52 LECTURE VII. 



Systems of wheels and pinions of various kinds resemble, in their mecha- 

 nical properties, either a series of levers, or the combination of cylinders 

 which constitutes the wheel and axis ; but the form of the teeth may pro- 

 duce a difference in their action, which will be mentioned when the prac- 

 tical construction of wheelwork is discussed. 



Sometimes the axis connected with a winch is composed of two cylinders, 

 one end of the rope being uncoiled from the smaller, while the other end 

 winds round the larger ; the weight being supported by a pulley running 

 in its angle. Here the conditions of equilibrium are easily determined 

 from the place of the centre of gravity, and the effect of the machine is the 

 same as if the weight were attached to a rope coiled round a simple 

 cylinder, of a diameter equal to half the difference of the diameters of the 

 double axis. The machine is, however, much stronger than such a 

 cylinder would be, and does not require so great a curvature in the ropes 

 employed. (Plate IV. Fig. 51.) 



The laws of the equilibrium of pullies have been referred, by some 

 writers on mechanics, to those of the lever ; but the comparison is both 

 unnecessary and imperfect ; in the simple case of two equal weights at- 

 tached to a thread passing over a single pulley, which is the only one that 

 allows us to recur to the properties of the lever, the conditions of equili- 

 brium are axiomatically evident, without any further reasoning ; and in 

 more complicated cases the calculations proceed on perfectly different 

 grounds. We are, therefore, to consider a pulley as a cylinder, moving 

 on an axis, merely in order to change the direction of a thread, without 

 friction ; for whatever is demonstrable of pullies or their combinations, 

 would be equally true of as many perfectly smooth grooves, which do not 

 bear the most distant analogy to the lever. 



Now when the direction of a thread is altered, by passing over any per- 

 fectly smooth surface, it communicates the whole force acting on it ; for 

 the resistance of a surface, without friction, can only be in a direction 

 perpendicular to itself and to the thread, and the operation of any force 

 remains undisturbed by a resistance which is always in a direction per- 

 pendicular to it. 



A fixed pulley, therefore, has no effect in gaining a mechanical ad- 

 vantage ; but by means of a moveable pulley it is obvious that a weight 

 may be supported by two forces, each equivalent to half the weight, 

 applied in a vertical direction to the extremities of the thread ; and these 

 forces may be derived from two weights, if the thread be made to pass over 

 two fixed pullies in a proper position ; and if one of the ends be attached 

 to a fixed point, and the other remain connected to its weight, the equi- 

 librium will continue unimpaired, each portion of the thread still support- 

 ing one half of the original weight ; so that, by means of a single moveable 

 pulley, one body may retain in equilibrium another of double its weight. 

 (Plate IV. Fig. 52, 53.) 



The modes of arranging pullies are very various, but the advantage 

 which they procure may always be estimated, from the consideration that 

 every part of the same thread must be equally stretched ; and where there 

 is only one thread, the weight will be divided equally among all the por- 



