WHEEL. 



ber of teeth in the axis F D, is to the number of teeth in 

 the circumference of the wheel M, as the circumference of 

 that to the circumference of this ; the revolutions of the 

 flower wheel M, are to the revolutions of the fwifter N, as 

 the number of teeth in the axis to the number of teeth in 

 the wheel M, which it catches. 



5. If the facftum of the radii of the wheels G E, D C, 

 be multiplied into the number of revolutions of the flowed 

 wheel, M, and the produtt be divided by the faftum of the 

 radii of the axes which catch into them, G H, D E, &c. 

 the quotient will be the number of revolutions of the fwifteft 

 wheel O. .E. ^r. If G E = 8, D C = 12, G H = 4, 

 D E = 3, and the revolution of the wheel M be i ; the 

 number of revolutions of the wheel O will be 8. 



6. If a power move a weight by means of divers 

 wheels, the fpace pafled over by the weight, is to the fpace 

 of the power, as the power to the weight. Hence, the 

 greater the power, the quicker is the weight moved ; and 

 vice ver/a. 



7. The fpaces pafled over by the weight and the power, 

 are in a ratio compounded of the revolutions of the floweft 

 wheel, to the revolution of the fwifteft ; and of the peri- 

 phery of the axis of that, to the periphery of this. Hence, 

 fince the fpaces of the weight and the power are recipro- 

 cally as the fuftaining power is to the weight ; the power 

 that fuftains a weight will be to the weight, in a ratio com- 

 pounded of the revolutions of the floweft wheel, to thofe of 

 the fwifteft, and of the periphery of the axis of that, to 

 the periphery of this. 



8. The periphery of the axis of the floweft wheel, with 

 the periphery of the fwifteft wheel, being given ; as alfo the 

 ratio of the revolutions of the one, to thofe of the other ; 

 to find the fpace which the power is to pafs over, while the 

 weight goes any given length. 



Multiply the periphery of the axis of the floweft wheel 

 into the antecedent term of the ratio, and the periphery of 

 the fwifteft wheel into the confequent term ; and to thefe 

 two produfts, and the given fpace of the weight, find a 

 fourth proportional : this will be the fpace of the power. 

 Suppofe, e. gr. the ratio of the revolutions of the floweft 

 wheel, to thofe of the fwifteft, to be as 2 to 7, and the 

 fpace of the weight 30 feet ; and let the periphery of the 

 axis of the floweft wheel be to that of the fwifteft, as 3 to 

 8 : the fpace of the power will be found 280. For 2x3: 

 7 X 8 :: 30 : 280. 



9. The ratio of the peripheries of the fwifteft wheel, and 

 of the axis of the floweft ; together with the ratio of their 

 revolutions, and the weight, being given : to find the power 

 able to fuftain it. 



Multiply both the antecedents and the confequents, of 

 the given ratios into each other, and to the produft of the 

 antecedents, the produft of the confequents, and the given 

 weight, find a fourth proportional : that will be the power 

 required. Suppofe, e. gr. the ratio of the peripheries 8:3; 

 that of the revolutions 7 : 2, and the weight 20CO ; the 

 power wiU be found 214^. For 7 x 8:2 x 3 :: 2000 

 : 214I. After the fame manner may the weight be found ; 

 the power, and the ratio of the peripheries, &c. being 

 given. 



10. The revolutions the fwifteft wheel is to perform while 

 the floweft makes one revolution, being given ; together 

 with the fpace the weight is to be raifed, and the periphery 

 of the floweft wheel ; to find the time that will be fpent in 

 raifing it. 



Say, As the periphery of the axis of the floweft wheel 

 is to the given fpace of the weight ; fo is the given number 

 of revolutions of the fwifteft wheel to a fourth proportional : 



Vol. XXXVIII. 



which will be the number of revolutions performed while 

 the weight reaches the given height. Then, by experiment, 

 determine the number of revolutions the fwifteft wheel per- 

 forms in an hour ; and, by this, divide the fourth propor- 

 tional found before. The quotient will be the time fpent 

 in raifing the weight. Wolf. Elem. Math. tom. ii. p. 214, 

 &c. 



Wheels of a Clock, &c. are the crown wheel, contratc 

 wheel, great wheel, fecond wheel, third wheel, ftriking 

 wheel, detent wheel, &c. See Clock and Watch. 



Wheels of Coaches, Waggons, &c. With refpeft to 

 thefe, the following particulars are collefted from the ex- 

 periments and reafonings of Camus, Defaguliers, Beighton, 

 Fergufon, Brewfter, &c. 



I. The ufe of wheels, in the draught of carriages, is 

 two-fold ; 1)12. that of diminifliing, or of more eafily over- 

 coming the refiftance arifing from the friftion of the car- 

 riage, and that of more readily furmounting obftacles, which 

 form angular prominences on the plane over which they are 

 drawn, and which muft be either deprefled by the weight of 

 the carriage, or render it iieceffary for the carriage, with its 

 load, to be lifted over them. They ferve in their firft ufe 

 to transfer the friftion from the under furface of the car- 

 riage, and the plane fupporting it, to the furfaces of the 

 axle and nave of the wheel. The common method of ac- 

 counting for this advantage is by faying, that the refiftance, 

 arifing from friftion in planes of equal afperity, increafes 

 with the velocity of the motion ; fo that this velocity muft 

 be compared with that of the power neceifary to move the 

 machine, and overcome the friftion ; and it is obvious, at 

 the fame time, that the velocity of a circular motion dimi- 

 niflies gradually from the circumference to the centre. See 

 Friction. 



But to this pofition it has been objefted, that the illuf- 

 tration is not applicable to the cafe : for, granting that, in 

 the friftion of fledges or flat furfaces, the refiftance increafes 

 in proportion to the velocity of their motion, this is not a 

 parallel cafe with that of a circular furface rolling over a 

 flat plane. On the contrary, the velocity of motion, in the 

 outer furface of a wheel, is greater than that of its nave, 

 moving under the axle ; while at fuch outer furface there is 

 little or no friftion at all ; whereas at the nave, moving 

 much flower, there is much more. Indeed, the friftion, 

 which the wheel would have againft its fupporting plane, if 

 it did not turn round its axis, is by its turning round trans- 

 ferred almoft wholly to the axis and nave ; whofe circular 

 motion is notwithftanding fo much flower. It is, indeed, 

 notorious, that the great friftion of the wheels of carriages 

 lies between the axle and nave ; and how then can it be pro- 

 perly aflerted, that fuch friftion is diminiflied at the axle, as 

 the velocity of the circular motion is there diminifhed ? Ac- 

 cordingly it has been alleged by a late writer, it&X friftion 

 is not diminiflied by the ufe of wheels, but merely trans- 

 ferred from the outer furface of the wheel to its nave and 

 axle ; and that in the cafe of a wheel rolHng along the 

 ground, the fpokes aft only as fingle levers, to overcome 

 the fridlion of the periphery againft the plane of its fupport, 

 the prominences, conftituting the roughnefs of the plane 

 over which it moves, being the fulcra upon which they 

 turn, and not the common centre of thefe fpokes, as others 

 have maintained, who fay that the wheel afts, in overcoming 

 friftion, as an axis in peritrochio. However, in obviating 

 the friftion of the wheels in loaded carriages, their fpokes 

 aft as double levers, refting on a fulcrum at each end. See 

 the author's method of illuftrating and evincing thefe prin- 

 ciples, in Jacob's Obf. on Wheel-Carriages, p. 23, &c. 



If carriages were to move along fmooth horizontal planes, 

 Z z wheels 



