Of THE VELOCITY AXD FRrCTIOX OF WHEEL \rORlt. 



J.» 



I — — ; — 1 ■=o>— r; (x— a).- 



3XXT-{-2aaxi 



0, 



x'-i^a'x — (*• — a).{3x::z-ia')zzo, *''4-a'a- — 3x' — 2a"i-l-3a.r 

 + 2a'=0, •2T' + (a''- — 3a).x— 20% x=:v'ia' + -i\{a' — 3a)') 



— J(a^ — 3a)=..(v'(aa+l0a+Q) — a+s). Mence, ifar: 



4 



I, yr: ., and when a is diminished without limit, 



r— -o ; when it is increased without limit, j-::r:2a ; for in 



2 



this case ^(aa+ioa+v) approaches infinitely near to 

 a+S- This proposition has not always been sufficiently 

 distinguished from the preceding one. 



Scholium. If the force accumulated during the opera- 

 tion of the machine, as that of a stream ol water collected 

 continually in a reservoir, there would be no limit to the 

 »dv»ntage of a slow motion. 



363. Theorem. If a weight be drawn 

 along a horizontal surface by a given force, 

 with a resistance in the direction of the sur- 

 face which is always a certain portion of the 

 pressure, the force will act with the greatest 

 advantage when the tangent of its inclina- 

 tion is to the radius as the resistance to the 

 pressure. 



Let AB represent the force, 

 and let BC be to CD as the 

 pressure to the friction, then 

 K. CD AD will represent the sura of 



the horizontal forces, AC being the efficient portion of the 

 forc» AB, and CD the diminution of the friction. But the 

 »ngle D is given, since the proportion of BC to CD is given, 

 and BCD is a right angle ; and AB being given, AD will 

 vary as the sine of the angle ABD, which is greatest when 

 ABD is a right angle; and ACB is then similar to BCD; 

 but BC is the tangent of the angle BAC, AC being the ra- 

 dius. The angle BAC is also the same at which the weight 

 would begin to slide along the given surface if it were in- 

 clined to the horizon (soo). 



SECTION Xlll. OF THE VELOCITY AND 

 FRICTION OF WHEEL WORK. 



364. Theorem. The angular motion 

 of two wheels may be made uniform at the 

 eame time by means of a right line sliding 



on an epicycloida! surface, or by two surfaces 

 which are involutes of circles, acting on each 

 otlier. 



Let A and B be the centres of 

 the wheels, and CD a portion of 

 an epicycloid, described by the 

 point D of the circle BDE, equal 

 in diameter to the radius of the 

 wheel B, in rolling on the wheel 

 A : then if the tooth of the wheel 

 B be terminated by the right line 

 BD, and touch CD in D, the line DE perpendicular to BD, 

 will pass through the point of contact of the circles, E (206) ; 

 and the force will be communicated in the direction DE, 

 so that the angular motion of each wheel will be the same, 

 as if it acted immediately at the end of the perpendicular 

 AF, and the angular motion of A will be to that of B, in 

 the constant ratio of BD to AF, or BE to AE. It is obvi- 

 ous, that BD cannot act in the same manner on CD be- 

 yond the line BA, unless its extremity be made epicycloidal, 

 and tl'.e corresponding part of the tooth of A a' right line. 

 Let each tooth now terminate in the curve described by the 

 evolution of a thread from its res- 

 pective circle : then the curve will 

 be always perpendicular to the 

 thread (l93), which is the tangent 

 of the circle, and the force will al- 

 ways act in the direction of the 

 circumference of the circles at E 

 and G, and the motion will be 

 uniform as before. 



365. Theorem, The relative velocity of 

 the teeth of two wheels, or the velocity with 

 which the surfaces slide on each other, varies 

 ultimately as the sum of the angular distances 

 of the point of contact from the line joining 

 the centres. 



Let A and B be the two centres, C 

 the point of contact, and CD the com- 

 mon ungent there ; and suppose the 

 teeth to move to the positions E, F, 

 and G to be the new point of contact ; 

 and let BD and BH be perpendicular 

 to CD and to AC produced ; then CE 

 and CF, the elements of the paths 

 of the points which were at C, will ba 

 perpendicular to AC and BC ; and the 

 difference of EG and FG, which re- 

 presejiu the friction, ultimately equa) 



