GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



I07 



If we consider <p = x as abscissa and ds = y as ordinate, the constant values 



of da give straight lines - = const, which pass through the origin of the 



coordinates. Fig. 86 contains these lines for "integer" values da = 

 2.10 -1 , i.io -1 , o, no -1 , 2.io _1 . . . We get the well-known sheet 

 for direct measurements of lengths. When the line-elements are 

 short, it will be convenient to measure them two by two between 

 the lines da = + iio~'and da = i.io -1 . The abscissa will 

 then give the length in ten-fold enlargement, with a corre- 

 sponding increased accuracy of the reading. Longer 

 elements can be measured without any enlargement 

 by use of the lines da = 



167. Curves of Equal Intensity and Curves 

 of Equal Transport. In order to give an 

 example of the use of this sheet, we shall 

 treat the following problem: to draw 

 curves for equal transport when the 

 vector-lines and the intensity 

 curves of a vector are given. 

 The curves of equal trans- / 

 port will be determinate 



Fig. 86. Divided sheet 

 for direct length-measure- 

 ments. 



only when the vector- 

 bands have been chosen 

 (section 119). If we make a 

 perfectly arbitrary choice of 

 these bands, letting narrow and 

 broad bands follow each other in 

 an irregular way, the curves of equal 

 transport will get an oscillating course, 

 see fig. 84 b. In order to get simple curves 

 of equal transport, we must therefore first make 

 a careful choice of the vector-bands. A simple way 

 of doing it will be to draw an arbitrary initial curve 

 C of regular shape, to divide it into elements ds' accord- 

 ing to a continuous law, and then to draw vector-lines 

 through the points of division. As a rule, we shall divide 

 the curve C into elements through which there goes unit trans- 

 port ; that is, A ' being the value of the vector at the points of 

 the curve C, and dn' the projection of the elements ds' of this curve 

 on the normal to the vector-lines, we determine these elements so that 

 for each of them 



(a) A' dn' = 1 



