Page:The Solar System - Six Lectures - Lowell.djvu/58

40

The Solar System

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striving by friction to set it at right angles to the line. The bulge, therefore, acts as a brake upon the Earth's rotation, and must continue so to act until the Earth's rotation and revolution coincide.

Tide-generating force.Now let us determine the tide-generating force^[1]:—

Let $M$= mass of the Earth;
$m$ =mass of the Moon;
$x$,$y$,$z$ = be coördinates of the Moon referred to the Earth's centre;
$r$=its distance;
$\xi ,\eta ,\zeta$ = the coördinates of the particle referred to the Earth's centre;
$\rho$ = its distance.

Then the Earth describes an ellipse round the centre of inertia of the Earth and Moon, and its acceleration is ${\frac {m}{r^{2}}}$ toward this centre.

To bring it to rest, we must apply to it an acceleration, $-{\frac {m}{r^{2}}}$, of which the accelerations along the coördinates are,—

$\textstyle -{\frac {m}{r^{2}}}.{\frac {x}{r}},-{\frac {m}{r^{2}}}.{\frac {y}{r}},-{\frac {m}{r^{2}}}.{\frac {z}{r}}$

Now

$\textstyle cos{z}={\frac {x}{r}}.{\frac {\xi }{\rho }}+{\frac {y}{r}}.{\frac {\eta }{\rho }}+{\frac {z}{r}}.{\frac {\zeta }{\rho }}$

and

$\textstyle r\rho \cos {z}=x\xi +y\eta +z\zeta ;$

1. 1 After G. H. Darwin. Article in the Encyclopedia Britannica on "Tides.")