on differentiation , each reverts to the other : &amp;formula; and &amp;formula; . the hyperbolic tangent , tanhat , is &amp;formula; and , starting at zero , never exceeds unity , however large t may become . the remaining three hyperbolic functions , sech , cosech and coth , are the reciprocals of the above three ratios respectively . fig 1.4 shows the whole family of curves . tables of the hyperbolic functions are available , but are not so readily available as those of the circular functions . a device by which the more extensive circular function tables may be used in conjunction with a subsidiary table ( the Gudermannian ) is described in appendix 2 . the general case , where the time constants of the two exponential terms are not the same , may be expressed as the product of another exponential and a hyperbolic function . thus : &amp;formula; . if a is positive , this expression will always diverge . if a is negative ( with b positive ) , the final value will always be zero , and this is the more usual in practice . fig 1.5 ( a ) shows the result of the sum of two negative exponentials , and fig 1.5 ( b ) the difference . the second is seen to start at zero , reach a maximum , and then decay . as Sallust remarked : omnia orta occidunt et aucta secuntur , or everything rises but to fall and increases but to decay . the time at which the maximum is reached is easily found to be &amp;formula; and there is a point of inflexion where &amp;formula; . this type of curve is encountered , for example , in radioactive cases where a substance A decays into another substance B , which , in turn decays into a stable end-product C . the curve shows how the amount of the second substance varies with time . intuitive estimation of transients . a demonstration will now be given of how the transient current resulting from switching operations may be obtained in simple cases , without resort to mathematics ( or very little ) . the following plausible assumptions are made : 1 . that an uncharged capacitor behaves as a short circuit at the instant of applying a steady p.d ; and after a long time , when fully charged , acts as a disconnexion or infinite impedance . 2 . that a pure ( resistanceless ) inductance behaves in the opposite way ; offering apparently infinite impedance at the instant of application of the direct voltage , and short circuit after a long time - that is , when the current is steady . 3 . that , in the interim period , the current changes according to a simple exponential law ; the time constant of which is either rc or l/r , where R , L or C may be simple or compound . 4 . that there can be no discontinuous jumps in either the voltage across a capacitor or the current in an inductor . the magnetic space constant m0 ( otherwise the permeability of free space ) has dimensions henry &amp;sol; metre , and the electric space constant , or permittivity of free space , e0 , farad &amp;sol; metre . the square root of the reciprocal of the product of these two , therefore , has the dimensions of velocity and this is the velocity of electro-magnetic waves , c , equal to 299792 km &amp;sol; sec , according to the latest evidence . it follows that &amp;symbol;(LC) has dimensions of time , and &amp;symbol;(L/C) dimensions of resistance . in fact , &amp;symbol;(L/C) is the well-known expression for the characteristic impedance of a loss-free transmission line . from this it is seen that l/r and cr both have dimensions of time , and this time is the time constant . any time constants we may encounter in the study of transients must be in the form of a certain inductance divided by a certain resistance , or a capacitance multiplied by a resistance , or else the square root of the product of an inductance and a capacitance . no other combinations are possible . let a simple series lr circuit be suddenly connected to a constant voltage source V , at time t = 0 . the initial current will be zero and after the transient has subsided will be v/r . at first sight , this is not a decaying exponential ; it decays upwards , so to speak . it may be easier to consider the voltage across the ( pure ) inductance L . the initial voltage across this part of the circuit is equal to V , and the final value will be zero . using the assumptions made above , the voltage across L in the transient period will be &amp;formula; and , because there are only two circuit elements , T is obviously equal to l/r . the voltage vr across the resistive part of the circuit when added to vl must always give V , hence &amp;formula; and the current in R ( and also L , of course ) is &amp;formula; the well-known result of a problem which is often given to beginners as an exercise in solving differential equations of the first order . by similar reasoning , the current through a cr series circuit is found to be &amp;formula; . the voltage across the resistor is &amp;formula; and that across the capacitor is , therefore , &amp;formula; , and so the charge in the capacitor at time t is &amp;formula; . theoretically , the current never does reach its final value ; the final value may be said to be attained when it falls short of the theoretical final value by an amount too small to be detected by the measuring instrument in use , or , in decay , when it has reached the r.m.s value of the noise level . for practical purposes , and as a rough guide , the current will have reached within one per cent of the final value in a time five times the length of the time constant ( see p 2 ) . this is roughly seven times as long as the half-life of radioactivity . one can not help feeling that , subconsciously or not , people who think in terms of half-life have the idea that all activity will have ceased in about twice that time . three-element circuits . it may well be argued at this point that the above type of reasoning is all very well for simple two-element circuits , but would fail if carried further . let us consider , therefore , the circuit of fig 1.6 in which the capacitor C has a leakage resistance R2 . the initial current on making the switch ( t = 0 ) is v/r1 , and the final current will be &amp;formula; . this fixes the limits between which the current must vary exponentially . the time constant of this exponential must be the product of a capacitance