MECHANICS. 



for attaining the same end is represent- 

 ed. A B is a wheel turned by a winch 

 H or otherwise ; a b is a beam, which 

 moves through the guides m n. At- 

 tached to this beam is a cross-piece 

 I) E, having a groove p q cut in it. 

 This groove receives a pin c, which 

 projects from the wheel A B. As the 

 wheel revolves, the pin c moves in the 

 groove from p to q and from q to p al- 

 ternately, and evidently raises and de- 

 presses the beam a b through the guides 

 m n. The vertical action of the pin c 

 on the sides of the groove a b is variable, 

 and gives the beam a b an unequable 

 motion. This defect may be removed 

 by forming the sides of the groove into 

 proper curves, so that the action of the 

 pin may be rendered perfectly equable. 



Another very ingenious contrivance is 

 represented in/#. 105. A B is a double 

 Fig. 105. 



rack, with circular ends fixed to a beam, 

 capable of moving in the direction of 

 its length. This rack is driven by a 

 pinion P, which moves in a groove m n 

 cut in the cross-piece. When the 

 pinion has moved the rack and beam 

 until it comes to the end B, the project- 

 ing piece a meets the spring s, and the 

 rack is pressed against the pinion. The 

 pinion, then working in the circular end 

 of the rack, will be forced down the 

 groove m n until it works in the lower 

 side of the rack, and moves the beam 

 back in the opposite direction ; and in 

 this way the motion is continued. 



Another elegant contrivance is repre- 

 sented in/o-. 106, where A B is a wheel 

 having teeth in the inner part of its rim. 

 This wheel is fixed so as not to revolve ; 

 C is a wheel of half the diameter of the 

 former, and having teeth on its outer 

 edge, which work in the teeth of the 

 fixed wheel. As the wheel C is turned 

 on its axis, it traverses the inner 

 circumference of the wheel A B, the 

 centre C describing a circle round the 

 centre of the fixed wheel. Now it may 

 be proved geometrically, .that any point 

 on the circumference of the wheel C 

 will move along a diameter of the wheel 



A 



A B in one revolution of the wheel C, 

 and will return along the same diameter 

 the next revolution : this may be proved 

 as follows: 

 Let C,/#. 107, be the centre of the 



Fig. 107. 



fixed wheel, and let A C be the initial 

 position of the revolving one. Let the 

 revolving wheel roll over the arc A B, 

 so as to assume the position C B. It is 

 evident, then, that an arc of the lesser 

 circle, which is equal to A B, has rolled 

 over A B. But by the established 

 principles of geometry, an arc of the 

 lesser circle, equal to the arc A B, sub- 

 tends at its centre an angle which bears 

 to the angle A C B the same proportion 

 as the diameter of the greater circle 

 bears to that of the lesser, that is, two 

 to one. Hence, the arc of the lesser 

 circle, which has rolled over A B, sub- 

 tends at the centre of the lesser circle 

 an angle equal to twice the angle ACB. 

 To the point where the lesser circle in 

 the position A B intersects C A, draw 

 the line B E, and from the centre of the 

 lesser circle draw D E. The angle 

 B D E is equal to twice the angle B C 

 A*, and therefore the arc B E of the 

 lesser circle is equal to the arc B A of 



; 

 * DAKLE y's Popular Geometry, Art. 71. 



