382 PROCEEDINGS OP THE AMERICAN ACADEMY. 



The example just discussed is in contrast with the case where the O 

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

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

 curves coincide, and if D^^U 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 5i and So are everywhere perpendicular to each other, 

 we may without loss of generality write Z2 = — m^, ruz = h; in which 

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



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 Si, $2 is immaterial if both 

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

 fixed. 



If Sj is the direction in which 12 increases most rapidly, and Sg the 

 direction of constant Q, 



D.D,n = A h 



r9n ^9h _9n ^ S/H /, 

 \_9x 9y 9y 9x\ j 



9^n 



9n\- /9n\ 



_ _ _ ^ 9nr9^_9Hf\ I 



9x-9yl\9xJ \9y)_\'^9x'9y\_9f 9x'A\''' ^ ^ 



- 



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

 at any point are 



1 5ri 1 5a , ?^/^ 



h'9i' l' ¥y' 9y\9x' 



So that the curvature of the line is 



d V9^ /5n~| 

 1 _ dx \_9y I 9x _\ 



9^n rfSny (9n\n 9n9nr9H}. 

 h^ l\9x) \9y)j'^9x'9y\_ 9y' 



