92
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 rotationperiods.^{[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 rotationperiod 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 rotationperiod.
 ↑ 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.