o o > UC-NRLF $B 531 TIM OS In ^&Tyl^ APPLICATIONS OF ALGEBRA DEALING WITH B AUTOMOBILES FOR USE IN CONNECTION WITH THE FIRST YEAR's WORK IN ALGEBRA BY THIRMUTfflS BROOKIVIAN AND OTHERS MEMBERS OF THE TEACHERS CLASS IN APPLIED MATHEMATICS UMVEBSITY PF CALIFORNIA, Sl'MMEK SE3SIOX, 1913 REVISED 19U. 1915 =1 CHARLES SCRIBNER'S SONS NEW YORK CHICAGO BOSTON APPLICATIONS OF ALGEBRA DEALING WITH AUTOMOBILES FOR USE IN CONNECTION WITH THE FIRST YEAr's WORK IN ALGEBRA BY THIRMUTfflS\BR0OKMAN AND OTHERS MEMBERS OF THE TEACHERS CLASS IN' APPLIED MATHEMATICS UNIVERSITY OF CALIFORNIA, SUMMER SESSION, 1913 REVISED 1914, 1915 i CHARLES SCRIBNER'S SONS y 11 NEW YORK CHICAGO BOSTON CAJORI Copyright, 1916, bt CHARLES SCRIBNER'S SONS CONTRIBUTORS Thirmuthis Brookman O. W. Baird .... Geo. T. Brooks . . . Lilly E. Burkhardt . Alex. R. Craven . . Flora E. Crowley . . Laura Gilbert . . . Catherine Lamberson F. J. Lawrence . . . B. a. Lindsay . . . Kate Mitchell Meek George E. Mercer Emily G. Palmer . . Georgia M. Simon . . Charles E. Taylor . Anna G. Wright . . San Francisco, Calif. High School, Nome, Alaska. High School, Hutchinson, Kansas. San Francisco, Calif. Lowell High School, San Francisco, Calif. High School, Williams, Calif. High School, Corona, Calif. Washington High School, Portland, Ore. High School, Inglewood, Calif. High School, Sparks, Nev. High School, South Pasadena, Calif. High School, Palo Alto, Calif. High School, Salem, Ore. High School, Tuolumne, Calif. High School, Berkeley, Calif. Mayfield, Calif. Digitized by the Internet Arciiive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/applicationsofalOObroorich PREFACE The purpose of this booklet is to vitalize first-year al- gebra by applying it to objects with which the pupils may readily become famihar in daily life. Among the many machines whose operations can be fairly comprehended by those whose knowledge is limited to elementary algebra, none seem of more far-reaching interest and importance than the automobile. These pages, therefore, develop the simpler algebraic formulas used in the operation of automobile engines, in the transmission of speed, and in problems dealing with automobiles on the road. The algebra used in the solution of these appHcations is the linear equation, direct proportion, and a Umited knowledge of pure quadratics. The applications of elementary algebraic principles herein are similar to those developed throughout the first year's work in algebra in Brookman's Practical Algebra for Beginners. It is expected that such problems of real life, which are of a nature to enlist the ready interest of many pupils, will replace the difficult manipulations of abstract symbols included in several of the current texts, and will also give more real significance to the algebraic equa- tion. It is hoped that this booklet will encourage the beginner to enter into the study of algebra with alertness and keen interest, because he reahzes that it gives him mastery over the practical formulas needed in actual experience. The writers wish to acknowledge their indebtedness to all who have given them suggestions and practical information, and especially to Mr. Alden McElrath of the Oakland head- quarters of the Cadillac Motor Company. San Franciscx), June, 1916. CONTENTS CHAPTER PAGE I. AUTOMOBILE ENGINES 1 1. General Description 1 2. The Gasolene-Engine 4 3. Problems on a One-Cylinder Engine . . 6 4. Horse-Power of Gas-Engines 7 5. Problems on Piston Pressure ..... 8 6. Formula for Energy per Minute .... 10 7. Problems on Revolutions of the Fly- Wheel . 11 8. Problems on Horse-Power 12 n. TRANSMISSION OF SPEED 14 9. Gears in Mesh 14 10. Problems on Gears 15 11. High-Speed Transmission 16 12. Problems on High-Speed Transmission . . 18 13. Gear-Box and Gears in Neutral .... 19 14. Intermediate Speed 22 15. Problems on Intermediate Speed .... 