F

take m, any small element of the plane CD, and let mr be a prism whose base is m and whose sides are parallel to AD and BC; of elementary prisms similar to which the whole solid ABCD may be supposed Now the volume of this prism, whose base ism and its height mr, equals mr×m = sec. 1 × (mr. cos. 1) × m = sec. 1× (mr . sin. mrn) m = sec. 1 × mn ×m.

to be made up.

Therefore the whole solid equals the sum of all such products as mn ×m, each such product being multiplied by the constant quantity sec., or it is equal to the sum just spoken of, that sum being divided by cos. 1. Let this sum be represented by mnm, therefore the volume of the solid is re

presented by

Σmn x m

COS. I

Now suppose CD to represent a thin lamina of uniform thickness, the weight of each square. unit of which is , then will the weight of the element m be represented by pm, and its moment about the plane ABN by mnm, and 2mm ×m will represent the sum of the moments of all the elements of the lamina similar to m about that plane. Now by Art. 15. this sum equals the moment of the whole weight of the lamina XCD supposed to be col-. lected in G, about that plane. Therefore

x CDxNG=pΣmn xm,

:. CD × NG = Σmn x m.

Substituting this value of Emn xn, we have volume of solid = sec. × CD × NG.

But the plane CD is the projection of AB, therefore CD = AB cos., . CD × sec. = AB;

.. vol. of solid ABCD = AB × NG = vol. of prism ABEF. Therefore, &c.

[Q. E. D.]

PART II.

DYNAMICS.

42. MOTION is change of place.

The science of DYNAMICS is that which treats of the laws which govern the motions of material bodies, and of their relation to the forces whence those motions result.

The SPACES described by a moving body are the distances between the positions which it occupies at different successive periods of time.

UNIFORM MOTION is that in which equal spaces are described in equal successive intervals of time.

The VELOCITY of uniform motion is the space which a body moving uniformly describes in each second of time. Thus if a body move uniformly with a velocity represented by V, and during a time represented in seconds by T, then the space S described by it in those T seconds is represented S S: by TV, or S=TV. Whence it follows that V = and T= T V

so that if a body move uniformly, the space described by it is equal to the velocity multiplied by the time in seconds, the velocity is equal to the space divided by the time, and the time is equal to the space divided by the velocity.

43. It is a law of motion, established from constant observation upon the motions of the planets, and by experiment upon the motions of the bodies around us, that when once communicated to a body, it remains in that body, unaffected by the lapse of time, carrying it forward for ever with the same velocity and in the same direction in which it first began to move, unless some force act afterwards in a contrary direction to destroy it.*

*This is the first LAW OF MOTION. For numerous illustrations of this fun. damental law of motion, the reader is referred to the author's work, entitled ILLUSTRATIONS OF MECHANICS, Art. 193.

The velocity, at any instant, of a body moving with a VARIABLE MOTION, is the space which it would describe in one second of time if its motion were from that instant to become UNIFORM.

An ACCELERATING FORCE is that which acting continually upon a body in the direction of its motion, produces in it a continually increasing velocity of motion.

A RETARDING FORCE is that which acting upon a body in a direction opposite to that of its motion produces in it a continually diminishing velocity.

An IMPULSIVE FORCE is that which having communicated motion to a body, ceases to act upon it after an exceedingly small time from the commencement of the motion.

44. A UNIFORMLY accelerating or retarding force is that which produces equal increments or decrements of velocity in equal successive intervals of time. If f represent the additional velocity communicated to a body by a uniformly accelerating force in each successive second of time, and T the number of seconds during which it moves, then since by the first law of motion it retains all these increments of velocity (if its motion be unopposed), it follows that after T seconds, an additional velocity represented by fT, will have been communicated to it; and if at the commencement of this T seconds its velocity in the same direction was V, then this initial velocity having been retained (by the first law of motion), its whole velocity will have become V+ƒT.

If, on the contrary, f represents the velocity continually taken away from a body in each successive second of time, by a uniformly retarding force, and V the velocity with which it began to move in a direction opposite to that in which this retarding force acts, then will its remaining velocity after T seconds be represented by V-fT; so that generally the velocity V of a body acted upon by a uniformly accelerating or retarding force is represented, after T seconds, by the formula

v=V±ƒT... . . (34).

