Apkil 2, 1915] 



SCIENCE 



493 



adopted and the results derived are at once 

 unique, " necessary " and " sufficient." Both 

 authors insist with much particularity that 

 the discrepancy between us is due to superior 

 methods of approximation followed by them 

 in integrating the fundamental equations of 

 motion, since we all agree on the forms of 

 these equations. 



But the subject is not thus easily disposed of. 

 A sense of humor should lead us to inquire 

 whether the parties concerned have all solved 

 the same problem. The answer to such an 

 inquiry in this case is that while all have 

 ostensibly treated the same problem, two differ- 

 ent problems have actually been solved. We 

 have thus developed a fresh illustration of a 

 common danger in mathematical physics, 

 namely, that of fixing attention on mathe- 

 matical perfection before adequate regard has 

 been given to physical requirements. 



It would be out of place in the columns of 

 this journal to enter into a review of the de- 

 tails of the investigations of Professor Moulton 

 and Professor Eoever. Such a review is, in 

 fact, neither desirable here nor essential in a 

 technical publication. The source of the dis- 

 crepancy referred to is so evident that it needs 

 only to he stated to be appreciated; and once 

 stated there is no ground for controversy in 

 this part of the subject. It appears desirable, 

 however, to refer in some detail to the general 

 considerations involved in deriving the orbits 

 of falling bodies as well as to those special 

 considerations which determine meridional 

 deviations. For this purpose it will be essen- 

 tial in a limited degree to use the abridged 

 language of analysis. 



But before adducing these considerations 

 I wish to plead guilty to an oversight in read- 

 ing Professor Eoever's earlier papers^ and to 

 submit a brief of extenuating circumstances. 

 At first reading of these papers it appeared to 

 me that he had neglected terms involving the 

 square of the angular velocity of the earth in 

 his equations of motion of the falling body. 



s "The Southerly Deviation of Falling Bodies," 

 Transactions of the American Mathematical So- 

 ciety, Vol. XII., pp. 335-53. "The Southerly and 

 Easterly Deviations of Palling Bodies for an TJn- 

 symmetrieal Gravitational Field of Force," Hid., 

 Vol. XIII., pp. 4S9-490. 



These terms do not appear explicitly in those 

 equations, but only implicitly through a spe- 

 cial potential function used by him for the 

 first time, apparently, in this connection. Not 

 being able to follow his derivation of these 

 equations (if, indeed, he may be said to have 

 derived them in the mechanical sense), I as- 

 sumed them to be identical in meaning, as 

 they are in form, with those published by sev- 

 eral earlier writers. This assumption was 

 supported by uncertainty as to meaning and 

 by lack of homogeneity of his expression for 

 the potential function introduced on page 342 

 of his first paper; and still more by his iden- 

 tification of astronomic with geocentric lati- 

 tude (on p. 339, same paper) by means of the 

 loose phrase "with sufficient approximation.'' 

 A similar lack of " accuracy and precision " 

 will be found in several parts of his latest 

 paper cited above. See, for example, his equa- 

 tions (/), wherein he confounds geocentric 

 with reduced latitude; also p. 199, where he 

 identifies his equations (38) and (41) with my 

 equation (26) and makes with respect to them 

 the surprising statement that " it is, of course, 

 evident that this function corresponds to some 

 distribution of revolution " in the earth's 

 mass. Concerning the absence of validity for 

 this latter statement some remarks are made 

 below. 



Now, to account for the discrepancy in ques- 

 tion, namely, our differing values for the 

 meridional deviation of the falling body, it is 

 only essential to observe that two different sur- 

 faces of reference have been used. Professors 

 Moulton and Eoever have referred the motion 

 to a geoid specified by a certain approximate 

 potential function, while I have referred the 

 same motion to Clarke's spheroid of revolution 

 (of 1866), which is determined by certain axes 

 (a, h) dependent on geodetic measurements. 

 These surfaces are not coincident to the order 

 of approximation adopted by either party, and 

 the discrepancy developed appears to be both 

 " necessary " and " sufficient " to restore con- 

 fidence in the mathematical mills of all 

 concerned.* 



4 It has been known since the earlier writings of 

 Airy that the geoid and the spheroid are not coin- 

 cident, but I was not aware that their inclination 