26 16. Problems on Low Speed 29 17. Reverse Speed 31 m. AUTOMOBILES IN MOTION 34 18. Road Problems 34 19. Formulas Concerning Automobiles ... 39 APPLICATIONS OF ALGEBRA DEALING WITH AUTOMOBILES FOR USE IN CONNECTION WITH THE FIRST YEAR S WORK IN ALGEBRA CHAPTER I AUTOMOBILE ENGINES 1. General Description. The relation of the different parts of an automobile may be seen in the following pic- ture of one of the latest types of machines. This descrip- tion, with slight variations, applies to all popular makes of cars. The parts italicized in the description should be lo- cated in the following diagram. The power which runs the car is generally obtained from gasolene or electricity. This book will consider only those machines which are run by gasolene. The motive power of such machines is generated in cylinders located in front of the car. The movement of the piston in each cylinder helps to turn the crank-shaft, which is beneath the cylinders. (The crank-shaft does not appear in the drawing.) The crank-shaft and the fly-wheel (usually behind the engine) are securely connected and so revolve together. Into the hollowed portion of the fly-wheel is fitted a cone clutch, by means of which the motion of the fly-wheel may be imparted to the machine. When the cone clutch is 1 Cylinders Flywheel Cone Clutch Main Tran6^ mission Shaft Gear Box — Rear Wheel ^Rear Axle Bevel Gear DIAGRAM SHOWING OPERATING MECHANISM Cylinder)! \ of gas- I engine Fly-wheel •e Cfme diUch Main transmiasion- shajl CADILLAC OPERATING MECHANISM 3 4 APPLICATIONS OF ALGEBRA disconnected or the hand-lever is in neutral position the en- gine may run while the automobile is standing still. This is called "idling." From the cone clutch extends the main transmission-shaft in two parts through the gear-box to the bevel-gears, which turn the rear axle of the machine. Some machines have the multiple disk or some other type of clutch in place of the cone. (If possible pupils should see the oper- ating mechanism of several different automobiles.) 2. The Gasolene-Engine. The engine is the most vital and also the most intricate part of the automobile, and an understanding of its construction and workings is of the utmost importance. Most motor-cars are equipped with en- gines of four, six, or eight cylinders, while the simplest form of engine has but one. IV E.V. LOADING COMPRESSING EXPLODING EXHAUSTING CYLINDER SHOWING FOUR POSITIONS OF PISTON The cylinder is closed at the top and fitted with a piston (P) connected with a rod {R) to the crank-shaft {C.S.). The accompanying picture of one type, the four-cycle gas- engine, shows four positions of the piston. Above the piston is the combustion chamber (CC), where there are AUTOMOBILE ENGINES 5 two valves, an inlet valve (/.F.) and an exhaust valve {E.V.). Find these parts in the accompanying illustration. /, II, III and IV show four different positions of the piston in a single cylinder. In /, if the machine is in motion, the piston is approaching the bottom of the cylinder. As the piston moves down- ward the inlet valve (I.V.) opens to admit a mixture of gasolene- vapor and air, the explosion of which is to furnish the motive power. This valve closes automatically when the piston is near the bottom of the stroke. In II the piston is approaching the top of the cylinder and compressing the gas in the combustion chamber to a frac- tion of its former volume. In III the dotted line shows the highest position which the piston reaches. Near this point an electric spark from the spark-plug (S.P.) explodes the compressed gas in the cylinder and the piston is immediately forced downward by the expansion of the gas. In IV the piston has again reached its lowest position, and as it moves upward the exhaust valve {E.V.) opens and the piston forces out the