382 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The example just discussed is iu contrast with the case where the Q 

 family are a set of parallel curves of any kind, and h in consequence (if 

 not constant) is a function of U alone, so that the h curves and the O 

 curves coincide, and if D 2 C1 vanishes anywhere, it must be where h! 

 vanishes. A simple example of this is furnished by the field of attrac- 

 tion within a very long cylinder of revolution, the density of which is a 

 function of the distance from the axis alone. 



If the directions s x and s 2 are everywhere perpendicular to each other, 

 we may without loss of generality write l 2 = — m x , m 2 == l y ; in Which 

 case the coefficients of 9Q/9x, 9tl/9y in (4) become 



( 



9m 2 9m 2 \ -, ( 1 9h , 9l 2 



L • -7T- + m 2 • -=— and — / 2 • 7=- + m 2 • — J : (13) 



9x Sy J \ dx 3y J 



these vanish if the v curves form a family of straight lines, or the u 

 curves a family of straight or curved parallels. The order of differentia- 

 tion with respect to the orthogonal directions s 1} s 2 is immaterial if both 

 the u and the v curves are straight lines, that is, if the directions are 

 fixed. 



If s x is the direction in which Q increases most rapidly, and s 2 the 

 direction of constant Q, 



D< D<Q = D S h 



'2 1 



9 2 n 



po 9h _ 9n m 9fi~\ /, 

 [_9x 9y 9y 9x_\ j 



9x • 9y 



9n\ 2 _ [9n\ 2 

 9*)" Vy J J 



9n 9n \~9°-n 9 2 n~] } 

 + 9x"9yhf-^i\ IL 



Now the direction cosines and the slope of the line of the gradient vector 

 at any point are 



l 9a i 9a , 9ni9n 

 h'te' h'9y' 9y\3x- 



So that the curvature of the line is 



d po i9nl 

 1 _ dx\_9y 1 9x J 



r Kf/i)T 



j J^r[3aY_(9n\ 2 l 9n 9nf9Hi_9^f\ ) U* (15) 

 ~ \9x'9y\\9x) \9y ) y 9x 9y \_9y 2 9x*]\l K } 



