When they are all put in series, so that the battery has a much greater resistance than the rest of the circuit, the current rises much more rapidly, because of the smallness of the time constant, although it never attains the same ultimate maximum as when grouped in the other way.  That is to say, if there is self-induction as well as resistance in the circuit, the amateur rule does not tell you the best way of arranging the battery.  There is another mode of regarding the matter which is helpful.  Self-induction, while the current is growing, acts as if there were a sort of spurious addition to the resistance of the circuit; and while the current is dying away it acts of course in the other way, as if there were a subtraction from the resistance.  Therefore you ought to arrange the battery so that the internal resistance is equal to the real resistance of the circuit, plus the spurious resistance during that time.  But how much is the spurious resistance during that time?  It is a resistance proportional to the time that has elapsed since the current was turned on.  So then it comes to a question of the length of time for which you want to work it.  What fraction of a second do you require your signal to be given in?  What is the rate of the vibrator of your electric bell?  Suppose you have settled that point, and that the short time during which the current is required to rise is called t; then the apparent resistance at time t after the current is turned on is given by the formula: 

        R_{t} = R x e^{(R/L)t} + ( e^{(R/L)t} — 1 )

TIME CONSTANTS OF ELECTROMAGNETS.

I may here refer to some determinations made by M. Vaschy,[1] respecting the coefficients of self-induction of the electromagnets of a number of pieces of telegraphic apparatus.  Of these I must only quote one result, which is very significant.  It relates to the electromagnet of a Morse receiver of the pattern habitually used on the French telegraph lines.

                                              L, in quadrants. 
    Bobbins, separately, without iron cores. 0.233 and 0.265
    Bobbins, separately, with iron cores. 1.65 and 1.71
    Bobbins, with cores joined by yoke,
       coils in series 6.37
    Bobbins, with armature resting on poles. 10.68

[Footnote 1:  “Bulletin de la Societe Internationale des Electriciens,” 1886.]

It is interesting to note how the perfecting of the magnetic circuit increases the self-induction.