GRAPHICAL DIFFERENTIATION AND INTEGRATION. 1 25 



Let a be a scalar which depends upon coordinates and time. Now let a be the 

 value of this scalar at a certain point at a time t , and a, its value at the same point 

 at the time/,. The quotient 



/ \ a i An 



(a) <P = j-~r 



will then represent the average value which the differential-quotient 



W * = Tt 



has at this point during the interval of time t T t . If this interval is sufficiently 

 short we can consider the value of the quotient (a) as identical with the value of the 

 differential quotient (b) at the time 



ic) t = t + ^ 



If we know the field of the scalar a at two moments t Q and /,, which are separated 

 by a sufficiently short interval of time t 1 t , we can form the field of the derivative 



(b) at the time (c) in this manner: 



We form by graphical subtraction the field of the difference 



id) a. <*o 



and afterwards perform the division of this field by the constant factor 



() h-k 



The problem is thus reduced to algebraic problems which we have already 

 treated. The only difficulty will be that the fields a and a, may too closely resemble 

 each other. Their equiscalar curves may cut each other under too small angles and 

 it may be difficult to get a good drawing of that set of diagonal curves which repre- 

 sents the difference (d). It will be important to remark, however, that the errors 

 will take precisely the same character as in the previous cases of differentiation: 

 the curves representing the derivative will get an oscillating course, and these 

 oscillations can be smoothed out afterwards. But in order to avoid these errors from 

 the beginning, it will be important not to choose too short an interval of time (e). 

 On the other hand it must not be chosen too long if it is to be allowed to identify, 

 within the margin of allowable departures, the finite difference-quotient (a) with 

 the differential-quotient (b) at the time (c). 



The reversed problem, that of the pure time-integration, will be solved with 

 the same ease. Let the field of a be given at the time t a , a = a ; and let the value 

 of the derivative <p be known at any time t which is subject to the condition t a <t<t t . 

 If then the interval of time /, t Q is sufficiently short, we can identify the value of 

 <p at the time / with the average value <p during the interval of time t t t . We then 

 find the value of a at the time t, by the formula 



(/) a, = 0.+5P&-0 



Thus we have to perform the following graphical operations: first to multiply the 

 field of the derivative V by the interval of time t l t , and then to perform the 



