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The velocity of the stream estimated in the direction EP

= v.cos APE, .. R = 1 pv2. (cos APE)2 K.

COR. 1. The resolved part of the impelling force estimated in the direction of the stream

= R.cos APE = 1 pv2. (cos APE)3 K.

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COR. 2. The resolved part of the impelling force estimated in a direction perpendicular to the stream, and in the plane APE, = R. sin APE = pv2. (cos APE). sin APE. K.

70. A cylinder having the curve BPC (fig. 34.) for its base, is immersed in a stream flowing in the direction AE; to find the force with which the stream impels the cylinder in the directions AE and MA.

Draw AN perpendicular to AE; PM, QN parallel to AE; PE a normal to BP at P. Let a be the altitude of the cylinder, MP = x, AM=y, MN = dy, R the impelling force, or resistance, on that part of the cylinder which stands on BP, estimated in the direction AE, therefore ultimately d, R.dy resistance on that part of the cylinder which stands on PQ

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= 1⁄2 pv2 (cos AEP)3a. PQ = 1⁄2 pv2 (cos AEP)2ady,


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and tan AEP = − d„î, ··dyR= 1⁄2pv2 1+(d ̧x)2 '

1 and R = 1⁄21⁄2 pv2 a fy

1 + (dy x) * *


So also, if S be the resistance on the part BP of the cylinder, estimated in the direction MA,

dyS. dy = 1 pv2 (cos AEP)2 sin AEP.a.PQ

= pv2. cos AEP. sin AEP.a.dy;

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71. A solid is generated by the revolution of the curve BPC (fig. 34.) round AE; to find the force with which it is impelled by a stream moving in the direction AE.

Let R be the resistance on that part of the solid which is generated by the revolution of BP round AE; then, retaining the construction and notation of the preceding Article, we have ultimately,

d, R. dy = 1 v2. (cos PEA)3.2π. AM. PQ = p22π

убу 1 + (d,x)2

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72. To find the resistance on a sphere.

Let the centre of the sphere be the origin of the co-ordinates, a the radius of the, sphere, and, therefore, x2 + y2 = a2 the equation to its generating circle. Then, y + xdyx = 0,

a2 = x2 + y2 = x2 {1 + (d,x)2} = (a2 − y3) {1 + (d,x)2},

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COR. 1. The resistance on a circular plate, the radius of which is a, = p2πa2, therefore the resistance on a sphere is half the resistance on a circular plate of the same radius as the sphere.

= σ

COR. 2. If the density of the sphere = σ, its mass

Ta3, and the retarding force arising from the resistance


of the fluid (resistance) ÷ (mass of the sphere)


3 P v2


16 σα




ART. 73. ALMOST all bodies expand by heat, and contract by cold. This property furnishes the only known mode of comparing and recording the temperatures to which any body is exposed. The expansions of mercury, or air, combined with that of the glass vessel in which they are contained, are usually employed for this purpose.

74. The common mercurial thermometer is a glass tube of uniform bore, having a bulb at one end, which, with part of the tube, is filled with mercury; the other end is usually sealed, the space between it and the mercury being a vacuum.

To fill the thermometer with mercury, the air must be partly expelled from the bulb by holding it over the flame of a lamp, and then, the other end, which is open, immersed in mercury. As the bulb cools, the mercury will be forced into it by the pressure of the atmosphere. If a paper funnel be now tied round the open end, and filled with mercury; and the mercury in the bulb be heated till it boils, the remainder of the air will be driven out, and its place supplied by mercurial vapour: this condenses on cooling, and the mercury will descend from the funnel and fill the instrument completely. When it has cooled down nearly to the highest temperature intended to be measured by it, the open end must be sealed; and as it continues to cool, the mercury will descend leaving a vacuum in the upper part of the tube.

75. To graduate a thermometer.

Let the bulb, and that part of the tube which is occupied by the mercury, be immersed in melting snow, and make a mark on the tube, opposite to the extremity of the column of mercury, when it is stationary. This is the freezing point. Next let the thermometer be surrounded by the vapour of boiling water, and make a mark on the tube at the place where the extremity of the column of mercury rests, when it is stationary. This is the boiling point. The space between the freezing and boiling points is, in the centigrade thermometer, divided into 100 equal parts, called degrees; the freezing point being called 0o, and the boiling point 100o.

In Fahrenheit's thermometer the space is divided into 180 parts. The freezing point is marked 32o; and the boiling point 212o.

In Reaumur's thermometer the freezing point is marked 0o, and the boiling point 80°.

76. The temperature of melting snow is found to be the same under all circumstances. The temperature of steam, however, varies with the atmospheric pressure. 100° of the centigrade thermometer denotes the temperature of steam, when the pressure of the atmosphere is equal to that of a column of mercury at 0o, 0,76 metres, or 29,9218 inches high. A variation of 1,045 inches in the height of the mercury in the barometer, occasions a change of 1o in the temperature of steam.

77. To compare the scales of two differently graduated thermometers.

Let of the thermometer (4), and yo of the thermometer (B), denote the same temperature; and let y = mx +n: then, if a, b; a', b' be known corresponding values of x and y,

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Let Co of the centigrade thermometer, F of Fahrenheit's, Roof Reaumur's, denote the same temperature; then if we suppose the temperature indicated by the boiling point to be the same in all three,

|| C = }} (F - 32) = | R.

78. The apparent expansion of mercury in glass, between


0° and 100o, is of its volume at 0°, the difference between

this and

10 555'


the actual expansion of mercury between the same temperatures, arises from the expansion of the glass.

79. When a solid expands by heat, its cubic expansion is three times its linear expansion.

Let the cubic and linear expansions of the solid for one degree of heat, be E, e respectively; and let V be the volume of the solid, a the distance between two given points in it, at 0°; then, at the temperature To, the volume of the solid will be V (1+ET), and the distance between the two given points a (1+eT). But V (1+ET): V= {a(1+eT)}3 : a3, .'. 1+ ET=1+3eT nearly, since et is very small, .'. E = 3e,

BRAMAH'S PRESS. (Fig. 35).

80. AB, CDE are two strong hollow cylinders communicating with each other by means of a pipe BD; M, Q two solid cylinders working in water-tight collars at A and C. The cylinder M, the diameter of which is much larger than that of Q, supports a platform F, on which the substance to be pressed is placed. Q is capable of being moved up and down by means of a lever HL having its fulcrum at H. D is a valve opening upwards; B a valve opening into the cylinder AB; E a cistern filled with water; I a cross beam, the ends of which are fastened to the upright posts G, H.

Suppose to be in its lowest position, and the space between the solid and hollow cylinders to be filled with water;

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