September 5, 1913] 



SCIENCE 



517 



sign in the case of Gauss. The effect of 

 the attraction of the earth presents diffi- 

 culty, for the earth is not centrobaric, 

 though many authors have assumed it to be 

 such. Gauss and Laplace undoubtedly 

 understood the nature of this difficulty; 

 Laplace's paper (referred to above), is, 

 indeed, entirely satisfactory even now so 

 far as its generalities are concerned. But 

 the necessary observational knowledge, 

 since accumulated, was not available to 

 these pioneers. Each of them was justi- 

 fied, perhaps, in assuming that the effect 

 of the square of the angular velocity would 

 be negligible and that the attraction would 

 be sensibly what has been generally, but 

 now quite vaguely and inappropriately, 

 called "gravity" or "acceleration of grav- 

 ity," and expressed by the letter g. But 

 this attraction varies certainly with the 

 latitude of the position of the falling body 

 and possibly also with its longitude, and it 

 is not identical with the resultant accelera- 

 tion due to the attraction and to the rota- 

 tion of the earth. In respect to both of 

 these points the details of the papers of 

 Gauss, Laplace and Poisson along with the 

 papers of their followers, are all, so far as 

 I am aware, not only obscure, but inade- 

 quate. Closely related to the question of 

 the earth's attraction of a falling body is 

 the distinction between its varying geocen- 

 tric latitude and the constant geographical 

 latitude of the plumb line to which the 

 orbit of the body is referred. This dis- 

 tinction is essential to a correct determina- 

 tion of the meridional deviation, but its 

 fundamental importance does not appear 

 to have been recognized hitherto. 



Failure on the part of the earlier au- 

 thors to perceive the essential roles of these 

 elements and a tendency to avoid the com- 

 plications they entail in dealing with the 

 differential equations of motion, account 

 completely for the obscurities and the eon- 



fusion which initially beset the modern 

 reader who attempts to understand the 

 present extensive literature of this subject. 

 The admirably conceived investigation of 

 Laplace, since published as Chapitre V., 

 Tome IV., of his Mecanique Celeste, pre- 

 sents additional difficulties by reason of 

 his autocratic and unnecessary neglect of 

 terms, without assigning their relative mag- 

 nitudes, and by reason of his ready sup- 

 pression, after the fashion of his day, of 

 the identity of any quantity by calling it 

 unity. Following Gauss, many recent au- 

 thors also after neglecting terms of the 

 second order in their equations of motion, 

 have proceeded to get such terms by a 

 purely mathematical process which has no 

 warrant in the physical circumstances of 

 the case. It has been necessary, therefore, 

 in order to remove the prevailing uncer- 

 tainties of the subject, to reinvestigate it, 

 avoiding precedent and visualizing the con- 

 ditions of the problem in the light of the 

 more recent developments of physical 

 geodesy rather than in the light of the 

 foundations of this science laid so largely 

 and so effectively by Gauss, Laplace and 

 Poisson a century ago. 



Accordingly, the equations of motion of 

 the falling body are established without 

 neglect of any terms which belong to them, 

 and no terms in the integration of these 

 equations are neglected without precise 

 specification of their relative magnitudes. 

 The energy method of Lagrange is followed 

 in establishing the equations of motion, 

 partly because it is specially adapted to the 

 case and partly because it does not appear 

 to have been used for this purpose hitherto. 

 The position of the body is defined by ref- 

 erence to four sets of axes, and the equa- 

 tions of motion for each of three of these 

 sets are derived and integrated so as to 

 include all terms of the second order. 

 These latter depend not only on the square 



