In this formula i_{t} means the strength of the current after the lapse of a short time t; E is the electromotive force; R, the resistance of the whole circuit; L, its coefficient of self-induction; and e the number 2.7183, which is the base of the Napierian logarithms.  Let us look at this formula; in its general form it resembles Ohm’s law, but with a new factor, namely, the expression contained within the brackets.  The factor is necessarily a fractional quantity, for it consists of unity less a certain negative exponential, which we will presently further consider.  If the factor within brackets is a quantity less than unity, that signifies that i_{t} will be less than E / R. Now the exponential of negative sign, and with negative fractional index, is rather a troublesome thing to deal with in a popular lecture.  Our best way is to calculate some values, and then plot it out as a curve.  When once you have got it into the form of a curve, you can begin to think about it, for the curve gives you a mental picture of the facts that the long formula expresses in the abstract.  Accordingly we will take the following case.  Let E = 2 volts; R = 1 ohm; and let us take a relatively large self-induction, so as to exaggerate the effect; say let L = 10 quads.  This gives us the following: 

________________________________________
|             |              |         |
|  t_{(sec.)} | e^{+(R/L)t}  |  i_{t}  |
--------------+--------------+---------|
|      0      |      1       |  0      |
|      1      |      1.105   |  0.950  |
|      2      |      1.221   |  1.810  |
|      5      |      1.649   |  3.936  |
|     10      |      2.718   |  6.343  |
|     20      |      7.389   |  8.646  |
|     30      |     20.08    |  9.501  |
|     60      |    403.4     |  9.975  |
|    120      |  16200.0     |  9.999  |
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In this case the value of the steady current as calculated by Ohm’s law is 10 amperes, but Helmholtz’s law shows us that with the great self-induction which we have assumed to be present, the current, even at the end of 30 seconds, has only risen up to within 5 percent. of its final value; and only at the end of two minutes has practically attained full strength.  These values are set out in the highest curve in Fig. 54, in which, however, the further supposition is made that the number of spirals, S, in the coils of the electromagnet is 100, so that when the current attains its full value of 10 amperes, the full magnetizing power will be Si = 1000.  It will be noticed that the curve rises from zero at first steeply and nearly in a straight line, then bends over, and then becomes nearly straight again, as it gradually rises to its limiting value.  The first part of the curve—­that relating to the strength of the current after very small interval of time—­is the period within which the strength of the current is governed by inertia (i.e., the self-induction) rather than by resistance.  In reality the current is not governed either by the self-induction or by the resistance alone, but by the ratio of the two.  This ratio is sometimes called the “time constant” of the circuit, for it represents the time which the current takes in that circuit to rise to a definite fraction of its final value.