it is found desirable to remove the twist from the belt before it enters upon the follower, then a guide-pulley, GP, must be used as shown by the second figure. When the shafts are parallel, but not in the same plane, then the power must be transmitted by aid of two guide-pulleys, as seen from an inspection of the third figure. Or, should the shafts not be parallel, but in the same plane, two guide-pulleys are necessary, as in the fourth figure. Guide-pulleys, if supported by spindles running in adjustable bearings or brackets, may be made serviceable as tighteningpulleys for the purpose of taking up the slack of the belt, and thus giving the necessary grip for transmitting more power with a steadier drive than can be obtained without them. Shape of Pulley Face.-The student will have observed that the faces or rims of the fast and loose pulleys, as well as those of the stepped cones in the previous set of figures, are slightly curved. This convex curvature, or double coning, is purposely done in order to ensure that the belt may ride easily and fairly in the centre line of the pulley face without inclining to either side. A flat band, if placed on the smaller end of a revolving straight conical pulley, will naturally tend to rise to the larger end of the cone. Consequently, if each half of the face of a pulley is coned (or, which amounts to the same thing, if the rim of the pulley be curved so as to have its largest diameter in the middle of its face), each half of the breadth of the belt will have an equal tendency towards the middle of the pulley's rim. When very fast driving and sudden severe stresses are brought to bear upon a machine, as in the case of circular saws, morticing machines, and emery-wheel grinders, it is found necessary to fit the pulleys with side flanges, in addition to curving their rims, in order to prevent the belts from sliding off the pulley's face to one side or to the other. LECTURE XI.-QUESTIONS. 1. In machinery, where one pulley drives another by means of an endless belt, there is a difference of tension in the two parts of the belt. Why is this? The pulley on an engine shaft is 5 feet in diameter, and it makes 100 revolutions per minute. The motion is transmitted from this pulley to the main shaft by a belt running on a pulley, and the difference in tension between the tight and slack sides of the belt is 115 lbs. What is the work done per minute in overcoming the resistance to motion of the main shaft? (S. and A. Exam. 1891.) Ans. 180,550 ft.-lbs. 2. Deduce from the "principle of work" a formula for the brake horse-power transmitted by a belt. The pull on the driving side of a belt is 200 lbs. and on the following side 100 lbs., whilst the belt has a velocity of 990 ft. per minute. Find the number of units of work performed in two minutes and the B.H.P. transmitted. Ans. 198,000 ft.-lbs., 3 B.H.P. 3. State and prove the rule for estimating the relative speeds of two pulleys connected by a belt. Also, the velocity ratio between the first driver and the last follower in belt gearing, where there are two or more drivers and a corresponding number of followers. [A main shaft carrying a pulley of 12 inches diameter and running at 60 revolutions per minute, communicates motion by a belt to a pulley of 12 inches diameter, fixed to a countershaft. A second pulley on the countershaft, of 8 inches dia. meter, carries on the motion to a revolving spindle which is keyed to a pulley of 4 inches diameter. Sketch the arrangement and find the number of revolutions per minute made by this last pulley. (S. and A. Exam. 1891.)] Ans. 123'53. 4. Two pulleys are connected by a driving belt, and the sum of their diameters is 30 inches; one pulley makes 2 revolutions while the other makes 3 revolutions; find their respective diameters. Ans. 18", 12". 5. An engine works normally at 106 revolutions per minute. At that speed it was found that it drove by belting a dynamo at 420 revolutions per minute, but to show off the electric lights at their normal candle power the dynamo had to be run at 460 revolutions per minute. At what speed was the engine being driven? Ans. 116 revolutions per minute. 6. A pulley of 3 feet radius rotates at 100 revolutions per minute and transmits motion to another pulley of 36 inches diameter. If there is 10 per cent. slip on the belt what will be the speed of the follower? What will be the net driving pull on the belt if 5 B.H.P. is transmitted by it? Ans. 180 revolutions per minute; 97.2 lbs. 7. Sketch an arrangement of pulleys and bands for obtaining a reversing motion from a shaft driven at a constant rate in one direction, and describe the action of the combination. 8. Sketch a combination of fast and loose pulleys as used for setting in motion, or stopping machinery. Explain the construction adopted for retaining a flat belt upon a pulley, pointing out where the fork is to be applied, and why. 9. Sketch and describe a good form of slow forward and quick return for a shaping machine. 10. Sketch and describe an arrangement for driving the table of a planing machine by means of a screw, so that the table may travel 50 per cent. faster in the return than in the forward or cutting stroke. (S. and A. Exam. 1888.) II. What is the object of using guide-pulleys in machinery? Mention instances of their use, and show how the directions of their axes are ascertained. 12. Describe, with a sketch, the mode of reversing the motion of the table in planing machine, a screw being employed to drive the table. 13. What is the object of using speed pulleys? Show their application in a foot lathe, and the manner in which the motion of the workman's foot is converted into the circular motion of the mandril. Sketch the arrangement. The diameter of the largest pulley on the crank shaft being 2 feet, and that of the smallest pulley on the mandril being 3 inches; find the number of rotations of the mandril for each complete rotation of the crank shaft when these pulleys are connected by a band. Ans. 8 revolutions. [N.B.-Refer to the figures in Lecture XVI. of the foot-driving gear for the screw-cutting lathe, before answering this question.] 14. When an ordinary train of wheels is employed for obtaining an increased speed of rotation, how are the wheels arranged? Sketch an arrangement of four pulleys with bands for driving a fan at 1500 revolutions per minute from a shaft making 100 revolutions per minute. (S. and A. Exam. 1893.) LECTURE XII CONTENTS.-Velocity Ratio of Two Friction Circular Discs-Pitch Surfaces and Pitch Circles-Pitch of Teeth in Wheel Gearing-Rack and Pinion Velocity Ration in Wheel Gearing-Example I.-Principle of Work applied to Wheel Gearing-Examples II. III.—Questions. Velocity Ratio of Two Circular Friction Discs.-If two truly centred circular discs or cylindrical rollers, having their shafts parallel to each other and free to turn in fixed bearings, be brought into firm contact; then, if one of them be driven round, and if there be no slipping, the other one will rotate in the opposite direction with the same circumferential speed or surface velocity (see the next figure). Consequently, their velocity ratio will be in the inverse ratio to their diameters. This may be proved in exactly the same way as we found the velocity ratio of two pulleys driven by a belt in Lecture XI. Then, The surface velocity of D1 = Surface velocity of F1 i.e. Or, F1 i.e. The Driver's diameter x its speed Follower's diam? × its speed. i e. = F, Diameter of Follower Speed of the Follower Diameter of Driver This velocity ratio may also be proved in the following way :Let the two circles centred at A and B represent a cross section of the two friction discs in contact at C; and let them move by rolling contact through the angles and respectively in the same time. Since the magnitude of an angle in circular measure is |