MECHANICS. 



are respectively equal to the radii of the 

 several wheels, and whose shorter 

 arms are equal to the radii of the 

 several axles, and therefore the con- 

 dition of equilibrium is that the power 

 multiplied by the product of the radii 

 of all the wheels is equal to the weight 

 multiplied by the radii of all the axles. 

 Thus if R, R', R" be the radii of the 

 wheels, and r, r 1 , r" those of the axles, 

 we have 



P x R R' R" = W x r r' r". 

 (62.) The force of these observations 

 will more readily appear, by consider- 

 ing the combination of wheels and axles 

 in fig. 41. The radius of the wheel 

 on which the power acts being R, and 



Fig. 41. 



w 



that of the axle r, it is equivalent to a 

 lever of the first kind, whose fulcrum is 

 its centre. The first axle acts on the 

 circumference of the second wheel with 

 the leverage of its radius r, and the 

 second wheel is also a lever of the first 

 kind, the fulcrum being the centre. The 

 radius r of the first acts upon the radius 

 R' of the second wheel, which depresses 

 the radius r' of the second axle, and 

 thereby "depresses the radius R" of the 

 third wheel, which raises the radius r" 

 of the third axle, and thereby raises the 

 weight. 



(63.) In this case the wheels and 

 axles are supposed to work by the fric- 

 tion of their surfaces. In light work, 

 where the pressure on the machinery is 

 not very considerable, this method of 

 connecting the wheel-work is often 

 adopted with advantage. The friction 

 of the surfaces is increased by cutting 

 the wood so that the grains of the sur- 

 faces in contact shall run in opposite 

 directions. Also by glueing upon the 

 surfaces of the wheels and axles buffed 

 leather. A saw-mill in which the 

 wheels act by friction has been used at 

 Mr. Taylor's, of Southampton, for 

 nearly twenty years, and is found to 

 work well. 



(64.) There are, however, other ways 

 of transmitting the force of each axle 



to the circumference ef te SHQC 

 wheel. A very common method is 

 ropes or straps passing round the c 

 cumferences of the wheel and axle, 

 act one upon the other. The action is 

 in this manner transmitted by the ten- 

 sion of the rope or strap, and rendered 

 effective by friction with the circumfe- 

 rences on which it is rolled. This 



Fig. 42. 



method of connecting wheels and axles 

 is represented in fig. 42. When the 

 wheel and the axle from which it re- 

 ceives motion are intended to revolve 

 in the same direction, the strap is 

 not crossed, but applied as between 

 the axle R and the wheel G. But when 

 the wheel is to revolve in a direction 

 contrary to the revolution of the axle, 

 the strap is crossed as between the 

 axle D and the wheel F. This latter 

 method of applying the strap has the 

 advantage of having more surface to 

 act upon, and therefore having more 

 friction. 



(65.) But by far the most usual way 

 of transmitting the action of the axles 

 to the succeeding wheels, is by means 

 of teeth or cogs raised on their surfaces. 

 When this is the case, the cogs on the 

 surface of the wheel, are generally called 

 teeth, and those on the surface of the 

 axle are called leaves ; the axle itself is 

 in this case called a pinion. 



As the leaves of the pinion succes- 

 sively pass between the teeth of the 

 wheel, they are perfectly equal and 

 similar to them. Hence the circum- 

 ferences of the wheels and pinions are 

 proportional to their respective numbers 

 of teeth and leaves ; and since the cir- 

 cumferences are as the radii, it follows, 

 that the numbers of teeth or leaves are 

 proportional to the radii. Hence, in 

 the condition of equilibrium determined 

 in (61.), we substitute the number of 

 teeth and leaves for the radii of the 

 wheels and axles. Thus, then, the con- 

 dition of equilibrium is, that the power 

 multiplied by the product of the num- 

 bers of teeth in all the wheels is equal 



