The ventilation circuit of a mine is a complex network of underground airways. In all but the smallest of mines, airflow through the circuit is induced by mechanical means. The relationship between the volume of air flowing and the associated pressure loss in any airway is non-linear. Slow through the circuit can be predicted by numerical analysis or by using an analogue computer. Numerical analysis can be performed manually, or aided by a linear analogue, or programmed for a digital computer. Analogue computers and linear analogues axe reviewed briefly and the mathematical theory underlying linear analogues is included in a general development of methods of numerical analysis. Three sets of linear equations, approximating to the non-linear network equations, are considered. Standard numerical methods can be used to solve the linear equations, but additional steps are required in order to obtain a solution of the original nonlinear equations. The coefficient matrices of the linear equations are sparse so that elimination, which applies particularly to the 'linear conversion' equations, is not suited to a digital computer. The 'balancing flow' equations, which have previously received but little attention in mine ventilation studies, are considered in this thesis because, compared with both the 'linear conversion' and 'balancing pressure’ equations, they have the advantage of not requiring selection of meshes. However, the author has developed a computer program for the selection of meshes, suited for a rapid solution of the 'balancing pressure' equations. Since the program includes consideration of the pressure-flow characteristics of fans, which would be a more complex procedure with the 'balancing flow' equations, the 'balancing pressure' method with automatic mesh selection has been preferred. Techniques are given for manipulating data to enable a wider range of problems to be solved using the single computer program.