GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



IOI 



We represent this function by marking the points where it has certain integer values, 

 a OI a,, a 2 , . . . a, a +I , . . . The expression " integer " must be taken in the same 

 generalized sense of the word as before (section 147). The differences between the 

 values of a in consecutive points will then also be expressed by "integer" numbers, 

 and they must be small enough to be considered as differentials, da = a +t a n . 

 The distance between the points will be the corresponding differentials of line ds, and 

 the problem of differentiation will consist in forming the values of the quotient 



(b) <p(s) = d 



ds 



at the different points of the line s. 



Fig. 81. Divided sheet for linear differentia- 

 tion and integration. (Measurement of recipro- 

 cal lengths.) 



In order to construct a convenient auxiliary for the formation of the value of <p 

 in one operation, we solve equation (b) with respect to da 



(c) da = ipds 



We measure off <p x along the axis of abscissae and ds = y along the axis of ordinates 

 of a rectangular system of coordinates. To each positive or negative integer value 

 of da, viz, da = . . . 2, 1, o, 1, 2, . . . will then correspond an equilateral 

 hyperbola xy = const. The diagram of fig. 81 contains these curves together with a 

 number of ordinates, one for each millimeter. Now let a value of the differential 

 da be given, say da = 4. The abscissae of the hyperbola da = 4 then gives the values 



