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 resistance 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

U1={

P2

1+ (sin. 4, +2 sin. 42 :)} U2+WS, -tan. 4... (352).

a

2

THE GOVERnor.

π

272. This instrument is represented in the figure, under that form in which it is most commonly applied to the steam

E

F

Q

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 spindle-which 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=4, APD=0,. 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

sent the centrifugal force upon the ball (equation 102),

W

and a2a2 sin. cos. its moment about the point A; also 9

the centrifugal force of the rod BD produces the same effect as though its weight were collected in its centre of gravity (Art. 124.), whose distance from A is represented by (a−b), so that the centrifugal force of the rod is represented by

w

g

-a2 (a—b) sin. 0, and its moment about the point A by

w a2 (a—b)2 sin. cos. 9. On the whole, therefore, the sum of

g

the moments of the centrifugal forces of the rod and ball are

represented by {Wa2+w(a—b)2} sin. cos..

Now if μ

represent the weight of each unit in the length of the rod, w=μ(a+b); therefore Wa2+4w(a−b)2 = Wa2+}μ(a2—b2) (a−b). Let this quantity be represented by W1a2,

then will W1a2sin. cos. represent the sum of the moments of the centrifugal forces of the rod and ball about A. Moreover, the sum of the moments of the weights of the rod and ball, about the same point, is evidently represented by Wa sin.+w1⁄2(a—b) sin. 0, or by {Wa+1⁄2μ(a2 — b2)} sin. 0; let this quantity be represenred by W,a sin. 4,

Also the moment of Q about A=Q. AH=Qb sin. (4+◊ ̧). Therefore, by the principle of the equality of moments, observing that the centrifugal force of the rod and ball tend to communicate motion in an opposite direction from their weights and the pressure Q,

a2

W1a'sin. cos. =Qb sin.(0+9 ̧)+ W2a sin. §.

Now P is the resultant of the pressures Q acting in the directions of the rods PD and PE, and inclined to one another at the angle 20,; therefore (equation 13),

P=2Q cos. 1;

.. Q sin.(0+,) = 1 psin. (6+0)P(sin.+cos. 9 tan. 03.

cos.

But since the sides b and c of the triangle APD are opposite

to the angles 0, and 0, therefore sim

sin., b

=

therefore cos. 1

sin. 0

... Q sin. (0+0,) = + P { sin. 0+ sin. cos. (1-in. 20)

{

Substituting this value in the preceding equation, dividing by sin. 9, and writing (1-cos.20) for sin. 29, we obtain

which equation, of four dimensions in terms of cos. 0, being solved in respect to that variable, determines the inclination of the arms under a given angular velocity of the spindle. It is, however, more commonly the case that the inclination of the arms is given, and that the lengths of the arms, or the weights of the balls, are required to be determined, so that this inclination may, under the proposed conditions, be attained. In this case the values of W, and W, must be substituted in the above equation from equations (353) and (354), and that equation solved in respect to a or W.

The values of b and c are determined by the position on the spindle, to which it is proposed to make the collar descend, at the given inclination of the arms or value of 6. If the distance AP, of this position of the collar from A, be represented by h, we have h=b cos. + c cos. 01,

b2

.:. h=bcos. 9 + c ( 1 − 62 sin.24)'. . . . (356);

- . . .

from which equation and the preceding, the value of one of the quantities b or c may be determined, according to the proposed conditions, the value of the other being assumed to be any whatever.

If N represent the number of revolutions, or parts of a revolution, made per second by the fly-wheel, and yN the number of revolutions made in the same time by the spindle of the governor, tben will 2YN represent the space a described per second by a point, situated at distance unity from the axis of the spindle. Substituting this value for a in equation (355), and assuming b=c, we obtain

Let P (1+) and P (1-2) represent the values of P corresponding to the two states bordering upon motion (Art. 140.) and let N(1+) and N(1) be the corresponding values of N; so that the variation either way of th from the mean number N of revolutions, may be upon the point of causing the valve to move. If these values be respectively substituted for P and N in the above formula, it is evident that the corresponding values of h will be equal. Equating those values of h and reducing, we obtain

By which equation there is established that relation between the quantities W1, a, P, m which must obtain, in order that a variation of the number of revolutions, ever so little greater than the th part, may cause the valve to move. Neglectingas small when compared with n,

which expression, representing that fractional variation in the number of revolutions which is sufficient to give motion to the valve, is the true measure of the SENSIBILITY of the governor.

E

273. The joints E and D are sometimes fixed upon the arms AB and AC as in the accompanying figure, instead of upon the prolongations of those arms as in the preceding figure. All the formulæ of the last Article evidently adapt themselves to this case, if b be assumed =0 (in equations 353, 354). The centrifugal force of the rods EP and DP is neglected in this computation.

THE CARRIAGE-WHEEL.

274. Whatever be the nature of the resistance opposed to the motion of a carriage-wheel, it is evidently equivalent to that of an obstacle, real or imaginary, which the wheel may be supposed, at every instant, to be in the act of surmounting. Indeed it is certain, that, however yielding may be the material of the road, yet by reason of its compression before the wheel, such an immoveable obstacle, of exceedingly small height, is continually in the act of being presented to it.

275. The two-wheeled carriage.

Let AB represent one of the wheels of a two-wheeled carriage, EF an inclined plane, which it is in the act of ascending, O a solid elevation of the surface of the plane, or