would 'receive an equal steadiness of motion from a fly wheel of one eighth the weight; the mean radii of the wheels being the same. If, in equation (342), we substitute for I its value 2-beR', or 2-KR (representing by K the section be of the rim), and if we suppose the wheel to be formed of cast iron of mean quality, the weight of each cubic foot of which may be assumed to be 450 lb., we shall obtain by reduction by which equation is determined the mean radius R of a flywheel of cast iron of a given section K, which being applied to an engine of given horse power H, making a given number of revolutions N per minute, shall cause its angular 1 n velocity not to deviate more than th from the mean; or conversely, the mean radius being given, the section K may be determined according to these conditions. 269. In the above equations, m is taken to represent the number of effective strokes made by the piston of the engine whilst the fly-wheel makes one revolution, and ʼn to represent Let now the axis of the fly-wheel be supposed to be a continuation of the axis of the crank, so that both turn with the same angular velocity, as is usually the case; and let its application to the single-acting engine, the double-acting engine, and to the double crank engine, be considered separately. 1. In the single-acting engine, but one effective stroke of the piston is made whilst the fly-wheel makes each revolution. In this case, therefore, m=1, and ein. "===0·3183098= 1 η sin. 18° 33'; therefore, cos. ·9480460, also -= ช 18° 33' 180° Substituting in equations (345) and (346), by which equations are determined, according to the proposed conditions, the weight W in tons of a fly-wheel for a single-acting engine, its mean radius in feet R being given, and its material being any whatever; and also its mean radius R in feet, its section (in square feet) K being given, and its material being cast iron of mean quality; and lastly, the section K of its rim in square feet, its mean radius R being given, and its material being, as before, cast iron. 2. In the double-acting engine, two effective strokes are made by the piston, whilst the fly-wheel makes one 2 revolution. In this cases therefore, m = 2 and sin. ŋ=-= T 0-636619 sin. 39° 32'; therefore, cos. =7712549 = Substituting in equations (345) and (346), by which equations the weight of the fly-wheel in tons, the mean radius in feet, and the section of the rim in square feet are determined for the double-acting engine. 3. In the engine working with two cylinders and a double crank, it has been shown (Art. 263.) that the conditions of the working of the two arms of a double crank are precisely the same as though the aggregate pressure 2P, upon their extremities, were applied to the axis of the crank by the intervention of a single arm and a single connecting rod; a the length of this arm being represented by instead of a 12 and its direction equally dividing the inclination of the arms of the double crank to one another. Now, equations (345) and (346) show the proper dimen sions of the fly-wheel to be wholly independent of the length of the crank arm; whence it follows that the dimen sion of a fly-wheel applicable to the double as well as a single crank, are determined by those equations as applied to the case of a double-acting engine, the pressure upon whose piston rod is represented by 2P,. But in assuming Nm. 2P,a=33000H, we have assumed the pressure upon the piston rod to be represented by P,; to correct this error, and to adapt equations (345) and (346) to the case of a double crank engine, we must therefore substitute H for II in those equations. We shall thus obtain by which equations, the dimensions of a fly-wheel necessary to give the required steadiness of motion to a double crank engine are determined under the proposed conditions. THE FRICTION OF THE FLY-WHEEL. the 270. W representing the weight of the wheel and limiting angle of resistance between the surface of its axis and that of its bearings, sin. will represent its coefficient of friction (Art. 138.), and W sin., the resistance opposed to its revolution by friction at the surface of its axis. Now, whilst the wheel makes one revolution, this resistance is overcome through a space equal to the circumference of the axis, and represented by 2p, if p be taken to represent the radius of the axis. The work expended upon the friction of the axis, during each complete revolution of the wheel, is therefore represented by 2p W sin. ; and if N represent the number of strokes made by the engine per minute, and N therefore the number of revolutions made by the fly-wheel 2 per minute, then will the number of units of work expended per minute, upon the friction of the axis be represented by Νπρ W sin. ; and the number of horses' power, or the fractional part of a horse's power thus expended, by If in this equation we substitute for W the weight in lbs. of the fly-wheel necessary to establish a given degree of stea cliness in the engine, as determined by equations (347), (348), and (349), we shall obtain for the horse power lost by friction of the fly-wheel, in the single-acting engine, the double-acting engine, and the double crank engine, respectively, the formulæ THE MODULUS OF THE CRANK AND FLY-WHEEL. 271. If S, represent the space traversed by the piston of the engine in any given time, and a the radius of the crank, W the weight of the fly-wheel in lbs., and p the radius of its axis, then will 2a represent the length of each stroke, S1 the 2a number of strokes made in that time, and 2pW sin. . S, Φ 2a' or W S, sin. the work expended upon the friction of the fly-wheel during that time, which expression being added to the equation (329) representing the work necessary to cause the crank to yield a given amount of work U, to the machine driven by it (independently of the work expended on the friction of the fly-wheel), will give the whole amount of work which must be done upon the combination of the crank and fly-wheel, to cause this given amount of work to be yielded by it, on the machine which the crank drives. Let this amount of work be represented by U,, then in the case in which the directions of the driving pressure and the reBistance upon the crank are parallel (equation (329), and the friction of the crane guide is neglected, we obtain for the modulus of the crank and fly-wheel in the double-acting engine Р U,= { 1 + "(); sin. 9, + 2 sin. 9.) U, +WS, sin. (352). THE GOVERNOR. 272. This instrument is represented in the figure, under that form in which it is most commonly applied to the steam در E F B H A engine. BD and CE are rods jointed at A upon the vertical spindle AF, and at D and E upon the rods DP and EP, which last are again jointed at their extremities to a collar fitted accurately to the surface of the spindle and moveable upon it. At their extremities B and C, the rods DB and EC carry two heavy balls, and being swept round by the spindlewhich receives a rapid rotation always proportional to the speed of the machine, whose motion the governor is intended to regulate-these arms by their own centrifugal force, and that of the balls, are made to separate, and thereby to cause the collar at P to descend upon the spindle, carrying with it, by the intervention of the shoulder, the extremity of a lever, whose motion controls the access of the moving power to the driving point of the machine, closing the throttle valve and shutting off the steam from the steam engine, or closing the sluice and thus diminishing the supply of water to the water-wheel. Let P be taken to represent the pressure of the extremity of the lever upon the collar, Q the strain thereby produced upon each of the rods DP and EP in the direction of its length, W the weight of each of the balls, w the weight of each of the rods BD and CE, AB=a. AD=b, DP=c, FAB=8, APD=4,. Now upon either of these rods as BD, the following pressures are applied: the weight of the ball and the weight of the rod acting vertically, the centrifugal force of the ball and the centrifugal force of the rod acting horizontally, the strain Q of the rod DP, and the resistance of the axis A. If a represent the angular |