=

vers-1x+2x-x2+C.
2

When x =0, A = 0 .. C = 0, and when x = 2r,

semi-cycloidal area = · I'π =

3r
2

3

2

.. cycloid = 32 = 3 times area of generating circle.

(5.) To find the area of the curve, whose equation is

a2 (y2--x2) + (y2+a12)2 = 0.

In order to transform this equation from rectangular to polar co-ordinates, we must put y = r sin 0, and x = r cos 0; then, by substitution in the proposed equation, we have

Again; let A' denote the polar area, or space between the radius vector and the curve; then

And between the limits ra and r = 0, or between 0 and 0=

area of curve =

and if ✈ make a complete revolution, the entire area will be = a2.

.. surface of spherical segment = 2xax = circumf. x height of segment. .. surface of sphere =2πα. 2α = 4πa2 = circumf. x diameter.

(2.) To find the surface of a paraboloid.

(3.) To find the surface of a cone, and also the surface of a conic frustum Put a height of whole cone; r = radius of base of frustum; a2+r2=c2; b= height of top cone; ' radius of top of frustum; b2+r12=c22; and, taking the vertex as the origin of co-ordinates, we have y = the equation of the line generating the surface; whence

When x = a; surface of whole cone = ra2+r2 = πrc Sim. we have surface of top cone = πr'√b2+r22 = m1'c' .. surface of frustum of cone = « (rc — r'c')

rx

a

(1.) To find the content of a cone, and also that of a conic frustum.

Let a altitude of whole cone; r radius of base of frustum

b = altitude of top cone; = radius of top of frustum

= }} ( ar2 + arr'— br2+ar"?— brr'—

(ar2+arr— 2_brr'—br12)

́since arꞌ = br, or arrꞌ = br2, and ar12 = brr'; whence

π

segment = (6a—2x) x2 = (3d—2x) x2, if d=2a.

6

π

(3.) Find the volume of the paraboloid.

Here y2 = 4mx, is the equation to the generating parabola.

.. volume of paraboloid = volume of circumscribing cylinder.

(4.) Find the content of the prolate spheroid formed by the revolution of a semi-ellipse round its major axis.

(5.) Find the content of the oblate spheroid formed by the revolution of a semi-ellipse round its minor axis.

Hence prolate spheroid : oblate spheroid :: ab2 : a2b : : b ; a

4

:

4

.. sphere on major axis : prolate spheroid ::

πα :

παρ' : : α? : δε

3

3

.. oblate spheroid sphere on minor axis ::

¤a2b; }; #b3 : : a2 : b2.

4

MECHANICS.

DEFINITIONS AND FUNDAMENTAL NOTIONS.

1. Mechanics is the science which treats of the laws of rest and motion of bodies, whether solid or fluid, and is usually divided into the four following branches:

(1.) Statics, which treats of the laws of forces in equilibrium.
(2.) Dynamics, which treats of the laws of motion of solid bodies.
(3.) Hydrostatics, of the laws of the equilibrium of fluid bodies.
(4.) Hydrodynamics, of the laws of motion of fluid bodies.

2. Force or power is the cause which produces, or tends to produce, motion in a body, or which changes, or tends to change, motion.

3. A body is a portion of matter limited in every direction, and is therefore of a determinate form and volume.

4. All bodies have a tendency to fall to the earth; and the force which they exert in consequence of this tendency is called their weight.

5. When forces are applied simultaneously to a body, and produce rest, they balance each other, or destroy each other's effects; and therefore such forces are said to be in equilibrium.

6. The measure of a force, in statics, is the weight which that force would support.

7. The quantity of matter of a body is proportional to its weight.

8. The density of a body is measured by the quantity of matter contained in a given space.

9. Gravity is that force by which a body endeavours to fall downwards. 10. Specific gravity is the relation of the weights of different bodies o equal magnitude, and is therefore proportional to the density of the body.

STATICS.

THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING ON A

MATERIAL PARTICLE.

11. Def. The resultant of any number of forces is that single force which is equally effective with, or equivalent to, all the forces, and these forces are termed component or constituent forces.