PEIRCE. — ON CERTAIN CLASSES OP VECTORS. 297 



If a set of orthogonal curvilinear coordinates in the plane <f> = be 

 defined by the functions, u = fx(x, r), v =f 2 (x, r), and if 



U= i(x, r), V= v (x, r), * = 0, (7) 



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

 in the directions in which u, v, cj> 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 



~r-' 1 9u (4r) + * (V - ) i ' (8) 



T , ^ 9 2 u 9u 9 2 u , . o ( 9u\ % /9u\ 2 ,.„ N 

 .here L(u) =^+ — + - and «.• = (^ + (^) . (10) 



The components, (i$T u , K v , K$) of the curl of Q are 



°'°> -**••*. I k (£)-£© 1- (11) 



It is to be noticed that 



cV 1 5m 7 <9A U 5?< 3 2 « 9u 9 2 u 



oT) ««• ^T = a" * ^ + 



5w h 2 ' 9r' " ' cV 5?" cV 2 5x cV * eta: ' 



, 2 9K _9K ^9u 9K 9u L(u) _ 9_ f r . h u \ 

 u ' 9u ~ 9r ' 9r + 9x 9x' h* ~ 9u g V K )' 



L(v) 9 fr.h v \ 

 and -—- = =- log — ; — • (12) 



h v 2 9o °\ K ) K J 



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

 vector has no component perpendicular to these curves, and £7" is every- 

 where zero, so that 



Divergence Q = h* . ~ (£\ + L{v) (£\ , (13) 



K ' K 9 frV\ 



= — '9-v{TJ> (U) 



