July 23, 1915] 



SCIENCE 



123 



problem which he originally proposed. The 

 solution of the problem which he now states in 

 his note that he has solved corresponds to a 

 meridional deviation different from that orig- 

 inally defined. This deviation is of the form 

 Ah + Bh^, while that originally defined was 

 of the form Ch^, in which h is the height of 

 fall and A, B, are constants. I will also 

 show that a formula for this new meridional 

 deviation may be obtained without integrating 

 the equations of motion at all, and that this 

 formula yields a result differing but slightly 

 from the result given by Dr. Woodward, but 

 given by him for the deviation originally de- 

 fined. In this article I will also reply to cer- 

 tain criticisms made by Dr. Woodward con- 

 cerning my work. 



1. In the sixth paragraph of his note' Dr. 

 Woodward says : 



Now, to account for the discrepancy in question, 

 namely, our differing values for the meridional 

 deviation of the falling body, it ia only essential 

 to observe that two different surfaces of reference 

 have been used. Profesors 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 (o, 6) dependent on geodetic meas- 

 urements. 



In reply to this statement I should like to 

 say that in order to determine the path (orbit) 

 of the falling body a potential function is 

 needed; a surface of reference is not enough. 

 When once the potential function is chosen 

 the geoid (or level surface) is determined. 

 That the geoid, and not the spheroid, was orig- 

 inally contemplated by Dr. Woodward as the 

 surface of reference, appears from the state- 

 ment made below equations (2) of his paper 

 in the Astronomical Journal (Nos. 651-52). 

 For, of the points P^ and P^ from which, re- 

 spectively, the body is let fall and the devia- 

 tions measured, he says: 



It is important to specify how this point P, is 

 located with reference to the initial point Pq. 

 Imagine a basin of mercury at the point Pi. The 



e Science, No. 1057, pp. 493. 



surface of the mercury will be the level, or equipo- 

 tential (or horizontal) surface through this point; 

 and if it is located as here assumed the line join- 

 ing the two points Po and Pi will be normal to the 

 surface of the mercury. 



Now, the surface of the mercury is surely a 

 portion of the geoid and not of the spheroid. 

 The position of the point Pj besides depending 

 on that of P„, depends on the potential func- 

 tion, and, furthermore, on the same potential 

 function as that which is used in the differ- 



GA. 



Pig. 1. 



ential equations of motion of the path of the 

 falling body. Dr. Woodward now states that 

 he used for his surface of reference the 

 spheroid (of Clarke) instead of the geoid. If 

 these two surfaces differ ever so slightly from 

 one another — and they do differ according to 

 equatioas (2) and (8) of his note'' — the quan- 

 tities which are determined by using the sphe- 

 roid for reference are not the same as the 

 quantities -q, ^ (measured from P^) which he 

 originally defined as the easterly and merid- 

 ional deviations of the falling body. There- 

 fore, the problem which he now states that he 

 has solved is not the one which he originally 

 proposed. 



7 Science, No. 1057. 



