sent and

W

the centrifugal force upon the ball (equation 102), a'a' sin. cos. its moment about the point A; also 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 ¥¤2(ɑ—b) sin. §, and its moment about the point A by

W a2(ab)' sin. cos. 8. On the whole, therefore, the sum of the moments of the centrifugal forces of the rod and ball are represented by {Wa2+w(a−b)} sin. cos. 4. Now if

a2

represent the weight of each unit in the length of the rod, w = (a+b); therefore Wa+w(a - b) = Wa2 + (a2 — b2) (a—b). Let this quantity be represented by W1a',

g

..W1=W+u1

· W,1 = W + ↓u ( 1 − 2) (a−b) .

....

(353);

then will "W,a'sin. 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 Bill. A + wz(a−b) sin. e, or by Wa+u(aa—b3)} sin. ; let this quantity be represented by W,a sin. 0,

.:. W‚=W+bμa( 1 − 2 ) . . . . (354).

Also the moment of Q about A=Q. AHI-Q sin. (+). 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,

a'

-W1a' sin. cos. =Qb sin. (8 +a ̧)+W,a sin. 4.

Now P is the resultant of the pressures Q acting i.. the directions of the rods PD and PE, and inclined to one another at the angle 24,; therefore (equation 13),

P=2Q cos., ;

:: Q sin. (4+4,) = &P

sin. (+,)

=P{sin. +cos. A tan. }.

cos.

But since the sides b and c of the triangle APD are opposite to the angles 4, and 6, therefore

sin., b

=- ; therefore

sin.

.. Q sin. (6 +6,) = +P { sin. 6 + sin. è cos. è (1–2 sin.)}.

Substituting this value in the preceding equation, dividing by sin. 4, and writing (1-cos. 4) for sin. 4, we obtain

which equation, of four dimensions in terms of cos. e, 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 . If the distance AP, of this position of the collar from A, be represented by h, we have h=b cos. + c cos. 4,

367

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 7N the number of revolutions made in the same time by the spindle of the governor, then will 27 N 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

Eliminating cos. 4 between these equations, and solving in

respect to h,

Let P (1+) and P (1-1) represent the values of P corresponding to the two states bordering upon motion (Art. 140) and let N (1+) and N (1-1) 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 W2, 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. Neglect ing as 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.

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 an obstacle which it is at any instant in the act of passing over,

KA

P the corresponding traction, W the weight of the wheel and of the load which it supports.

Now the surface of the box of the wheel being in the state bordering upon motion on the surface of the axle, the direction of the resistance of the one upon the other is inclined at the limiting angle of resistance, to a radius of the axle at their point of contact (Art. 141.). This resistance has, moreover, its direction through the point of contact O of the tire of the wheel with the obstacle on which it is in the act of turning. If, therefore, OR be drawn intersecting the circumference of the axis in a point c, such that the angle CeR may equal the limiting angle of resistance o, then will its direction be that of the resistance of the obstacle upon the wheel.

and

Draw the vertical GH representing the weight W, through H draw HK parallel to OR, then will this line represent (to the same scale) the resistance R, and GK the traction P (Art. 14.);

Let R radius of wheel, p=radius of axle, ACO=7, ACW

inclination of the road to the horizon,

direction of the traction to the road.

Now W80=WCO+

inclination of

presented by a, then W80=1+7+a, and

Also PLW=++; PLW-W8O= +ɩ+8; therefore PLW —W8O=¦—(n+a—ê) ;

2

2