*THE POPULAR SCIENCE MONTHLY.*

Then, by dynamical principles, we have—

f | : | f' | :: | r | : | r' | |||||

t^{2} |
t'^{2} |
||||||||||

and | g | : | g' | :: | Q | : | Q' | ||||

r^{2} |
r'^{2} |

Now, for these two planets we have— | ||||||||||

r |
= | 3962∙8 miles, | and | r' |
= | 2100 miles. | ||||

t |
= | 86164 seconds, | " | t' |
= | 88643 seconds. | ||||

Q | = | 1 | and | Q' | = | 1 | of mass of the sun. | |||

326690 | 3090000 |

Substituting these numbers in foregoing proportions, and performing the arithmetical operations, and we have—

f |
: | f' |
:: | 1 | : | 0∙500704, | and |

g |
: | g' |
:: | 1 | : | 0∙376482 |

Hence we have | f |
: | f' |
:: | 1 | : | 0∙500704 | or 1 : 1∙32996. But for the earth, | |

g' |
g' |
0∙376482 |

f |
= | 1 | ; hence we have | 1 | : | 1 | : | f' |
:: | 1 | : | 1∙32996. Consequently for |

g |
289 | 289 | g' |

Mars we have | f' |
= | 1∙32996 | = | 1 | Now, according to the elegant |

g' |
289 | 217 |

theorem of Newton, if the rotating planets were *homogeneous liquid masses,* their *ellipticities* would be ^{5}⁄_{4} of ^{1}⁄_{289} = ^{1}⁄_{231} for the earth, and ^{5}⁄_{4} of ^{1}⁄_{217} = ^{1}⁄_{174} for Mars. These are the *greatest possible* values of the *ellipticities* for these two planets with their *present* rotation-periods.^{[1]}

In the case of the earth, we know that it is much *smaller;* being about ^{1}⁄_{300} instead of ^{1}⁄_{231}. Hence, for Mars also, we should expect an ellipticity *smaller* than ^{1}⁄_{174}; whereas, as we have seen, nearly all the measurements indicate a much *greater* ellipticity.

It is evident that a more *rapid* rotation of the planet would *augment* its ellipticity; hence the question naturally suggests itself: Might not this great ellipticity of Mars have been the result of solidification having taken place when his rotation-period was much *shorter* than it is at present? This explanation is not free from serious difficulties. For, if aqueous and aërial agencies were in action after solidification took place, they would have tended to make the shape of the planet conform to its new rotation-period.

- ↑ That the values of ellipticity deduced from the assumption of an
*homogeneons liquid mass*in the rotating planet must be*maxima*is evident from the consideration that, if the*density augmented*from the surface toward the center of the planet (which must, from the compressibility of matter, be the*real*condition of things), it would render the computed ellipticity*smaller*. The problem of the*theoretical figure*of a rotating planet is greatly*complicated*as soon as we abandon the assumption of*homogeneousness.*