## 1 Introduction  
[PERSON] solved the problem of distribution of flow in networks of pipes in his article \"Analysis of Flow in Networks of Conduits or Conductors\" [1] published on 13 November 1936.  
Networks of pipes are nonlinear systems since the relation between flow and pressure is not linear. On the contrary, the relation between current and voltage in electrical networks with regular resistors is governed by the linear Ohm's law. Electrical circuits with diodes as well as hydraulic networks are nonlinear systems where resistance depends on current and voltage, i.e., on flow and pressure, respectively [2]. Nonlinear electrical circuits are electrical circuits containing nonlinear components. Nonlinear components can be resistive, capacitive, and inductive.  
The distribution of flow in a network of pipes depends on the known inputs and consumptions at all nodes, on the given geometry of pipes, and on network topology. A stable state of flow in a network must satisfy [PERSON]'s laws, which are statements of the conservation of mass and energy. Although in theory an indefinite number of flow distributions that satisfy that the conservation of mass is possible, only one distribution from this set also satisfies the conservation of energy for all closed paths formed by pipes in the network. This state is unique for the given network and in- and outflows [3].  
Since the relation between flow and pressure is not linear, [PERSON] used a relation between an increment of flow and an increment of pressure, as this relation is linear for the given quantity of flow. If, however, the increments are fairly large, this linear relation is somewhat in error, such as for gas compressible flow. However, if the pressure drop in pipes is minor, such as in a municipality network for natural gas distribution, the Hardy Cross method can be used without significant errors [4; 5; 6]. Moreover, the Hardy Cross method can also be used for water pipe networks (district heating [7] and cooling networks [8]) and ventilation systems [9; 10] (a related formulation is presented in Appendix).  
The Hardy Cross method is an iterative method, i.e., a method using successive corrections [4]. [PERSON] and [PERSON] in the 1930s, writing in Russian, offered similar methods [11; 12]. Probably because of the language barrier and the political situation in Soviet Russia, [PERSON] was not aware of [PERSON] and [PERSON]'s contributions.  
Today, engineers use the most improved version of the Hardy Cross method (the \(\Delta Q\) method [13]; for \(\Delta p\), see [14]), which analyzes the whole looped network of pipes simultaneously [15].  
As a novel approach presented for the first time here, we tested multi-point iterative methods [16; 17] that can be used as a substitution for the Newton-Raphson approach used by [PERSON]. This approach, however, did not in this case reduce the number of required iterations to reach the final balanced solution.  
One example of the pipe network for distribution of gas is analyzed using the original Hardy Cross method [1] in Section 3.1, its related equivalent from Russian literature [11; 12] in Section 3.2, the improved version of the Hardy Cross method [15; 17; 18] in Section 3.3, and finally the approach which uses multi-point iterative methods instead of the commonly used Newton-Raphson method in Section 3.4.