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its descent, would have caused its motion at that instant to pass from a state of variable to a state of uniform motion. If the resistance of the air upon its descent be taken into account, then the moving force upon the body at any instant is measured by that single pressure which, being applied up wards, would, together with the resistance of the air ai thai instant, just balance the weight of the body.

A moving force being thus understood to be measured by a pressure, * being in fact the unbalanced pressure upon the moving body, the following relations between the amount of a moving force thus measured, and the degree of acceleration produced by it will become intelligible. These are laws of motion which have become known by experiment upon the motions of the bodies immediately around us, and by observation upon those of the planets.

93. When the moving force upon a body remains constantly the same in amount (as measured by the equivalent pressure) throughout the motion, or is a uniform moving force, it communicates to it equal additions of velocity in equal successive intervals of time. Thus the moving force upon a body descending freely by gravity (measured by its weight) being constantly the same in amount throughout it: descent (the resistance of the air being neglected), the body receives from it equal additions of velocity in equal successive intervals of time, viz. 32 feet in each successive second of time (Art. 44.).

94. The increments of velocity communicated to equal Vodies by, unequal moving forces (supposed uniform as above) are to one another as the amounts of those moving forces (measured by their equivalent pressures).

Thus let P and P, be any two wequal moving forces upo: two equal bodies, and let them act in the directions in which the bodies respectively move; let them be the only forces tending to communieate motion to those bodies, and remain constantly the same in amount throughout the motion. Also let f and f, represent the additional velocities which these two forces respectively communicate to those two equal bodies in each successive second of time; then it is a law of the motion of bodies, determined by observation and experiment, that P:P,::f:fi.

* Pressure and moving force are indeed but different modes of the operatior of the same principle of force.

If one of the moving forces, as for instance P,, be the incight W of the body moved, then the value f, of the increment of velocity per second corresponding to that moving force is 32' (Art. 41.) represented by g,

::P:W::f:g,

W
. ::P= .....(71).

95. If the amount or magnitude of the moving force does not remain the same throughout the motion, or if it be a Variable moving force, then the increments of velocity commmunicated by it in equal successive intervals of time are not equal; they increase continually if the moving force increases, and they diminish if it diminishes.

If two unequal moving forces, one or both of them, thus variable in magnitude, become the moving forces of two equal bodies, the additional velocities which they would communicate in the same interval of time to those bodies, it at any period of the motion from variable they become uniform, are to one another (Art. 94.) as the respective moving forces at that period of the motion.

Thus let f and f, represent the additional velocities which would thus be communicated to two equa! bodies in one second of time, if at any instant the pressures P and Pi, which are at that instant the moving forces of those bodies, were from variable to become constant pressures, then (Irt. 94.),

P:P,::f:f, This being true of any two moving forces, is evidently true, if one of them become a constant force. Let P, represent the weight W of the body, then will fi be represented

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::P:W::f:g. Let the moving force P be supposed to remain constant during a number of seconds or parts of a second, represented by At, and let a V be the increment of velocity in the time at on this supposition. Vow f represents the increment of velocity in each second, and AV the increment of velocity in at seconds : moreover the force P is supposed constant during st, so that the notion is uniformly accelerated during that time (Art. 41.).

::.ft=av, :f= Now this is true (if the supposition, that P remains constant during the time At, on which it is founded, be true), however small the time at may be. But if this time be infinitely small, the supposition on which it is founded is in all cases true, for P may in all cases be considered to remain the same during an infinitely small period of time, although it does not remain the same during any time which is not

„AV - AV infinitely small. Now when at is infinitely small ;

edV generally therefore fri.

If V increase as the time t increases, or if the motion be accelerated, then 7 is necessarily a positive quantity. If, on the contrary, V diminishes as the time increases, then dV com is negative; so that, generally,

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the sign + being taken according as the motion is accelerated or retarded. Substituting this value of f in the last proportion we have in the case, in which P represents a variable pressure,

P=+* *** ..... (73).

The principles stated above constitute the fundamental relations of pressure and motion.

96. The velocity V at any instant of a body moving with a variable motion, being the space which it would describe in a second of time, if at that instant its motion were to become uniform, it follows, that if we represent by At any * number of seconds or parts of a second, beginning from that instant, and by AS, the space which the body would describe

* Note (r) Ed. App.

in the time At, if its motion continued uniform from the com. mencement of that time, then, .

AS

Vatras, ::V=it Now this is true if the motion remain uniform during the time At, however small that time may be, and therefore if it be infinitely small. But if the time at be infinitely small, the motion does remain witorm during that time, however variable may be the moving force; also when át is infi

AS ds
nitely small, 2= Therefore, generally,

dS
• V= .....(74)."

The equations (73) and (74) are the fundamental equations of dynamics: they involve those dynamical results which have been discussed on other principles in the preceding parts of this work. *

THE DESCENT OF A BODY UPON A CURVE.

*97. If the moving force Pupon a body varies directly as its

distance at any time from a given point towards which it falls, then the whole time of the body's falling to that point will be the same, whatever may be the distance from which it falls.

Let A be the point from which the body falls, and B a

point towards which it falls along the path APB, which may be either curved or straight; also let the body be acted upon at each point P of its path, by a force in the direc

tion of its path at that point which varies as * Thus if the latter equation be inverted, and multiplied by the former, wo obtain the equation

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its distance BP, measured along the path from B; the time of falling to B will be the same, whatever may be the dis. tance of the point A from which the body falls.

For let BP=S, and let the force impelling the body towards B be represented by CS, where c is a constant quantity; suppose the body, instead of falling from A towards B, to be projected with any velocity from B towards A, and let v be the velocity acquired at P, and V that at A, and let BA=S,, then by equation (47),

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Suppose now the velocity of projection from B to have been such as would only just carry the body to A, so that V=0,

::v=o8,'—S)..... (75). Now by equation (74),

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and if £T represent the whole time in seconds occupied in the ascent of the body from B to A,

1 Si

•S
T=1

Jis?_S?
leg(S

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It is clear that the time required for the body's descent from A to B is equal to that necessary for the ascent from B to A, so that the whole time required to complete the ascent and descent is equal to T, and is represented by the formula

..... (76)

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