Page:Popular Science Monthly Volume 16.djvu/102

Then, by dynamical principles, we have—

	f	:	f'	::	r	:	r'
					t2		t'2
and	g	:	g'	::	Q	:	Q'
					r2		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.

1. 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.