GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



I05 



As we shall make an extensive use of the process of differentiation described, it 

 will be important to direct the attention to the character of the errors which will 

 enter, and the methods of diminishing their influence. Let us for this purpose con- 

 sider the derivatives of the two fields which are given by fig. 84 a and b. In both 

 cases the lines a = const, have the same general course and the same average distance 

 from each other; but on the first figure the distance varies in a regular way, and 

 in the second it shows small irregularities in its variations. The curves which repre- 

 sent the field of the differential quotient are then seen to be very different in the 

 two cases. In the first case they have a regular course, while in the second they show 

 great sinuosities. 



Now a free off-hand drawing which should represent a field as that of the first 

 figure will in consequence of the unavoidable errors get more or less the character 

 of the second figure. Thus the irregularities in the drawing of the given field will cause 

 oscillations in the course of the curves representing the field of the derivative. But as the 

 oscillations will go equally to both sides, they will be easy to reduce afterwards. 



Fig. 84. Regular course (A) and oscillating course (B) of the curves representing the differential-quotient f = 



ds 



A good method of diminishing them from the beginning will be to measure the line- 

 elements not one by one, but two by two or even more of them at a time. On the 

 divided sheet we can always find the proper hyperbolae for doing this. But the final 

 correction will always consist in reducing those sinuosities which are seen to arise 

 from errors in the drawing and not from the true nature of the given field. By this 

 correction a posteriori of the field of the derivative, we get a determination of this 

 field which by far exceeds the accuracy of the single measurements upon which the 

 process of differentiation depends. 



For the process of integration, the irregularities in the drawing of the given 

 field will cause no errors of greater importance. The process of integration itself 

 involves a formation of averages, by which the consequences of the irregularities 

 in the drawing are reduced. 



166. Other Forms of the Problem of Linear Differentiation and Integration. 

 Instead of constructing an auxiliary sheet for the determination of the differen- 



