Substituting this value, and also that of U, (equation 204) in equation (193), and assuming

C1=(1+A)W+2Aw+2B and C,= ~(W+2w)

a

β

which is the MODULUS of the machine, all the various elements, whence a sacrifice of power may arise in the working of it, being taken into account.

THE FRICTION OF CORDS.

186. Let the polygonal line ABC... YZ, of an infinite number of sides, be taken to represent the curved portion of a cord embracing any arc of a cylindrical surface (whether circular or not), in a plane perpendicular to the axis of the cylinder; also let Aa, Bb, Cc, &c. be normals or perpendiculars to the curve, inclined to one another at equal angles, each represented by A. Imagine the surface of the cylinder to be removed between each two of the points A, B, &c. in succession, so that the cord may be supported by a small portion only of the surface remaining at each of those points, whilst in the intermediate space it assumes the direction of a straight line joining them, and does not touch the surface of the cylinder. Let P, P1 represent the tension upon the cord before it has passed over the point A; T, the tension upon it after it has passed over that point, or before it passes over the point B; T, the tension upon it after it has passed over the point B, or before it passes over C; T, that after it has passed over C; and let P,

2

represent the tension upon the cord after it has passed over the nth or last point Z.

Now, any point B of the cord is held at rest by the tensions T, and T, upon it at that point, in the directions BC and BA, and by the resistance R of the surface of the cylinder there; and, if we conceive the cord to be there in the state bordering upon motion, then (Art. 138.) the direction of this resistance R is inclined to the perpendicular bB to the surface of the cylinder at an angle RBb equal to the limiting angle of resistance .

Now T1, T2, and R are pressures in equilibrium; therefore (Art. 14.)

=

but T,BR=ABb=RBb-(π-AaB)—RBb)=

π ΔΙ

+ $

COS.

COS.

(+

2 sin. sin.
2

ᎪᎾ

2

COS

COS.

(++) (+9

cos. cos. - sin.

or dividing numerator and denominator of the fraction in the

Suppose now the angles AaB, BbC, &c. each of which equals A, to be exceedingly small, and therefore the points A, B, C, &c. to be exceedingly near to one another, and

exceedingly numerous. By this supposition we shall manifestly approach exceedingly near to the actual case of an infinite number of such points and a continuous surface; and if we suppose ▲ infinitely small, our supposition will coincide with that case. Now, on the supposition that A is exceed

ingly small, tan.

ΔΙ

2

tan. is exceedingly small, and may

be neglected as compared with unity; it may therefore be neglected in the denominator of the above fraction. Moreover,

A being exceedingly small, tan.

Δθ Αθ

=

2 2

T—T2=tan. © . A6•; ... T, =T,(1+tan. 4. A§).

Now the number of the points A, B, C, &c. being represented by n, and the whole angle AdZ between the extreme normals at A and Z by 0, it follows (Euclid, i. 32.) that 0=n. A◊ ; Ө therefore A= =- ;

Similarly,

n

* If we consider the tension T as a function of 9, of which any consecutive values are represented by T, and T2, and their difference or the

increment of T by AT, then T

T

= tan. p. 40; therefore

1 AT Τ ́ ΔΗ

=

- tan. , and integrating

between the limits 0 and 0, observing that at the latter limit T=P2,

and that at the former it equals P1, we have log. ()

P

=0 tan. &;

therefore P1 =Petan.

Multiplying these equations together, and striking out factors common to both sides of their product, we have

Now this relation of P, and P, obtains however small ▲ be taken, or however great n be taken. Let n be taken infinitely great, so that the points A, B, C, &c. may be infinitely numerous and infinitely near to each other. The supposed case thus passes into the actual case of a continuous surface, 123

the fractions

n n n

&c. vanish, and the above equation be

But the quantity within the brackets is the well known expansion (by the exponential theorem) of the function tan.,

... P1=P, tan...... (205).

Since the length of cord S1, which passes over the point A, is the same with that S, which passes over the point Z, it follows that the modulus (Art. 152.) of such a cylindrical surface considered as a machine, and supposed to be fixed and to have a rope pulled and made to slip over it, is

It is remarkable that these expressions are wholly independent of the form and dimensions of the surface sustaining the tension of the rope, and that they depend exclusively

upon the inclination or AdZ of the normals to the points A and Z, where the cord leaves the surface, and upon the co-efficient of friction (tan. ), of the material of which the rope is composed and the material of which the surface is composed. It matters not, for instance, so far as the friction of the rope or band is concerned, whether it passes over a large pulley or drum, or a small one, provided the angle subtended by the arc which it embraces is the same, and the materials of the pulley and rope the same.

In the case in which a cord is made to pass m times round such a surface, 6=2mm;

And this is true whatever be the form of the surface, so that the pressure necessary to cause a cord to slip when wound completely round such a cylindrical surface a given number of times is the same (and is always represented by this quantity), whatever may be the form or dimension of the surface, provided that its material be the same. It matters not whether

it be square, or circular, or elliptical.

187. If P',P", P,"", &c. represent the pressures which must be applied to one extremity of a rope to cause it to slip when wound once, twice, three times, &c. round any such surface, the same tension P, being in each case supposed to be applied to the other extremity of it, we have

P1'= P ̧2 tan., P," Ps4rtan., P,"

=

=

Per tan. e, &c. &c.

=

So that the pressures P, P.", P", &c. are in a geometrical progression, whose common ratio is 2 tan., which ratio is always greater than unity. Thus it appears by the experiments of M. Morin (p. 153.), that the co-efficient of friction between hempen rope and oak free from unguent is 33, when the rope is wetted. In this case tan. p=33 and 27 tan. ₫ 2 x 3.14159 x 33=2·07345. The common ratio of the progression is therefore in this case 2.07345, or it is the number

=