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and the initial conditions 



■ X SV Q 



a = a 



p = 



p' = 



at t = 



These differential equations cannot be solved exactly in terms of known 

 mathematical functions. 



In order to handle these differential equations numerically, it is 

 desirable to rewrite them as integral equations. This is to avoid 

 numerical differentiation, in which one subtracts two numbers. If both 

 numbers contain random errors, the percentage error in the difference 

 will be larger than in either number. However, in integration one 

 adds numbers, thereby decreasing the fractional error due to random 

 errors. Rewritten, the four equations become 



.v x — 



rt 



{-k + x{e-p-p'} + k_p)dt 



p = f {-l + ap - k_p + k + {e-p-p'}] 



Jo 



p' = {l+ap — m + ap')dt 

 Jo 



a — a Q = - {l + ap + m + ap') 

 Jo 



dt 



dt 



Experimental curves can be measured for the time dependence of x, p, 

 p', and a. The mathematical problem is to choose the constants 

 k + , k_, l + , and m + in such a fashion as to obtain an optimum fit. 



The integral equations can be solved in 1 second by an analog com- 

 puter. Each concentration is represented by the electrical potential 

 between a specified binding post and ground. Likewise, the reaction- 

 time units are represented by other time units. If, then, the potential 

 on any of these binding posts is connected to the vertical axis amplifier 

 of an oscilloscope and the horizontal axis to a sweep generator, a curve 

 will appear of potential V, against sweep time r. This is now interpreted 

 that Fis analogous to some concentration x, the height of the deflection 

 in centimeters being proportional to the concentration in millimoles per 

 liter. The horizontal distance on the oscilloscope tube face measured 



