It will be seen, from a general observation and study of all of the diagrams, that the best and most gradual curves are made by the front wheel in descending from, and by the rear wheel in advancing upon, the obstacle; hence it follows that the front wheel works against momentum more in ascending and the rear wheel more in descending. CHAPTER IX. SPRINGS IN RELATION TO THE CURVES OF TRANSLATION, MOMENTUM, AND CONCUSSION. It was a pet scheme of the writer’s to treat of the matter of the annoyance to the rider resulting from a shock or jolt and change in momentum in the various styles of bicycles in a purely mathematical form, and to some extent it can be done; but it is found that so many considerations enter that the question becomes almost interminable. The aim was to find a formula with the sizes of wheels, distances between centres, and position of saddle as variables, which would, when applied, give us a result representing the sum total of annoyance felt by the rider in passing over an obstacle or any depression, rut, or ditch of given height or depth on any combination of wheels likely to be used in one machine. The difficulty in the question is in determining just what that annoyance results from or consists in; no doubt the initial impact, change of direction, and sudden reduction of momentum, and also the duration of the shock, all enter into the grand total. From a theoretical stand-point there need be no loss of power and consequently no annoyance in running over an obstacle, since all the momentum lost in a forward direction ought to be transmitted vertically in mounting the obstacle, thereby establishing a potential energy which would again be transformed into momentum forward as the wheel rolls down from the elevation. Neither should a rut have to be avoided, since by running into it we gain a momentum that should carry us out; hence, as per theory, the cycler should not worry about riding over rough roads, for in mounting each obstacle he only loans a bit of power in going up, which will be returned to him in going down, and in running down into a rut momentum will be loaned to him sufficient to bring him out. But, alas! he does not fancy the thing; somehow he has a like prejudice against rough roads that he has to hills, and as this prejudice cannot arise from purely theoretical considerations, we must look for some violation of nature’s laws, or some cause why such laws are not directly applicable. In my judgment there is a reasonably definite connection between the annoyance felt by the cycler in riding over a rough road and the actual loss of energy, though not a similar one in all respects to that which applies in regard to hills. A shock produced by a sudden check or deviation of the momentum is not only hurtful in causing a direct loss of kinetic energy, which the rider has stored up and to regain which he must afterwards do work, but also in