Page:Somerville Mechanism of the heavens.djvu/118

Chap II.] $c : c' :: \frac{r}{t^2} : \frac{r'}{t'^2}$,

Thus the centrifugal forces are as the radii divided by the squares of the times of revolution.

95. With regard to the Earth the times of rotation are everywhere the same; hence the centrifugal forces, in different latitudes, are as the radii of these parallels. These elegant theorems discovered by Huygens, led Newton to the general theory of motion in curves, and to the law of universal gravitation.

Motion of Projectiles,

96. From the general equation of motion is also derived the motion of projectiles.

Gravitation affords a perpetual example of a continued force; its influence on matter is the same whether at rest or in motion; it penetrates its most intimate recesses, and were it not for the resistance of the air, it would cause all bodies to fall with the same velocity: it is exerted at the greatest heights to which man has been able to ascend, and in the most profound depths to which he has penetrated. Its direction is perpendicular to the horizon, and therefore varies for every point on the earth's surface; but in the motion of projectiles it may be assumed to act in parallel straight lines; for, any curves that projectiles could describe on the earth may be esteemed as nothing in comparison of its circumference.

The mean radius of the earth is about 4000 miles, and MM. Biot and Gay Lussac ascended in a baUoon to the height of about four miles, which is the greatest elevation that has been attained, but even that is only the 1000th part of the radius.

The power of gravitation at or near the earth's surface may, without sensible error, be considered as a uniform force; for the decrease of gravitation, inversely as the square of the distance, is hardly perceptible at any height within our reach.

97. Demonstration,—If a particle be projected in a straight line $$MT$$, fig. 26, forming any angle whatever with the horizon, it will constantly deviate from the direction $$MT$$ by the action of the gravitating force, and will describe a curve $$MN$$, which is concave towards the horizon. and to which $$MT$$ is tangent at $$M$$. On this particle there