Now, the terms of equation (180), represented in the above equations by A and B, are all of one dimension in the exceedingly small quantities D, E, sin. . If, therefore, the values of p, and p, given by these equations be substituted in the

2nT

sin. RLde (equation 193), then all the terms

0

of that expression which involve the quantities A and B will be at least of two dimensions in D, E, sin. o, and may be neglected. Neglecting, therefore, the values of A and B in equations (202, 203), we obtain

P1+p1=W+P2+2w, and p,-p,-W+P,-2μα,0;

:. af (p1+p,) d0=a, {W+P2+2w0} 2n« =

() { (2n-a,) P,

+(2nxa,) (W+2w) } = () {S,P,+S,(W+2w)}

a, ƒ (p.—P.)(a cos. 0 — ß sin. 0)d0 = a, f {W+P,−2μa,0}

2nπ

(a cos. 0—ß sin. 0)d0=a,(W+P,) / (a cos. 6¬ß sin. 0)No0—

:. a. ['p.—P.)(« cos. 0—ß sin. 0)d0=ßa,(W+P,)—2ßμa,

U

(2n≈a,)=Ba,2+ßa,(W—2μS,); observing that P,:

S2

a

U

=

U.

SRLd0=a (“;) {U,+S,(W +210)} —ßa, ; —ßa,(W—2μ4S,);

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

C,

C ̧ = (1+A) W+2 Aw+2B and C1 = ~(W+21) () +2μa,

cos. Od0*0 cos. + sin. 6. Now, substituting 2nñ for 6,

=-0 cos. 0 +

0

these integrals become respectively 0 and 2nπ.

Church's Diff. and Int. Cal. Art. 140.

which is the MODULUS of the machine, all the various ele ments, 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 are 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, 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, 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 RB6 equal to the limiting angle of resistance .

Now T,, T,, and R are pressures in equilibrium; there

fore (Art. 14.)

T, sin. T,BR

Tsin. T,BR

but T,BR=ABb-RBb=(-AaB)—RBb

T,BR=CB6+RBb=2(≈—BbC)+RBb)=

T

ΔΑ

ΔΗ

+ &

)}

Cos.

;

2

T1—T,
Ꭲ,

or dividing numerator and denominator of the fraction in the

Suppose now the angles Aab, BC, &c., each of which equals 4, 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 a infinitely small, our supposition will coincide with that case. Now, on the supposition that is exceed

ΔΗ

ingly small, tan.. tan. is exceedingly small, and may be neglected as compared with unity; it may therefore be neglected in the denominator of the above fraction.

over a being exceedingly small, tan.

ΔΗ

ΔΗ

= 2

More

= tan. ❤ . 48*; .. T,=T, (1+ tan. ❤ . ▲8).

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 e, it follows (Euclid, i. 32.) that

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

* If we consider the tension T as a function of 8, of which any consecutive values are represented by T, and T2, and their difference or the increment of

1 AT T ΔΟ tan., and integrating between the limits O

= -tan. ; therefore,

and 8, observing that at the latter limit T=P2, and that at the former it equals