spent gas. As the piston reaches the top the exhaust valve closes, the inlet valve opens, the piston returns again to the position shown in /, and the cycle is completed. In order to obtain increased power for running the ma- chine, the one-cylinder engines have been replaced by those containing more cylinders. As the number of cylinders increases, the number of explosions in the engine to every revolution of the fly-wheel also increases and produces a more continuous power and a more smoothly running engine. However, the ratio between the number of explosions in one cylinder and the revolutions of the crank-shaft remains the 6 APPLICATIONS OF ALGEBRA same. In the four-cylinder car four explosions, one in each cylinder, cause two revolutions of the fly-wheel; likewise, in the six-cylinder car, six explosions, one in each cylinder, cause only two revolutions of the fly-wheel. Questions 1. As the piston passes from one extreme position to another show the number of degrees through which the point A on the crank-shaft passes. 2. As the piston completes the four phases of one cycle, show that point A on the crank-shaft passes through 720°. 3. During the four phases of one cycle how many revo- lutions does the fly-wheel make.^^ How many explosions does the one-cylinder engine make? 4. How many revolutions would the fly-wheel make for 10 explosions in a one-cylinder engine ? How many for 120 explosions ? 5. When the fly-wheel turns 40 times, how many explo- sions occur in the cylinder ? 3. Problems on a One-Cylinder Engine. In the follow- ing problems consider the engine as having one cylinder only. 1. (a) If a gas-engine makes 420 explosions per minute, how many revolutions will the fly-wheel make in the same time? (b) How many revolutions will the fly-wheel make in 5 minutes ? 2. If the fly-wheel of a gas-engine makes 1200 revolu- tions per minute, how many explosions will there be? 3. (a) If a gas-engine of a motor-cycle makes 484 ex- plosions per minute, and the fly-wheel turns 11 times while AUTOMOBILE ENGINES 7 the rear wheel turns 3 times, how often will the rear wheel turn in a minute? (b) If the circumference of the rear wheel of a motor- cycle is 8 ft., how far does it travel in a minute? (c) How long will it take to travel half a mile ? 4. (a) The circumference of the rear wheel of a motor- cycle is 7^ ft. If the rear wheel makes 3 revolutions to 11 of the fly-wheel, how many revolutions will the fly-wheel make in travelling one-half mile ? (b) How many explosions will the engine make in trav- elling 3 miles? 5. (a) The diameter of the rear wheel of a motor-cycle is 2.5 ft. The fly-wheel turns 55 times while the rear wheel is turning 16 times. How many feet will the motor-cycle have moved when the engine has made 2000 explosions ? (b) If the fly-wheel makes 1500 revolutions per minute, what will be the speed in miles per hour ? 4. Horse-Power of Gas-Engines. The explosion in the cylinder produces a mean effective pressure of from 40 to 70 pounds per square inch in all directions. To obtain the number of foot-pounds of energy developed by a single explosion in a cylinder, three things must be considered : P, the average pressure of the gas in pounds per square inch. a, the area of the piston in square inches. I, the length of the stroke in inches. If d is the diameter of the piston or cylinder, the area of the piston is obtained by the formula Tvd?- 3.1416(^2 ^„r,,, a = — r- or a = -, = .7854^^ 4 4 Since the area of a circle equals the product of t and the square of the radius, explain how this formula is obtained. 8 APPLICATIONS OF ALGEBRA 5. Problems on Piston Pressure. 1. If the diameter of a piston is 4 in., what is the area? 2. Find the area of a piston whose diameter is 5 in. 3. Find the diameter of a piston whose area is 19.635 sq. in. 4. What is the diameter of a piston if its area is 15.9 sq. in. ? 