318 



SCIENCE 



[N. S. Vol. XXXVIII. No. 975 



of the angular velocity of the earth, but on 

 its attraction and on the difference between 

 the geocentric and the geographic latitudes 

 of the point in which a line drawn through 

 the initial position of the body and normal 

 to some plane of reference below pierces 

 this plane. The three sets of equations of 

 motion just referred to are expressed in 

 terms (1) of the polar coordinates of the 

 body (r, ij/, A), r denoting radius vector 

 from the center of the earth, i/f geocentric 

 latitude and A. longitude from a principal 

 equatorial axis of inertia of the earth; 

 (2) of the rectangular coordinates (^, *;, 

 t,), with origin at the point of intersection 

 of that plumb line through the initial posi- 

 tion of the body which is perpendicular to 

 "the horizontal plane of reference below, 

 with distance i measured in this horizontal 

 plane and parallel to the meridian plane 

 through the initial position of the body, 

 positively towards the equator, with dis- 

 tance t; positive towards the east and nor- 

 mal to the initial meridian plane, and with 

 distance £, positive upwards and parallel to 

 the normal at the origin; (3) of the ortho- 

 gonal coordinates (•>?, />, o-), giving the dis- 

 tance -q of the body east of the initial 

 meridian plane, the distance p of 7/ from 

 the earth's axis of figure and the distance 

 o- of the body from the plane of the earth's 

 equator. It is thus practicable not only to 

 approach the problem by different routes 

 and to check all steps in the processes of 

 solution, but also to see at once wherein the 

 results reached differ from the conflicting 

 results hitherto published. 



Of the three sets of equations of motion, 

 that for the last, or that for the coordinates 

 7], p, 0-, is the simplest. The integrals of 

 this set (new to the subject, so far as I am 

 aware) give the distance o- to a high order 

 of approximation as a simple harmonic 

 function whose amplitude is the initial 

 value of 0-; while the distances r] and p are 



given with equal precision by sums re- 

 spectively of two simple harmonic func- 

 tions of two different angles. It is re- 

 markable also that the diminution of the 

 radius vector r and the easterly deviation 

 ■)] are each expressed with precision by a 

 single hyperbolic term. In general, the 

 system of coordinates r, tj), \ is most con- 

 venient for the purposes of computation. 

 But the equations for interconversion of 

 all of the sets of coordinates are given in 

 detail in the mathematical paper referred 

 to. 



It is shown that the meridional deviation 

 specified by the ordinate i is always nega- 

 tive, or that this deviation is always to- 

 wards the adjacent pole in either hemi- 

 sphere instead of towards the equator as 

 hitherto supposed. For a fall of 10 seconds, 

 or 490.24 meters (in vacuo), in latitude 

 45° the meridional deviation would be 3.03 

 centimeters, and the easterly deviation 16.85 

 centimeters. These two deviations are pro- 

 portional approximately to the square and 

 to the cube, respectively, of the time of fall. 



My investigation is subject to two volun- 

 tary restrictions and to one limitation de- 

 pendent on our present lack of observa- 

 tional information in geodesy. The first 

 restriction lies in the neglect of the effect of 

 atmospheric resistance on the orbit of the 

 falling body. This effect is known from 

 the work of Laplace, Poisson and others 

 to be very small, since the path of the body 

 throughout its fall is everywhere very 

 nearly normal to the stratification of the 

 air. For such falls as may be practicable 

 for observation this effect is negligible, espe- 

 cially in comparison with the effects of cur- 

 rents of air and of lateral displacement due 

 to the rolling of the smoothest spheres.^ 

 The other restriction lies in solving the 



^ I consider it quite impracticable to make any 

 conclusive experiments on the deviation of spheres 

 falling in air. 



