General types of balancing problems are classified into three groups: Those requiring perfect balance, those in which the machine inherently is balanced sufficiently well for practical purposes, and those requiring a distinct compromise, such as single-cylinder engines. It is a method of securing an optimum condition for the last class that is presented. The method consists, first, of the selection of a suitable criterion for the particular problem, such as minimizing the average potential energy of vibration in a spring mounting, which tends to secure the least possible transmission of vibration. Another criterion may reduce the vibration of the machine itself to a minimum. The selection of a suitable criterion is a matter for the judgment of the designer. When the selection is made, the attainment of the optimum sought is carried out by mathematical analysis. This general method is particularly useful for balancing spring-mounted machines. An illustrative problem is given for this case, showing the effects of working to several different criteria.