Maximum-Minimum Properties. When we turn to stationary distributions, i.e., such as no longer change in time, we do indeed find such characteristics. Stationary processes have certain maximumminimum properties, i.e., a given parameter of these processeS has not just any magnitude but the smallest or the greatest possible. A few examples may make this clear: if we have a number of different circuits between the poles of the same electric battery the currents will distribute themselves so as to produce a minimum amount of energy within the system. To take the simplest case of

two part circuits only. Then Kirchhoff's law states that ':1 =!.=., '2 rl

where i1 and ;2 stand for the intensities of the two part currents, and r1 and r2 for the corresponding resistances in the part circuits. Now it is quite easy to show mathematically that these currents, i1 in the circuit with resistance r1, and i2 in circuit with resistance r2, produce less heat than if i1 were greater or smaller, and consequently "2 smaller or greater, than is demanded by Kirchhoff's law. (The sum of the two intensities must be constant, since the total intensity of the circuit depends only upon its electromotive force and its total resistance. )

Another example is the soap bubble. Why has it the shape of a sphere? Of all solids the sphere is that whose surface is smallest for a given volume, or whose volume is largest for a given surface. The soap bubble, therefore, solves a maximum-minimum problem, nor is it difficult to see why. The soap particles attract each other, they tend to take up as little space as possible, but the pressure of the air inside forces them to stay on the outside, forming the surface membrane of this air volume. So they must form as thick a surface lamella as they can, and the smaller the surface the greater can its thickness be, if the amount of mass is constant. At the same time potential energy of this membrane will be as small as possible.