GRAPHICAL DIFFERENTIATION AND INTEGRATION. 103 



initial point a length ds equal to the ordinate which the hyperbola da = 4 has for 

 the value <p of the abscissa. Using the value <p = <p, which <p has in the region of 

 this new point, we measure off in the same manner the length ds, which leads to the 

 point where a has the value a +8, and so on. Inasmuch as a is integer, we find 

 in this manner points for integer values of the function a. If we wish to proceed 

 by other steps da, we use other hyperbolae. 



The marking off of the successive points can be made without removing the 

 sheet from the paper. Thus in the case of da = 4 the sheet is placed with the hyper- 

 bola 4 on the point from which the length ds is to be measured. The new point can 

 then be marked through the slit in the sheet. 



We have spoken above of the value which the given function <p has in the 

 "region" of the point from which the length ds is to be measured. This "region" 

 will have a maximal extent equal to the length of the line-element ds. The use of 

 one value or another of <p from this region will give no appreciable difference in the 

 lengths ds obtained, if these lengths are sufficiently short; but the greater the 

 lengths ds used, the more important it will be not to choose an arbitrary value 

 of <p in this region, but the average value of <p along the element ds. As the approximate 

 length of ds is seen at once, it will cause no difficulty to find a sufficiently approximate 

 value of this average value of <p, and to use it for the final determination of ds. 



Evidently the function a(s) which we determine by the process described will 

 be that which is expressed analytically by the integral 



(d) a = a -\- ( <p(s) ds 



Fig. 82 can be used to examplify this graphical integration. We then consider 

 the divisions <p = 5, 4, 3, . . .on the lower line as given, and find by use of the 

 divided sheet the divisions a = 12, 16, 20, . . .on the upper line. 



165. Application to Two-Dimensional Scalar Fields. The application of the 

 described process of linear differentiation to scalar fields in two dimensions will be 

 the most important graphical differential operation. It will return in most of the 

 more complex differentiation-problems. 



Let the two-dimensional scalar field be represented by a system of equiscalar 

 curves 



(a) a = a Q a = a, a = a 2 . . . 



where a o; a,, a 2 , . . . are integer values in the widened sense of the word as 

 defined above. Let further a system of curves s be given which cut through the 

 field. (Compare fig. 83.) The scalar a will then have a definite value at each point 

 of a curve 5. On each of these curves the scalar a will appear as a function of the 

 length of arc 5. We can therefore perform a linear differentiation along each curve s, 

 using the divided sheet as described in the preceding section. In this manner we 

 find the value of the derivative 



(6) * = ds 



