130 W. D. Lambert — Mechanical Curiosities 



spaced at equal distances apart for equal differences of 

 the gravity potential. As a second approximation, we may 

 consider the lines along which gravity acts as straight 

 and as all meeting at a common center, that is, the center 

 of the earth. On this assumption the earth is a sphere 

 and the level surfaces of the gravity potential are spheres 

 concentric with it. So many of our ideas about gravity 

 are based, more or less unconsciously, on the conception 

 of parallel planes or concentric spheres that the conse- 

 quences of the fact that these conceptions are only approx- 

 imations to the truth often seem like paradoxes.^ Much 

 of this paper will be devoted to these apparent paradoxes. 

 The next step in the series of approximations would be 

 to take the level surfaces as spheroids of revolution 

 having a common axis, the polar axis of the earth. Now 

 spheroid is a rather indefinite term, more general than 

 ellipsoid. All ellipsoids of revolution that have their 

 axes nearly equal are spheroids of revolution, but not all 

 such spheroids are ellipsoids. This fact is illustrated 

 by the extreme case of the level surface marking the 

 outermost limit of the atmosphere.* This case was 

 treated by Laplace.^ The equatorial radius of this sur- 

 face is 6.6 times the radius of the earth, this distance 

 being such that the centrifugal force of rotation just 

 balances the earth's attraction at the equator of the sur- 

 face ; the polar radius is two-thirds of the equatorial 

 radius. The most curious feature of the surface (see 

 fig. 1) is at the equator where instead of continuous curva- 

 ture, as on an ordinary spheroid, there is found what 

 seems to be a sharp ridge but is really the intersection of 

 two nappes of the surface at an angle of 120° ; the portions 



^ The familiar paradox that the Mississippi river runs uphill, because its 

 mouth is farther from the center of the earth than its source, may be said 

 to arise from overlooking the fact that the sea level surface and other level 

 surfaces are spheroids and not spheres. The misconception is based on a 

 too literal acceptance of the statement that bodies tend to fall towards the 

 center of the earth. It would be just as logical to say that the ocean lies 

 on a slope, or rather on two slopes, because at the equator it is 13 miles 

 farther from the earth 's center than it is at the poles, as it is to say that 

 the Mississippi runs uphill. 



* The limit meant is one derived from the classical mechanics of masses. 

 No account is taken of the motion of the molecules of the atmosphere accord- 

 ing to the kinetic theory of gases nor of other questions of molecular 

 dynamics. 



^Mecanique Celeste, Book III, Chap. VII, Sect. 47, or in Bowditch's 

 translation, Vol. II, p. 519. See also Eesal, Mecanique Celeste, 2d ed., p. 

 322, and Helmert, Hohere Geodasie, Vol. II, p. 100. 



