hence, the values of t at A and B being Q and R,

log Q = C - μ1, log R = C - μO2

... (3),

which expresses the relation which must subsist between Q and R under the circumstances of the problem.

hence the whole pressure along the curve is equal to

but, when 0=0,, it is clear that the pressure along the curve

is zero; hence

and therefore the whole pressure from 0, to 0, is equal to

2

In addition to this pressure along the curve there are the pressures at the extremities A and B.

Euler; Nov. Comment. Petrop. 1775, p. 316.

Poisson; Traité de Mécanique, Tom. I. ch. 3.

(4) A heavy uniform string rests on a complete cycloid, the axis of which is vertical and vertex upwards, the whole length of the string exactly coinciding with the whole arc of the cycloid to find the law of the pressure at any point of the cycloid.

The pressure at any point varies inversely as the curvature.

(5) A heavy chain ABC (fig. 83), is fixed at its highest end to the circumference of a circular section of a rough horizontal cylinder, moveable about its axis: having given the lengths of the two portions AB, BC, of the chain, to determine the moment of a force about the axis of the cylinder which shall maintain the equilibrium.

Let AB-a, BC=b, r= the radius of the cylinder, and m = the mass of a unit of the chain's length. Then the required moment is equal to

mrg (r sin+b).

(6) Two equal weights P, P', are connected by a string which passes over a rough fixed horizontal cylinder: to compare the forces required to raise P, accordingly as P is pushed up or P' pulled down.

If Ρ

be the force in the former and p' in the latter case,

(7) If a weight P attached to one end of a fine cord, which is laid over a rough horizontal cylinder, can support a weight nP attached to the other end, to determine the weight which it can support when the cord is wrapped r times round the cylinder.

The required weight is equal to

n2r+1 P.

(8) A string is wrapped round a smooth elliptic cylinder in a plane perpendicular to its axis and is acted on by two forces

which tend from the foci, vary inversely as the square of the distance, and are equal at equal distances: to compare the tensions of the string at the ends of the major and minor axes.

The tension at an end of the major is to that at an end of the minor axis in the ratio of 1+e to 1, where e is the eccentricity of a transverse section of the cylinder.

(9) The extremities of a light thread, the length of which is 7a, are fastened to those of a uniform heavy rod, the length of which is 5a; and, when the thread is passed over a thin round peg, it is found that the rod will hang at rest provided that the point of support be anywhere within a space a in the middle of the thread to determine the coefficient of friction between the thread and the peg; and, when the rod hangs in a position bordering upon motion, to find its inclination to the horizon and the tensions of the two parts of the string.

If μ represent the coefficient of friction, W the weight of the rod, S, T, the tensions of the longer and shorter parts of the string respectively, and the inclination of the rod to the horizon;

2

μ=- log, SW, T=W, cos 0 = 3.

π

(10) A thin inextensible cord, in which the density of the material increases in geometric as the distance from one extremity increases in arithmetic progression, is laid directly across a rough horizontal cylinder, the circumference of a vertical section of which is equal to twice the length of the cord: to determine the coefficient of friction, supposing the cord to be only just supported when its two extremities are both in the horizontal plane through the axis of the cylinder.

Taking, in accordance with the hypothesis, ae* as the expression for the density at a distance s from one end of the string, and denoting the radius of the cylinder by r,

SECT. 5. Extensible Strings.

If a uniform extensible string of given length be stretched by any force, it is found by experiment that the extension of the string beyond its natural length is proportional to the force. From this it is easily seen that, if the string be of variable length, the extension will vary as the product of the force and the natural length of the string. Hence, if a denote the natural length of the string, and a' the length under the action of a stretching force P, we shall have

where a is a constant quantity depending upon the quality of the string, called the modulus of elasticity.

This theory was first announced by Hooke, in the form of an anagram, among a list of inventions at the end of his Descriptions of Helioscopes, published in the year 1676. The anagram is ceiiinosssttuu, from which may be extracted the proposition, "ut tensio sic vis." He afterwards published a work entitled De Potentia Restitutiva or Spring, in which the theory was developed at large with experimental illustrations. Hooke's theory forms the basis of a memoir by Leibnitz, in the Acta Eruditorum for the year 1684, entitled Demonstrationes Nova de Resistentia Solidorum. For additional information on the subject the student is referred to s'Gravesande's Element. Physic. Lib. 1. c. 26.

(1) An elastic string AC (fig. 84) is suspended from its extremity A, and a weight is attached to it at a point B; the natural lengths of AB, BC, being given, to find the length of the string AC in its present circumstances.

Let m denote the mass of a unit of length of the string in its natural state; a, b, the natural lengths of AB, BC, and a', b', their lengths under the circumstances of the problem; c the length of a portion of the natural string, the weight of which

is equal to the weight attached to B; let P be any point in AB, and p an adjacent point; let AP=x, Pp = dx; lett be the tension at P and t+dt at p.

Then, by Hooke's Principle, the weight of Pp will be

mg

mg

(a + b + c) { 1 + m2 (a + b + c)} = (b + c) { 1 + m2 (b + c)} + a',

+

Again, if Q be any point in BC, BQ=y, and T=the tension at Q, we shall have, as before,