than the 1th part, may cause the valve to move. Neglect

n

ingas small when compared with n.

n=2m (1+ W2,a);

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,

K

R

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 e, such that the angle CCR may equal the limiting angle of resistance o, then will its direction be that of the resistance of the obstacle upon the wheel.

W

F

Draw the vertical GH representing the weight W, and 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.);

P GK sin. GHK

sin. GHK

W ̄GH ̄sin. GKH ̄sin. (PGH—GHK

sin. WSO

sin. (PLW-WSO)

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

inclination of the road to the horizon,

direction of the traction to the road.

COs, but WCO=1+n, and

Now WSO=WCO+

inclination of

presented by a, then Ws0=1+n+a, and

Also PLW=++; therefore PLW-WsO=2(n+a−0);

α

when the direction of traction is parallel to the road, 0=0,

..P=W{sin. + cos. i tan. (n+α)}

a

. . . (362).

If the road and the direction of traction be both horizontal ==0, and

P=W tan. (n+a). .. (363).

In all cases of traction with wheels of the common dimensions upon ordinary roads, ACO or is an exceedingly small angle; a is also, in all cases, an exceedingly small angle (equation 360); therefore tan. (n+a)=n+a very nearly. Now if A be taken to represent the arc AO, whose length is determined by the height of the obstacle and the radius of the wheel, then

Substituting the value of a from equation (360),

276. It remains to determine the value of the arc A intercepted between the lowest point to which the wheel sinks in the road, and the summit Ô of the obstacle, which it is at every instant surmounting. Now, the experiments of Coulomb, and the more recent experiments of M. Morin,* appear to have fully established the fact, that, on horizontal roads of uniform quality and material, the traction P, when its direction is horizontal, varies directly as the load W, and inversely as the radius R of the wheel; whence it follows (equation 365), that the arc A is constant, or that it is the same for the same quality of road, whatever may be the weight of the load, or the dimensions of the wheel.† The

* Expériences sur le Tirage des Voitures, faites en 1837 et 1838. (See APPENDIX.)

In explanation of this fact let it be observed, that although the wheel sinks deeper beneath the surface of the load as the material is softer, yet the obstacle yields, for the same reason, more under the pressure of the wheel, the arc A being by the one cause increased, and by the other diminished. Also, that although by increasing the diameter of the wheel the arc A would be rendered greater if the wheel sank to the same depth as before, yet that it does not sink to the same depth by reason of the corresponding increase of the surface which sustains the pressure.

constant A may therefore be taken as a measure of the resisting quality of the road, and may be called the modulus of its resistance.

η

The mean value of this modulus being determined in respect to a road, whose surface is of any given quality, the value of ʼn will be known from equation (364), and the relation between the traction and the load upon that road, under all circumstances; it being observed, that, since the arc A is the same on a horizontal road, whatever be the load, if the traction be parallel, it is also the same under the same circumstances upon a sloping road; the effect of the slope being equivalent to a variation of the load. The same substitution may therefore be made for tan. (n+a) in equation (362), as was made in equation (363),

277. The best direction of traction in the two-wheeled

carriage.

This best direction of traction is evidently that which gives to the denominator of equation (361) its greatest value; it is therefore determined by the equation

2

278. The four-wheeled carriage.

2

Let W1, W, represent the loads borne by the fore and hind wheels, together with their own weights, R1, R, their radii, P., P, the radii of their axles, and o,, 9, the limiting angles of resistance. Suppose the direction of the traction P parallel to the road, then, since this traction equals the sums of the tractions upon the fore and hind wheels respectively, we have by equation (366)

279. The work accumulated in the carriage-wheel.*

Let I represent the moment of inertia of the wheel about its axis and M its volume; then will MR2+I represent its moment of inertia (Art. 79.) about the point in its circumferences about which it is, at every instant of its motion, in the act of turning. If, therefore, a represent its angular velocity about this point at any instant, U the work at that instant accumulated in it, and μ the weight of each cubical unit of its mass, then (Art. 75.), U=4a2 (MR2 + 1) =

M

(aR)2+1a2-I. Now if V represent the velocity of the axis g of the wheel, &R=V;

..U=1 MV2+++ α

whence it follows, that the whole work accumulated in the rolling wheel is equal to the sum obtained by adding the work which would have been accumulated in it if it had moved with its motion of translation only, to that which would have been accumulated in it if it had moved with its motion of rotation only. If we represent the radius of gyration (Art. 80.) by K, I=MK'; whence substituting and reducing,

U= £M (1+1) V'... . (369).

The accumulated work is therefore the same as though the wheel had moved with a motion of translation only, but with

K

a greater velocity, represented by the expression (1+) V.

Ꭱ

* For a further discussion of the conditions of the rolling of a wheel, see a paper in the Appendix on the Rolling Motion of a Cylinder.

+ The angular velocity of the wheel would evidently be a, if its centre were fixed, and its circumference made to revolve with the same velocity as now.