5. The total force F pressing against the end of a piston equals the product of the area in square inches by the pres- sure per square inch. This may be expressed by the for- mula F = Pa, in which F is the total number of pounds; also by the formula „ P 3.1416 ^2 p7r^2 F = -. or F = —r—' 4 4 Explain how this formula is obtained. 6. If the pressure in a cylinder is 50 lbs. per sq. in. and the area of the piston is 16.4 sq. in., what is the total force (F) pressing against the piston? 7. If the pressure (P) is 55 lbs. per sq. in. and the diam- eter of the piston 5 in., what is the total force (F) pressing against the piston? 8. What is the total force against a piston during the power or explosion stroke of an engine, if the diameter of the cylinder is 4.5 in. and the average pressure is 65 lbs. per sq. in. ? 9. (a) Find the area of the piston if a pressure of 60 lbs. per sq. in. produces a total pressure of 756 lbs. (b) Find the diameter of the piston. 10. A pressure of 80 lbs. per sq. in. on a piston produces a total force of 570r8 lbs. What is the diameter of the piston ? AUTOMOBILE ENGINES 9 The total amount of energy generated during one explosion stroke may be expressed in inch-pounds; 820 in.-lbs. will lift 820 lbs. through 1 in. or 410 lbs. through 2 in., etc. Describe other illustrations of work done by inch-pounds. The total amount of energy {E) is the product of the total force (F) pressing against the piston and the length (Z) through which the piston acts. If, during one explosion stroke, a total average force of 2500 lbs. is exerted through a distance of 6 in., the total amount of energy generated is 15,000 in.-lbs., since 2500 X 6 = 15,000. 11. (a) How many inch-pounds are developed if F = 1900 and / = 8 in. ? (b) Explain the formula E = Fl. (c) Since F = Po, explain how the formula E = Fl becomes E = Pla. (d) 900 in.-lbs. lifts 900 lbs. through 1 In., or through j\ of a foot; hence 900 in.-lbs. = 75 ft.-lbs. Why.? (e) Explam the formula E = — = jy^ = -^g-, m which E denotes the number of foot-pounds of energy de- veloped during one explosion stroke. 12. How many foot-pounds of energy are generated by one explosion in an engine if the diameter of the cylinder is 4 in., the length of the stroke (Z) is 5 in., and the average pressure (P) is 62.5 lbs. per sq. in. ? 13. If d is 5.5 in., I is 4.6 in., P is 65 lbs. per sq. in., how many foot-pounds of energy are generated during each ex- plosion ? 14. If the area (a) of the piston of an engine is 18 sq. in., the length of the stroke 5 in., and the energy developed at each explosion 435 ft.-lbs., what is the mean average pres- sure (P) ? 10 APPLICATIONS OF ALGEBRA Find the value of the unknown in each of the following problems : DIAM. ENERGY PRESSURE LENGTH OF PISTON (during one explosion) Foot-Pounds 15 50 lbs. per sq. in. 4 in. 4.5 in. X 16 X " " " " 5 " 4 " 251.328 17 60 " " " " X " 4 " 345.576 18 50 " " " " 5 " 4.8 " X 19 48 " " « « 4.8 " X " 376.992 6. Formula for Energy per Minute. If n = the number of revolutions of the crank-shaft per minute, then n = the number of revolutions of the fly- wheel per minute. Why? And ^ = the number of ex- plosions in a single cylinder per minute. Why ? Since E = the number of foot-pounds of energy developed during one explosion stroke and E = -j^y then the number of foot-pounds of energy developed in a single cylinder in one minute. Why ? 12 '^ 2 Fn^^.nr. Ene^'gy Pe^ minute, -p _ Pla n _ Plan Formula. ^^^^ ^^^^^^^ ^ ' 1^ ^ 2 ' -2^' Illustrative Problem A one-cylinder gas-engine makes 420 explosions per min- ute, the inside diameter of the cyhnder is 4 in., the length of the stroke is 5 in. Find the pressure per square inch of the piston if the engine develops 109,956 ft. -lbs. of energy per minute. AUTOMOBILE ENGINES 11 E = 109,956 ft.