and a resistance . the capacitance is obviously C , but what are we to take as the resistance ? the answer is , that resistance which effectively appears across the terminals of C when the switch is closed ; this is clearly R1 and R2 in parallel , the voltage source having no internal resistance . so the time-constant is &amp;formula; and we can now sketch the current &amp;sol; time curve as in fig 1.7 . the exponential part is &amp;formula; and to this must be added &amp;formula; ; after a little manipulation the current can be written &amp;formula; . the final capacitor voltage &amp;formula; will be &amp;formula; , and its variation with time is &amp;formula; . the case of two capacitors and one resistor is amenable to similar treatment , though not quite so easily ( see fig 1.8 ) . here there is a little awkwardness due to the fact that the initial rush of current is very high ; theoretically infinite but lasting for zero time ( see under delta function in the next chapter ) . we shall side-step the current question and work , instead , in terms of voltage or quantity of charge , neither of which becomes infinite . when the switch is made , the capacitors immediately charge up to &amp;formula; and &amp;formula; volts respectively , and the quantity of charge on the plates of each is &amp;formula; coulombs . because of the presence of the resistance , C1 will discharge exponentially , with T = R(C1+C2) while C2 , following its initial charge at time t = 0 , will acquire further charge until its p.d reaches the source voltage , V , and it holds c2v coulombs . hence : charge in C1 &amp;formula; and charge in C2 &amp;formula; . the current taken from the supply , which is the same as that in C2 may be found by differentiating &amp;formula; with respect to time and is &amp;formula; plus , of course , the initial pulse of current . see p 43 , equation ( 2.8 ) . circuits comprising R , C and L are , in general , beyond this simple intuitive treatment , though there are exceptions . one of these is shown in fig 1.9 where a constant voltage source , V , is applied to two elementary circuits , lr1 and cr2 respectively . we shall suppose that the two time constants are the same ; l/r1 = cr2 or r1r2 = l/c . the initial current i0 will be v/r2 and the final current , &amp;formula; will be v/r1 . thus , during the transient period , the current will be switched over from the capacitive side to the inductive side at a rate governed by the common time constant . alternatively , we can make use of results already obtained on p 9 and write down the supply current immediately as &amp;formula; . in the special case where R1 = R2 = R , the term containing the exponential vanishes , so there is no transient and the current taken from the supply is constant and equal to v/r . in other words , the network is distortionless and free from phase shift for all frequencies ; provided always that R = &amp;symbol;(L/C) . analogies . in the elementary teaching of electricity use is often made of analogies with mechanical systems . electricity seems to be more difficult to understand than mechanics for most people , because the mind can readily picture mechanical processes , but electrical phenomena require the effort of abstract thought . as the understanding develops , the debt can be repaid , often with much interest , as problems in mechanical engineering are referred to their electrical counterparts for solution ; an example of this is in the theory of vibrations , both free and forced . the analogue of electro-motive force , E , is force , F , or mechano-motive force as it has been called : that which moves mechanical systems or particles , the unit being the newton ; though it is only fair to say that this unit is making but slow progress into mechanical circles . the magnetic circuit analogue , magneto-motive force , is not so good since , although we speak of flux , there is nothing which actually flows . in angular motion the equivalent is torque , Tq , measured in newton . metre or joule &amp;sol; radian . electric current has its analogue in velocity - linear , v , or angular , o , and consequently quantity of charge , the time-integral of current , corresponds to linear displacement x , or angular displacement th . mass ( kilogram ) or moment of inertia ( kilogram . metre2 ) is analogous to inductance . it is noteworthy that while there has never been any confusion in the mind of the electrician between electro-motive force and self-inductance , the tyro mechanician often finds difficulty in distinguishing force and mass , and tortures himself with big pounds and little pounds as well as slugs and poundals . the increased use of the newton might soften these difficulties . electrical &amp;formula; . mechanical &amp;formula; . rotational &amp;formula; ( I being the moment of inertia ) . figure 1.10 shows how current and angular and linear velocity increase with time in systems where the resistance or friction is zero . if the force is removed after a certain time , t1 , the current will go on flowing with circuit energy &amp;formula; , or the wheel will continue to rotate with angular energy &amp;formula; , or the particle will continue with constant velocity ( Newton &apos;s law ) , and kinetic energy &amp;formula; . if resistance is present , the current ( or velocity ) does not increase indefinitely but reaches a limit , as we have already seen ( p 9 ) . the initial slope is the same as for the resistanceless case and the final value is given by the resistance divided into the electro-motive force , or the mechanical resistance divided into the mechano-motive force and so on . alternatively , the electrical resistance in ohms ( or volt . ampere-1 or henry . second-1 ) is given by e/i where I is the final value of current : and similarly , mechanical resistance is &amp;formula; , where vT is the final or terminal velocity ; and rotational resistance is Tq/oT . it follows that the unit of mechanical resistance is newton . metre-1 . second , or kilogram . second-1 , and of rotational resistance is newton . metre . second , or kilogram . metre2 . second-1 . the terms mechanical ohm and rotational ohm are used by Olson , but these seem rather far-fetched , particularly as they are referred to c.g.s and not practical units . 