Page:Somerville Mechanism of the heavens.djvu/95

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60. the velocity of a moving body changes, the cause of that change is called an accelerating or retarding force; and when the increase or diminution of the velocity is uniform, its cause is called a continued, or uniformly accelerating or retarding force, the increments of space which would be described in a given time with the initial velocities being always equally increased or diminished.

Gravitation is a uniformly accelerating force, for at the earth's surface a stone falls feet nearly, during the first second of its motion,  during the second,  during the third, &amp;c., falling every second  feet more than during the preceding second.

61. The action of a continued force is uninterrupted, so that the velocity is either gradually increased or diminished; but to facilitate mathematical investigation it is assumed to act by repeated impulses, separated by indefinitely small intervals of time, so that a particle of matter moving by the action of a continued force is assumed to describe indefinitely small but unequal spaces with a uniform motion, in indefinitely small and equal intervals of time.

62. In this hypothesis, whatever has been demonstrated regarding uniform motion is equally applicable to motion uniformly varied; and X, Y, Z, which have hitherto represented the components of an impulsive force, may now represent the components of a force acting uniformly.

63. If the direction of the force be always the same, the motion will be in a straight line; but where the direction of a continued force is perpetually varying it will cause the particle to describe a curved line.

Demonstration.— Suppose a particle impelled in the direction mA, fig. 17, and at the same time attracted by a continued force whose origin is in o, the force being supposed to act impulsively at equal successive infinitely small times. By the first impulse alone, in any given time the particle would move equably to A: but in the same time the action of the continued, or as it must now be considered the impulsive force alone, would cause it to move uniformly through