-lbs. per minute, I = 5 in. 4 n = 2 X 420 = 840. Plan a = 47r = 12.5664 sq. in. Why? Then the formula E = becomes 109,956 = 24 PX 5X12.5664X840 24 or 2199. 12P = 109,956 and P = 50 lbs. per sq. in. 7. Problems on Revolutions of the Fly- Wheel. In the following problems n denotes the number of revolutions per minute of the fly-wheel, and E the number of foot-pounds developed in one cylinder per minute. 1. If the fly-wheel of a one-cylinder engine makes 400 revolutions per minute, the pressure (P) is 64 lbs. per sq. in., the length (l) is 4 in., and the area (a) is 16 sq. in., how many foot-pounds of energy are developed per minute ? 2. The fly-wheel of a gas-engine makes 960 revolutions per minute, P is 70 lbs. per sq. in., and I is 4.5 in. What is the area of the piston if 158,400 ft.-lbs. of energy are de- veloped per minute in a single cylinder ? Find the value of the unknown in each of the following problems, if E denotes the energy developed per minute in a single cylinder. P I a n E 3 66 5.6 14 1000 ' X 4 X 5.5 15 800 176,000 5 80 X 13.5 900 182,250 6 75 5 X 1200 243,750 7 70 5.2 18 X 300,300 8 72 4.4 20.5 1000 X 9 X 4.8 19.2 1500 345,600 12 APPLICATIONS OF ALGEBRA If c equals the number of cylinders in a gas-engine the number of foot-pounds of energy {E) developed per minute • • • Plo/ftc m the gas-engine is expressed by the formula E =■ . ATX. Explain. 10. A gas-engine has 4 cylinders, each having a diameter of 4 in. If the length of the stroke is 5 in., the average pressure on the piston is 60 lbs. per sq. in., and the number of revolutions of the fly-wheel per minute is 600, how many foot-pounds of energy are developed per minute ? 11. A two-cylinder gas-engine develops 360,000 ft. -lbs. of energy per minute, A is 16 sq. in., I is 4.5 in.^ and n is 800 revolutions per minute. What is the average pressure (P) per square inch against each piston ? If E denotes the energy developed in the gas-engine, find X in the following problems: P / a n c E 12 65 4.2 18 1000 4 X 13 70 X 15.5 1400 4 1,063,300 14 72 4.8 X 850 4 628,320 15 69 5 16 1300 X 1,196,000 16 74 5.3 17.5 900 8 X 8. Problems on Horse-Power. Let HP equal the horse- power of a gas-engine. Since 33,000 ft.-lbs. per minute is 1 horse-power, the horse-power of the engine is the number of foot-pounds developed per minute divided by 33,000. If P = pressure in pounds per square inch, I — length of stroke in inches, a = area of piston in square inches, c = number cylinders in engine, E = energy during one explosion stroke in foot-pounds. AUTOMOBILE ENGINES 13 n = number of revolutions of the fly-wheel per minute, __ E _ Plane _ PlT(Pnc then HF - ^^-q^ - 24 X 33,000 ~ 24 X 4 X 33,000 1. How many horse-power are developed by a four- cylinder engine if P is 60 lbs. per sq. in., Z is 5 in., a is 16.5 sq. in., and n is 1000 revolutions per minute? 2. A gas-engine has four cylinders. What is the aver- age pressure (P) per square inch if I is 4.8 in., a is 17.6 sq. in., n is 1350 revolutions per minute, and the horse-power is 48 ? 3. How many revolutions per minute will the fly-wheel of a four-cylinder gas-engine make while developing 40.5 horse-power, if P is 66 lbs., I is 4.5 in., and a is 20 sq. in. ? Find X in the following problems: P / d n c HP 4 72 5.5 4 1250 4 X 5 80 5.5 5 X 6 65.45 6 X 5 4 1400 4 31 7 77 X 4.8 1360 4 50.26 8 71 5.5 X 1000 4 40.3 9 68.5 4.8 5 1450 4 X 10 72 5 4 1500 X 68.5 The California State Automobile Registration Board dur- ing 1915 used the formula HP = .224((f -f- l)dc. By this formula find the horse-power in each of the problems 4, 5, 6, and 9 above. The formula used by the State Board in 1916 is HP = ^ d?c 5 By this formula find the horse-power in problems 4, 5, 6, and 9. CHAPTER II DRIVING GEAR TRANSMISSION OF SPEED 9. Gears in Mesh. The accompanying figure shows two gears in