Let the arms of the supposed lever, or the semi-diameters of the supposed a heel and axle, be represented by r and r', the power p being applied at the extremity of r, and the reaultance se at that of r'. By se r' the nature of the lever, p= in the case of equilibrium; therefore, when the power Is such as to produce motion, the motive force may be w r' expressed by p' if applied at the extremity of r. Now, in order that the momentum of the inertia of se at a dietanco from the fulcrum may be made equivalent to the momentum of inertia of a body at a distance r, on representing such body by p", we have whence : the whole inertia to be overcome, if to applied at a diatanco r from the fulcrum, will therefore be p+ and the accelerative force at the extremity of r will be ter 9+ r Opseer ----..
10 r's Or + w P+ But, by dynamics, the velocity of a body series with the force and time ; therefore, representing the velocity at the end of the arm r by r, e C in order to obtain the velocity at the end of the arm r, the expreaelon for must be reduced in the ratio of r to r'; therefore the velocity at the latter extremity varies with r r' w w r'" f.
This expression is to bo a maximum; therefore, on differentiating it, r' being the variable, and making the result equal to zero, there will be obtained the ratio of r to r' (which Is the same as that of the velocities of p and r) consistently with the condition that the velocity of w is a maximum.
In the theory of machine., the modification of motion and tho modification of force take place together, and are connected by certain laws; but in the etudy of the subject there is an advantage in first considering the principles of the modification of motion, which are based upon a branch of geometry called cinematic', and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the law,' of dynamics. The separation of cinematmes from dynemies is due mainly to Monge, Ampkre, and Willis. The modern view of the subject of applied mechanics may be best studied in the following works : Puncelet, ' likanique Industrielle ;' Morin, Notions Fondamentales de M6canique Willis, ' On the Principles of Mechanism ;' Moseley, 'Mechanics of Engineering and Architecture ;' Whewell, of Engineering ;' Rankine, On Applied Mechanics; and on Prime Movers.'