only party interested, but when man is directly concerned we can expect more rapid development. When we start our machines for a run it is considerable work to get up an initial velocity or momentum; however, after that there should be only the friction of the machine within itself and upon the road to be overcome, together with the friction against the air; that is to say, if inequalities in the road could be run over without a loss of momentum being caused thereby, there would not be nearly so much work in travelling upon the cycle as is now necessarily required. The principal parts of the cycle should be as rigid and firm as possible, so as not to respond at random in vibration to every little shock they should chance to receive, for the spring or elasticity wants to be such as can be controlled,—that is, made to store energy in the right way and give it out at the proper time with a desired effect upon the momentum. It must be remembered in this connection that useful energy can be stored in the machine only in the plane of horizontal motion and gravity; in other words, vertically and horizontally. Any elasticity at an angle to this plane can only be of use in reducing the concussion upon the rider in a lateral direction; and since, upon a single-track machine, but little if any shock can occur in such direction, it should be seen to that no undue side motion is permitted. In order to fully comprehend the loss of power that it is possible to save by proper springs, observe as a particular case the annexed diagram showing two thirty-inch wheels arranged substantially as in the present rear-driving Safety. Let _c_ be the centre of gravity, and let the line _c o_, drawn to the obstacle, pass through the centre of the front wheel and make an angle of forty-five degrees with the horizontal. [Illustration: Rover momentum.] The momentum _c l_ is split up into two equal components, one acting in the direction _c o_, and the other in the direction _c k_ perpendicular to _c o_, tending to turn the system about _o_ as a centre. The numerical value of the _c k_ component, calling _m_ the momentum, _m_ is ———, and its value in the forward direction _c o_ is √2 _m_ _m_ 1 _m_ ——— cos 45°= ——— ——— = ———, which is the forward momentum retained, √2 √2 √2 2 showing that in this case one-half of the forward momentum is saved and the other half lost. It is scarcely necessary to say that the use of an imaginary four-inch obstruction, in our study of momentum and concussion, is entirely arbitrary. Of course obstructions of all heights will evolve proportional results. This proportion would not, however, be linear; the nearest we can come to it is to say that the annoyance begins with an obstruction of zero height, and increases about as a trigonometrical sine increases when the angle grows larger.