GRAPHICAL DIFFERENTIATION AND INTEGRATION. Ill 



Or, let the linear differentiation be performed along the direction of the normal 

 n to the curves s. This derivative 



(c) h=f 



an 



will represent the change of direction per unit length when we proceed normally 

 from curve to curve instead of tangentially along one and the same curve s. It there- 

 fore shows how the different curves 5 diverge from each other. Equation (c) gives 

 the field of divergence of the given system of curves s. This field can also be found 

 by use of the divided sheet of fig. 81, and it will not be necessary to draw the normal 

 curves n, as the sheet can be placed with the same ease both with its ordinates 

 normal to and parallel to the given curves s. 



When we remember our definition of the positive normal n to a given direction 

 s (section 155) we see that formula (c) contains the following rule for the sign of the 

 divergence <5: 



Divergence of a system of curves will be positive or negative according as they 

 appear to an observer looking in the positive direction of the curves to 

 diverge or to converge (fig. 89). 



As will be seen at once, there is a close relationship between the fields of diver- 

 gence and of curvature. The field of divergence of a system of curves is the field 

 of curvature to the normal curves, and vice versa the field of curvature is the field 

 of divergence to the normal curves. 



The derivatives (b) and (c) are derivatives of the second order in reference to 

 the originally given system of curves. The two successive operations, consisting in 

 a directional and a subsequent linear differentiation, can be combined into one which 

 represents a differentiation of the second order and which can be performed by the 

 divided sheet of fig. 90. This sheet contains a set of concentric circles with integer 

 values (multiplied by a power of 10) of the curvature, i.e., integer reciprocal values 

 of the length of the radii and a set of divergent radii with equal and small angular 

 intervals. For continuous use the sheet is perforated at the points of intersection of 

 the circles with the central radius. 



This sheet can be placed directly upon the field of the system of curves S origin- 

 ally given. In order to find the field of curvature (fig. 91 a) we place it with the circles 

 tangential to and the radii normal to the curves 5. One after another of the curves 5 

 is followed, and the points are marked where these curves give complete osculation 

 with one of the circles of the sheet. In order to find the field of divergence (fig. 91 b) 

 we place the sheet with the circles normal to and the radii tangential to the curves s. 

 One after another of the curves 5 is followed, and the points are marked where the 

 circles osculate the normal curves, i. e., the points where one of the circles is normal 

 to the curves next to the considered curve s. As a supplementary condition we 

 have that the radii shall be tangential to the curves at the points where these radii 

 are cut by the circles. 



169. Partial Derivatives; Ascendant and Gradient. The two-dimensional 

 scalar a is a function of two coordinates which figure as independent variables. 



