it is easy to prove that the diver- 

 gence of Q is given by the expressions 



where L(u) = _ + _^^ + _ and /,.' = (^_ j + (_) . (,0) 

 The components, (^u, K^, K^) of the curl of Q are 



0,0,and*.-*.{|^(£)-|;g')j. (11) 



It is to be noticed that 



9r 1 9u 9h^ 9u 9'^if' 9u 9hc 



9u h^ 9r ' " ' 9r 9r 9r'^ 9x 9r • 9x^ 



" 9u 9r 9r 9x 9x' //„^ 9u ° 



(^")' 



, L(v) 5 , fr . h,\ , , 



If the lines of Q in the plane <^ = coincide with the u curves, the 

 vector has no component perpendicular to these curves, and U\?> every- 

 where zero, so that 



Divergence Q = hj^ . ^^ fPj + L{v) ff) , (13) 



