472 ADAMS : A USEFUL TYPE OF FORMULA 



1. The derivative is finite at the origin and increases with t, 

 rapidly at first and then more slowly, tending toward a constant 

 value as a limit (see fig. 2). Such a curve referred to the dotted 



line as the axis is given by — - = 1 — e~\ in general by ^ — = 



at dt 



6 (1 — e^^), C being a negative quantity ;3 or when referred to the 



dE 

 t — axis by equation — = a + 6 (1 — e*^'). Integrating and ap- 



plying the initial condition that E = 0, ^ = 0, we have, E = At -{- B 

 (1 - e"). Where A = a -\-h,B = —. This equation is of the 



desired form, and, as we have found by trial, is highly satisfactory 

 in a number of cases where the graph has the general form of 

 figure I, as for instance, the relations between temperature and 

 e.m.f. of copper-constantan thermoelements and between pres- 

 sure and resistance of manganin or of "therlo" wire. 



The calculation of the constants of this type of equation offers 

 no especial difficulty: in fact it involves no more time or labor 

 than in the case of the cubic equation: 



E = A't + B'P + C't' 

 a form of equation which is often used and is in general of less 

 utility. The method of evaluating the three constants, A, B 

 and C is as follows : Write down three simultaneous equations con- 

 taining pairs of corresponding values of E and t. Eliminate A 

 by addition or subtraction and B by division; we then obtain 

 an equation in C of the form : 



(1 - e^'^) - P (1 - e''^) El - P E_2 ^ ^ 



(1 - e'^^) - Q (1 - e''^) E2 - Q E3 

 where P = — ^^^^ Q = j-- Assume now several values of C and 



with the aid of tables of the exponential compute the corre- 

 sponding values of K. Then plot K against C and determine the 

 proper value of C by graphical interpolation. The value of C 

 being now known, B and finally A are readily calculated. 



dE 



3 Another suitable function is —- = a -{- b tanh ct which would lead to the equa- 



dt 



tion E = nT -\- h log cosh ct. 



