92 THE POPULAR SCIENCE MONTHLY. 



J J f f. 



Then, by dynamical principles, we have — 



r , r' 



and ^:^'::_._. 



Now, .for these two planets we have — 

 r = 3962-8 miles, and r' = 2100 miles. 

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

 Q = 3- ^ 6^6 5 ? and Q' = -g-owooo- of mass of the sun. 

 Substituting these numbers in foregoing proportions, and performing 

 the arithmetical operations, and we have — 

 /:/::l : 0-500704, and 

 (J : g'::l : 0-376482. 



Hence we have '-C'^ '' ^ ' ao^?!o1 - or 1 : 1-32996. But, for the earth, 

 g g 0*376482 



f \ If 



owQ » ^ence we have -— : ^ :: 1 : 1-32996. Consequently for 

 g /wOt/ /i'Oi/ g 



f 1-32996 1 

 Mars we have -^, — — -— ^ — = — ■. Now, according to the elesrant 

 g 289 217 '^ ^ 



theorem of Newton, if the rotating planets were homogeneous liquid 

 masses, their elli^yticities would be f of -^i-w = tsi ^^r the earth, and 

 f of 8TT == TTT ^or Mars. These are the greatest jyossible values of the 

 ellipticities for these two planets with their present rotation-periods.* 

 In the case of the earth, we know that it is much smaller ; being about 

 -5-^ instead of 3^. Hence, for Mars also, we should expect an ellip- 

 ticity smaller than j^ ; 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 aug- 

 ment its ellipticity ; hence the question naturally suggests itself : 

 Might not this great ellipticity of Mars have been the result of solidi- 

 fication having taken place when his rotation-period was much shorter 

 than it is at present ? This explanation is not free from serious diffi- 

 culties. For, if aqueous and aerial agencies were in action after solidi- 

 fication 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 homogeneous 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 com- 

 puted ellipticity smaller. The problem of the theoretical figure of a rotating planet is 

 greatly complicated as soon as we abandon the assumption of homogeneousness. 



