134 W. D. Lambert — Mechanical Curiosities 



elevations at the particular latitudes where the bench 

 marks are situated. This has the disadvantage, illus- 

 trated above, of giving different elevations to two ends 

 of a level lake. To allow for this lowering of the level 

 surfaces as the pole is approached and to obtain the 

 actual distances above sea level a correction must be 

 introduced into the direct result of spirit leveling. This 

 is called the orthometric correction}^ On one line of 

 levels from San Diego to Seattle, which runs much of the 

 way over high ground, the correction amounts to II/4 

 meters, a quantity far from negligible. 



The direction of gravity which defines the vertical is 

 always perpendicular to a level surface; that is, a line 

 whose tangent at any point takes the direction of gravity 

 at that point is an orthogonal trajectory of the family 

 of level surfaces. Such an orthogonal trajectory is 

 clearly not a straight line but, in the normal case here 

 considered, is convex toward the equator. (See fig. 2.) 

 The astronomic latitude of a place is defined as the angle 

 which the vertical at that place makes with the plane of 

 the equator. Evidently then the latitude at the top of 

 high tower is greater than the latitude of its base. The 

 reduction of astronomic latitude to sea level is part of the 

 ordinary routine of geodetic computations.^^ 



Let us now consider what appears to be a highly artifi- 

 cial problem. Suppose the earth smoothed off so that its 

 physical surface shall coincide exactly with a level sur- 

 face. A particle placed on such a surface is in relative 

 equilibrium even without the presence of friction. But 

 suppose a huge sphere to stand on this level surface. 

 (See iig. 3.) The vector representing the reaction of the 

 surface is perpendicular to it at the point of contact, 

 coinciding in direction with gravity at the point of contact. 

 But gravity acts on the sphere as if the latter were concen- 

 trated at its center, and thus the line of action of gravity 

 is not quite coincident Avith the direction in which the 

 earth's surface reacts. Gravity and the reaction are the 



" See for example Bowie and Avers, Fourth General Adjustment of the 

 Precise Level Net of the United States and the resulting Standard Eleva- 

 tions; U. S. Coast and Geodetic Survej, Special Publication No. 18, p. 49, 

 or Hosmer, Geodesy (New York, 1919), p. 254. 



" Bowie, Determination of Time, Longitude, Latitude and Azimuth (5th 

 Ed.), U. S. Coast and Geodetic Survey, Special Publication No. 14, p. 130. 

 Clarke, Geodesy (Oxford, 1880), p. 101. Helmert, Hohere Geodasie, Vol. II, 

 p. 98. 



