380 PROCEEDINGS OP THE AMERICAN ACADEMY. 



+ (l^ . ^ + nn • ^^ ^ + f/, • ^ + m, ' — ^^ — (7) 

 \^ 9x ^ 9y ) 9x \ 9x ^ 9y J 9y ' 



If the direction cosines of a plane curve at a point on it are I and w, 

 the curvature of the curve at P has the same absolute value as have the 

 expressions 



Ifl'^ m-—\ Ifl-^ + m- —\ (S-) 



m\9x 9y )^ l\9x 9y J 



If, therefore, two directions, 5i, S3, are defined by two curves which, at 

 a point, /*, common to both, have a common tangent and equal curvatures, 

 tlie second derivatives at P of a function fi taken with respect to the 

 two directions are equal. 



If at any point the curvature of the curve of the u family which 

 defines the direction s-^ is zero, the coefficients of 9Vi,/ 9x and 9^/9y in 

 the expression for £>/^Q, at the point vanish. If the u curves are a 

 family of straight lines, the last two terms of (7) disappear, but the 

 coefficients of the other terms are, in general, not constant. 



If there is no point in the region H at which both the quantities 9^j 9x, 

 5n I9y vanish together, and if the direction s is at every point of R that 

 in which fi increases most rapidly, Z)^ O =: h, where h is the gradient of 

 Q, that is, the tensor of the gradient vector. Now h is itself, in general, 

 a scalar point function, which, when equated to a parameter, yields a fam- 

 ily of curves the directions of which are usually quite different from those 

 of the lines of the gradient-vector. The normal at any point P to the 

 curve of this h family which passes through the point, has the direction 

 cosines 



where h' is the gradient of h. The angle between the direction, s, of the 

 gradient vector of O and the normal to the h curve has at every point 

 the value 



and the second derivative of 12 with respect to the direction s is, therefore, 