mesh, the smaller gear, or pinion (A), turns the driven gear (B). It is not difficult to see that the driving- gear rotates in one direction while the driven gear rotates in the opposite direction. (a) Count the number of teeth in each gear. (b) When the driving- gear (A) turns once, with how many teeth of the driven gear {B) has it been in mesh ? (c) How many times must the point c on the driving- gear return to its position in order to put each tooth of B in mesh once, or to cause one complete revolution oi B? Let t represent the number of teeth in A, t' represent the number of teeth in B, r represent the number of revolutions in Ay r' represent the number of revolutions in B, Show by counting that the following numerical ratios are true: DRIVEN GEAR 1. 7/ = ^ = o* '^^^^ equation is read: 'The number of teeth in A is to the number of teeth in 5 cw 16 is to 32, or as 1 to 2." 14 TRANSMISSION OF SPEED 15 2. Show by counting the revolutions that r 2 , r' 1 -,= jand-=-. t 1 r' 1 t r' . 3. Since r, = ?: and - = i^, then -, = — , since the value r 2 r 2 t' r t r' . 1 of each ratio, 7? and — , is ». This equation may be ex- Z T JL pressed by saying that the number of teeth in two gears in mesh is inversely proportional to the number of revolu- tions they make in a given time. t r' By clearing of fractions the equation p =^ — becomes tr = tV. In the accompanying figure < = 16, r = 2, /' = 32, r' = 1; the equation tr = t'r' becomes 16 X 2 = 32 X 1. This equation may be expressed by saying that the number of revolutions of the driving-gear multiplied by the num- ber of its teeth equals the number of revolutions of the driven gear multiplied by the number of its teeth. 10. Problems on Gears. 1. A driving-gear with 12 teeth makes 200 revolutions per minute. How many revolutions per minute will be made by the driven gear which has 24 teeth ? 2. A driving-gear with 20 teeth revolves 180 times per minute. How many revolutions per minute will the driven gear make if the number of its teeth is 30? How many if the number of its teeth is 24 ? 3. A gear having 16 teeth is making 240 revolutions per minute. How many teeth has a second gear in mesh with the first if it is making 120 revolutions per minute? How many teeth would the second gear have if it were making 160 revolutions per minute? 16 APPLICATIONS OF ALGEBRA Complete the following table by filling in the blank spaces. DRIVING-GEAR DRIVEN GEAR NO. OF NO. OF NO. OF NO. OF REV. TEETH REV. TEETH 4 180 16 20 5 360 12 540 6 750 , . 600 25 7 , , . 22 462 28 8 384 27 32 9 155 85 31 10 620 13 260 11 . . . 24 288 31 12 363 15 . . . 55 13 275 14 78 •• 11. High-Speed Transmission. The motion from the engine which turns the crank-shaft iC.S.) and fly-wheel (F. W.) reaches the rear axle by means of the main trans- mission-shaft, T. The front part of the main transmission-shaft is connected with the engine crank-shaft by means of the cone clutch (C.C.), the two conical surfaces of which are held firmly to- gether by springs, except when forced apart by the driver pressing the clutch pedal, as in the figure showing low speed (page 30) . With this exception the front part of the transmis- sion-shaft (T) revolves whenever the engine runs. The rear part (S) communicates its motion through the bevel pinion and gear (Z and J) to the rear axle and the rear wheels. The accompanying figure shows the high-speed transmis- sion which gives the greatest possible car speed for a given speed of the engine. This manner of connection is also called the direct drive (D.D.), One part of the clutch DD is fixed to the gear G. When this is brought into its extreme forward position, as shown in the figure, the clutch DD engages and T and S rotate as 17 18 APPLICATIONS OF ALGEBRA one shaft. This manner of connection is called the direct drive. It is also called