Fourth, and of almost as great importance as the reduction in weight which the five-cylinder radial arrangement permits, is its unusual qualities as regards vibration. Since these five-cylinder engines were built by the writer a very thorough treatment of their properties as regards balancing has been given in a treatise on the balancing of engines,[44] so no discussion of the mathematical formulæ involved in a study of the question of the inherent balancing properties of these engines will be here given. It is sufficient to call attention to the fact that in an engine having five cylinders arranged radially, all of the reciprocating parts are balanced for all forces of the first, second and third orders. As it is only the reciprocating parts which give any trouble in balancing any engine, the unbalanced rotating parts being readily balanced by placing an equal weight at an equal distance from the center of rotation, and on the opposite side thereof, it is readily seen that the properties of balancing which are inherent in this type of engine are unusual. A six-cylinder engine having a six-throw crank shaft is not nearly so thoroughly balanced as this type having its five cylinders radially arranged, for in the latter case all the moving parts are in one plane, while in the former case the moving parts are in six separate and parallel planes, and there is consequently considerable longitudinal vibration which can never be overcome. While this is true as regards the vibration due to moving masses, it is still more impressively true as regards vibration due to reaction arising from the force of the explosions in the engine cylinders, especially when the engine is running slowly and having heavy explosions.