132 W. D. Lambert — Mechanical Curiosities 



beyond the nodal line are inapplicable to the problem 

 for physical reasons.^ The equipotential surfaces just 

 within the limiting surface show a tendency to "sharp- 

 edgedness," a tendency that rapidly decreases as we 

 move inwards. The limiting surface is a spheroid by 

 courtesy only, but the surfaces with radii up to 3 or 4 

 times that of the earth are very passable spheroids, 

 although not exact ellipsoids ; in middle latitudes they 

 are depressed below ellipsoids having the same axes. 



In the problem just discussed the effect of the earth's 

 deviation from a spherical form was neglected. By intro- 

 ducing just the right distribution of mass we may make 

 one of the equipotential surfaces lying outside of 

 attracting matter an exact ellipsoid; the other surfaces 

 will not be absolutely exact ellipsoids but transcendental 

 surfaces very closely resembling ellipsoids, particularly 

 those not far from the exact ellipsoid.'^ The important 

 thing to notice, however, is not so much the question 

 whether the level surfaces are exact ellipsoids, but rather 

 the fact that these level surfaces are not similar; the 

 flattening increases with the distance from the center. 

 Any two given surfaces are farther apart at the equator 

 than at the poles. (See ^g, 2.) This corresponds to the 

 fact that gravity is less at the equator than at the poles, 

 the force at a point in a direction perpendicular to two 

 consecutive surfaces being inversely proportional to the 

 distance between them. For the earth a level surface 

 that is 1000 meters above sea level at the equator is about 

 995 meters above sea level at the poles, a variation of 5 

 parts in a thousand or one in two hundred. Gravity at 

 the pole is therefore greater than gravity at the equator 

 by about 1/200 part of itself .^ 



® A study of the mathematical equation giving the form of these equipo- 

 tential surfaces will explain the existence of the nodal line. The mathe- 

 matical equation defines surfaces not applicable to the problem in hand for 

 physical reasons. These surfaces are shown by dotted or dashed lines in the 

 figure. 



^ This refers to the surfaces outside of attracting matter. Within a homo- 

 geneous ellipsoid the equipotential surfaces, whether due to mass-attraction 

 alone or to mass-attraction combined with the centrifugal force, are exact 

 ellipsoids. 



On the subject of ellipsoidal and approximately ellipsoid level surfaces 

 see the author's Beport on StoJces' Theorem and related methods of deter- 

 mining the Figure of the Earth, probably soon to be issued as a U. S. Coast 

 and Geodetic Survey special publication; also Helmert. Hohere Geodasie, 

 Vol. II, p. 92. 



« More exactly 1/189 = 0.00529. 



