316 



SCIENCE 



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



SocietS Philomatique, in wliicli he invites 

 special attention to his conclusion that 

 there is no meridional deviation towards 

 the equator. In view of this discrepancy 

 between these preeminent authors it is a 

 surprising circumstance that nearly all 

 subsequent writers on the subject should 

 have followed Gauss; and it is still more 

 surprising that the more comprehensive 

 and more suggestive, though more difficult, 

 treatment of the problem by Laplace 

 should have been little noticed and less 

 followed by recent authors. Since the ap- 

 pearance of the papers just referred to by 

 Gauss and Laplace only one author, until 

 quite recently, appears to have considered 

 the subject worthy of an independent in- 

 vestigation. This author is Poisson, who 

 published in 1838 an important memoir on 

 the theory of gunnery (in the Journal de 

 I'Ecole Poly technique, Tome XVI.) of 

 which a freely falling body presents a 

 special case. As regards the meridional 

 deviation in question Poisson goes one step 

 further than Gauss and Laplace and leads 

 us to infer (correctly) that his investiga- 

 tion shows no deviation either towards or 

 away from the equator. 



My attention was called to this subject 

 about ten years ago, chiefly through the 

 communications concerning it published in 

 this journal by Professor Cajori and Pro- 

 fessor B. H. Hall. A casual reading of 

 the papers of Gauss, Laplace and Poisson 

 indicated that they ought all to agree es- 

 sentially, since they all limit themselves to 

 terms qf the first order of approximation 

 of the small quantities involved, especially 

 the angular velocity of the earth, which is 

 obviously a fundamental factor in any 

 solution of the problem. In the meantime, 

 other occupations have led me to neglect 

 this branch of geophysics until my atten- 

 tion was reattraeted to it by the suggestive 

 papers of Professor William H. Roever 



published recently in the Transactions of 

 the American Mathematical Society.^ A 

 preliminary survey of the subject indi- 

 cated that the obscurities and the discrep- 

 ancies presented by it could be removed 

 only by an independent investigation 

 founded on present-day knowledge of 

 geodesy. Such an investigation has been 

 made and is now available to the mathe- 

 matical physicist in Nos. 651-652 of the 

 Astronomical Journal (August 4, 1913) 

 under the title "The Orbits of Freely 

 Falling Bodies." The object of this com- 

 munication is to explain briefly for the 

 information of the general reader the 

 salient features of the subject, the sources 

 of its obscurities, the requirements of a 

 precise and correct determination of the 

 orbits in question, the new results reached, 

 and the reasons why they differ in certain 

 important respects from those hitherto 

 considered valid. 



The motion of a falling body depends on 

 three elements, namely: (1) the rotation of 

 the earth; (2) the attraction of the earth; 

 and (3) the difference between geocentric 

 and geographic latitude. The effect of ro- 

 tation is expressed in the equations of 

 motion of a falling body by terms involv- 

 ing both the first and the second powers 

 of the earth's angular velocity. In gen- 

 eral, following Gauss, Laplace and Poisson, 

 terms in the second power of this velocity 

 have been neglected. It turns out that the 

 meridional deviation is a term of the sec- 

 ond order in this velocity and other quan- 

 tities of the same order. Hence it failed 

 to appear in the investigations of the above- 

 named authors, or appeared only as a 

 mathematical fiction and with the wrong 



= "Tlie Southerly Deviation of Falling Bodies," 

 Vol. XII., No. 3, July, 1911; and "The Southerly 

 and Easterly Deviations of Falling Bodies in an 

 Unsymmetrical Gravitational Field," Vol. XIII., 

 No. 4, October, 1912. 



