MECHANICS. 



to the weight multiplied by the product 

 of the number of leaves in all the pinions. 

 Or, if N, N', N" be the numbers of 

 teeth in the several wheels, and n, n', n" 

 the numbers of leaves in all the pinions, 

 the condition of equilibrium is 



P x N N' N" = W x n n' n". 

 A system of tooth and pinion-work 

 is represented mjig. 43. In this case the 

 Fig. 43. 



wi 



power acts upon the first wheel by a 

 rope ; but in submitting it to the above 

 condition of equilibrium, it is only ne- 

 cessary to calculate how many teeth the 

 circumference would contain, and use 

 that number in the condition. 



(66.) It will be easy to show/that 

 complex wheel-work obeys the law of 

 virtual velocities (5) ; for since the 

 teeth are equal, the circumference of 

 each wheel moves with the same velo- 

 city as that of the circumference of the 

 pinion by which it is driven, which is 

 equally evident if they be connected by 

 straps or work by friction. Now, since 

 each wheel revolves in the same time with 

 its axle, the velocities of their circum- 

 ferences are as their circumferences, or, 

 what is the same, as their radii or num- 

 ber of teeth. Hence, the velocity of the 

 power, or the velocity of the circumfer- 

 ence of the first wheel, is to that of the 

 first axle as their radii. But the velo- 

 city of the circumference of the first axle 

 is equal to the velocity of the circum- 

 ference of the second wheel, which is to 

 that of the second axle as their radii ; 

 and by continuing this reasoning, we 

 shall find that the velocity of the power 

 is to that of the weight, as the product 

 of the radii of the wheels to the pro- 

 duct of the radii of all the axles ; and, 

 therefore, that the power multiplied by 

 the velocity of the power is equal to the 

 weight multiplied by the velocity of the 

 weight. 



This will be better understood by an 

 example. Let the number of teeth in 

 the first wheel be 100, and the leaves in 

 the first pinion 9 ; and let the teeth in 



the second and third wheels l:e 120 and 

 130, and the leaves in the respective 

 pinions be 7 and 1 1 . The velocity of the 

 circumference of the first wheel being 

 expressed by T ^, that of the circum- 

 ference of the second wheel will be T |^. 

 This velocity is to that of the circum- 

 ference of the second pinion or third 

 wheel, as 120 is to 7 ; and therefore the 

 velocity of the circumference of the 



. Again, this 



third wheel is 



velocity is to that of the circumference 

 of the last axle as 130 to 11. This 



velocity is therefore 9 x 7 X U 

 J 100 x 120 x 13' 



which verifies what we have just ad- 

 vanced. 



(G7.) In the construction of wheel- 

 work considerable attention ought to be 

 paid to their shape, as much of the 

 efficiency and permanency of the work 

 depends on this. Suppose that the 

 teeth were found as in/g-. 44. The tooth 



Fi%. 44. 



a b in driving a' b' would be moved 

 round the centre C, in a direction per- 

 pendicular to Cab, and would there- 

 fore press on the tooth a' b' obliquely to 

 the radius C' a' b' ; whereas, to produce 

 the best effect, the pressure should be 

 directed perpendicularly to that radius. 

 Besides this, the whole pressure of the 

 wheel is thrown upon one tooth, by which 

 the chances of fracture are much in- 

 creased, and the wear materially aug- 

 mented. Another defect which appears 

 manifest is, that during the motion the 

 direction of the pressure of a b on a' b' is 

 constantly changing while the teeth are 

 in contact ; and since the leverage by 

 which the wheel C' is turned by ab is 

 therefore variable, it is turned with an 

 unequable force. In the motion, the 

 corner of the tooth a b scrapes or rubs 

 the surface of the tooth a' b' -, and the 

 machine suffers a jolt when the tooth 

 a b , finally slipping from the tooth a' b', 

 falls into the angle formed at the point 

 where the tooth a' b' springs from the 

 circumference of the wheel. 



