12. A string fastened at A supports a weight by passing under a rough handle of any form, the loose end being held so that the parts on each side of the handle are parallel; find the least force which will prevent the weight from falling, and the greatest which will not draw it up.

13. An elastic band, whose unstretched length is 2a, is placed round four rough pegs A, B, C, D, which constitute the angular points of a square whose side is a; if it be taken hold of at a point P, between A and B, and pulled in the direction AB, shew that it will begin to slip round A and B at the same time if

14. An elastic string of variable thickness is extended by a given force, find the whole extension.

15. An elastic string whose density varies as the distance from one end, is suspended by that end and stretched by its own weight. If W be the weight of the string, l' its unstretched length, 7 its stretched length, shew that

16. A circular elastic string placed on a smooth sphere subtends when unstretched an angle 2a at the centre, and an angle 20 when in a position of equilibrium, shew that

where a =

radius of sphere, and c depends on the nature of the string.

17. An endless elastic string is placed over a smooth right cone, and repelled by a force in the vertex varying inversely as the distance.

If ' be the unstretched length, 7 the stretched length of the string, F the repulsive force on a unit of string at a unit of distance, then

18. A heavy elastic string surrounds a smooth horizontal cylinder, so that the surface of the cylinder is subject to no pressure at the lowest point; find the pressure at any point of the cylinder, and the tension of the string; its modulus of elasticity being equal to the weight of a portion of string the natural length of which is of the diameter of the cylinder.

19. A uniform heavy elastic string (natural length a) is stretched by forces applied at its ends, and then, being laid upon a rough inclined plane, is suffered slowly to contract itself. Shew that a point of the string, the natural distance of which from the upper end is

will not be affected by friction. a is the inclination of the plane.

20. A heavy elastic cord is passed through a number of fixed smooth rings. Shew that in the position of equilibrium its extremities will lie in the same horizontal plane. The same will also be the case if the cord rest upon any smooth surface.

21. A uniform elastic string just circumscribes a given circle, and is attracted by a force varying as the distance to a point within the circle. Find the tension at any point, supposing it to vanish at the point nearest to the centre of force, and shew that the force at the greatest distance whole pressure on the circle mass of the string

=

CHAPTER XIII.

ON ATTRACTIONS.

203. It appears from considerations which are detailed in works on Physical Astronomy, that two particles of matter attract each other with a force directly proportional to the product of their masses, and inversely proportional to the square of their distance.

Suppose then a particle to be attracted by all the particles of a body; if we resolve the attraction of each particle of the body into components parallel to fixed rectangular axes, and take the sum of the components which act in a given direction, we obtain the resolved attraction of the whole body on the particle in that direction, and can thus ascertain the resultant attraction of the body in magnitude and direction. We shall give some particular examples, and then proceed to general formulæ.

204. PROP. To find the attraction of a uniform straight line on an external point.

By a straight line we understand a cylinder such that the section perpendicular to its axis is a curve, every chord of which is indefinitely small.

Let AB be the rod, P the attracted particle; take A for the origin, and AB for the direction of the axis of x. Draw PL perpen

M and N be adjacent

points in the rod, AM=x,

=

MN Sx. If p be the density of the rod, and the area of a section perpendicular to its length, the mass of the element is pêx. Let m be the mass of P; then the attraction of the element MN on P is (Art. 203)

μαρκ δα
(PM)

where μ is some constant quantity. Hence, the resolved part of the attraction of the element parallel to the axis of x, is μαρκα ML μαρκ (α - α) δι

PM2 PM

or

{b2 + (a − x)2} }

Also the resolved part of the attraction of the element parallel to the axis of y, is

Let X and Y be the resolved parts of the attraction of the line, parallel to the axes of x and y respectively; then