high-speed transmission, since it gives the greatest possible car speed for a given speed of the engine. The direct-drive transmission does not use the counter-shaft R, but because gear F is in constant mesh with gear A the counter-shaft turns ready for use when intermediate, low, or reverse speeds are desired. Without changing the position of the gears the speed of the car may be regulated to some extent by increasing or decreasing the amount of gasolene used in the engine, or by advancing or retarding the spark. The following questions refer to the direct-drive position of the gears. 1. In this position which gears and shafts are rotating as the engine runs? Which are idling {i. e., turning without transmitting power) ? 2. What is the effect of disengaging the cone clutch ? 3. What is the effect of stopping the explosions in the engine ? What parts will continue to turn ? 4. What is the effect of disengaging D.D. when the engine is running and C.C. is engaged.'* 5. With both clutches engaged, one turn of the crank- shaft makes how many turns of the pinion / ? 12. Problems on High-Speed Transmission. In each of the following problems which concern four-cylinder engines the two clutches are connected to form the direct drive. 1. If a gas-engine makes 920 explosions per minute, how many times will the fly-wheel revolve per minute ? 2. If the clutch is thrown in, how many times will the small pinion-gear at the rear of the main driving-shaft re- volve per minute? TRANSMISSION OF SPEED 19 3. If the driving-pinion contains 13 teeth and is in mesh with a driven bevel-gear which contains 52 teeth, how many times will the driven gear turn when the driving-pinion turns once ? when the driving-pinion turns 460 times ? 4. If the fly-wheel makes 920 revolutions per minute, how many times will the rear axle turn if the bevel-gear ratio is 1 to 4 ? How far will the car travel in a minute if the circumference of the rear wheel is 9 ft. ? Complete the following table in problems 5 to 15 by filling in the blanks. Consider the bevel-pinion as having 13 teeth, the bevel-gear as having 52 teeth, and the circumference of the rear wheel as 9 ft. NO. REV. NO. REV. NO. REV. NO. REV. NO. FEET OF FLY- OF DRIV- OF BEVEL- OF REAR TRAVELLED WHEEL ING-PINION GEAR AXLE PER MIN. 5 900 .... 6 1000 . . . 7 1200 , . , 8 1280 . . . 9 840 10 325 11 . . . 275 12 310 13 290 14 . . . . . . 2385 15 2655 16. If the speed limit in the country is 25 miles per hour, which of the preceding problems have rates which exceed the speed limit ? 13. Gear-Box and Gears in Neutral. The purpose of the gear-box mechanism (G.B.) is to enable the two parts T and S of the transmission-shaft to be connected in either of four ways or to be entirely disconnected, the selection being made by the driver placing the hand-lever in one of 20 APPLICATIONS OF ALGEBRA five positions. When disconnected the lever and gear mech- anism are said to be neutral, as in the figure. Show why shaft S does not rotate when T rotates. (Study D.D.) The four methods of connection are called first or low speed, second or intermediate speed, third or high speed, and reverse speed. The words low and high do not refer to the actual running-speed of the car, but to the ratio of the speed of the car to that of the engine, which is determined by the speed ratio of the two parts T and *S of the transmission-shaft. The front part T of the transmission-shaft carries the gear F and one part of the clutch D.D. The other part of this clutch is fixed to the gear G. This clutch D.D. is en- gaged only in the direct-drive or high-speed transmission when T and S rotate as one shaft. Gears G and H are car- ried upon the square part of shaft S so as to rotate with it, but may slide along it. It is by sliding G and H into dif- ferent positions that the manner of connection of