The force of gravity is, in respect to the descent of bodies near the earth's surface, a constantly accelerating force, increasing the velocity of their descent by 324 feet in each successive second, and if they be projected upwards it is a constantly retarding force, diminishing their velocity by that quantity in each second. The symbol g is commonly used to

represent this number 32; so that in respect to gravity the above formula becomes v=VgT, the sign being taken according as the body is projected downwards or upwards.

A VARIABLE accelerating force is that which communicates unequal increments of velocity in equal successive intervals of time; and a variable retarding force that which takes away unequal decrements of velocity.*

C

45. TO DETERMINE THE RELATION BETWEEN THE VELOCITY AND THE SPACE, AND THE SPACE AND TIME OF A BODY'S MOTION. Let AM,, M,M,, M,M,, &c. represent the exceeding small successive periods of a body's motion, and AP the velocity with which it began to move, MP, the velocity at the expiration of the first interval of time, M,P, that at the expiration of the second, MP, of the third interval of time, and so on; and instead of the body varying the velocity of its motion continually throughout the period AM, suppose it to move through that interval with a velocity which is a mean between the velocity AP at A, and that M,P, at M,, or with a velocity equal to (AP+M,P,).

2

Since on this supposition it moves with a uniform motion, the space it describes during the period AM, equals the product of that velocity by that period of time, or it equals (AP+M,P)AM,. Now this product represents the area of the trapezoid AM,P, P. The space described then in the interval AM,, on the supposition that the body moves during that interval with a velocity which is the mean between those actually acquired at the commencement and termination of the interval, is represented by the trapezoidal area AM,P, P.

1

Similarly the areas P,M,, P,M,, &c. represent the spaces the body is made to describe in the successive intervals. M,M,, M,M,, &c.; and therefore the whole polygonal area APCB represents the whole space the body is made to describe in the whole time AB, on the supposition that it moves in each successively exceeding small interval of time with the mean velocity of that interval. Now the less the intervals are, the more nearly does this mean velocity of each interval approach the actual velocity of that interval; and if they be infinitely small, and therefore infinitely great in

Note (i) Ed. App.

number, then the mean velocity coincides with the actual velocity of each interval, and in this case the polygonal area passes into the curvilinear area APCB.

Generally, therefore, if we represent by the abscissa of a curve the times through which a body has moved, and by the corresponding ordinates of that curve the velocities which it has acquired after those times, then the area of that curve will represent the space through which the body has moved; or in other words, if a curve PC be taken such that the num ber of equal parts in any one of its abscissæ AM, being taken to represent the number of seconds during which a body has moved, the number of those equal parts in the corresponding ordinate M,P, will represent the number of feet in the velo city then acquired; then the space which the body has described will be represented by the number of these equal parts squared which are contained in the area of that curve.

46. TO DETERMINE THE SPACE DESCRIBED IN A GIVEN TIME BY A BODY WHICH IS PROJECTED WITH A GIVEN VELOCITY, AND WHOSE MOTION IS UNIFORMLY ACCELERATED, OR UNIFORMLY RETARDED.

D

P

M

F

B

Take any straight line AB to represent the whole time T, in seconds, of the body's motion, and draw AD perpendicular to it, representing on the same. scale its velocity at the commencement of its motion. Draw DE parallel to AB, and according as the motion is accelerated or retarded draw DC or DF inclined to DE, at an angle whose tangent equals f, the constant increment or decrement of the body's velocity. Then if any abscissa AM be taken to represent a number of seconds t during which the body has moved, the corresponding ordinate MP or MQ will represent the velocity then acquired by it, according as its motion is accelerated or retarded. For PR=RQ=DR tan. PDE=AM tan. PDE; but AM=t, and tan. PDE=f: therefore PR=RQ=ft. Also RM=AD=V, therefore MP= RM+PR=V+ft, and MQ=RM-RQ=V-ft; therefore by equation (34), MP or MQ represents the velocity after the time AM according as the motion is accelerated or retarded. The same being true. of every other time, it follows, by the last proposition, that the whole space described in the time T or AB is represented by the area ABCD if the motion be accelerated, and by the area ABFD if it be retarded.