April 2, 1915] 



SCIENCE 



495 



Bessel, 1841 690.6" 



Clarke, 1866 700.4" 



Harkness, 1891 688.2" 



Hayford, 1909 695.8" 



It appears essential in this connection to 

 call attention to a common misapprehension 

 with respect to the earth which Professors 

 Moulton and Eoever have helped to dissemi- 

 nate by their able contributions to the subject 

 before us. The potential function V which 

 appears in equation (1) above, may be devel- 

 oped in a series of spherical harmonics whose 

 first three terms are given in the second mem- 

 ber of the following equation: 



y = ^ + ^z^C-i(B + A)}(1 - 3sinV) 



3fc (^) 



+ ^,{B- A) cos2 ;/■ cos 2X. 



In this r, if/, \ are, respectively, the radius 

 vector, geocentric latitude and longitude of 

 the point, outside the earth, to which V ap- 

 plies. M is the mass of the earth, fc is the 

 gravitation constant and A, B, C are in order 

 of increasing magnitude the moments of 

 inertia of the earth with respect to a set of 

 principal axes originating at its centroid. 

 is commonly said to be the moment with re- 

 spect to the axis of rotation of the earth, but 

 in these days of " variation of latitudes " 

 and of " mathematical rigor," it should be said 

 to apply to the axis of figure nearest the axis 

 of rotation. A and B are then the moments 

 with respect to the principal axes in a plane 

 through the centroid and normal to the axis 

 of C, or in the plane of the equator as we 

 commonly say. 



The expression (4) has very remarkable 

 properties. It is equation (26) of my paper 

 cited above. The value of Y is the same 

 whether the latitude xp is positive or negative ; 

 and dependence on longitude vanishes if 

 B=^A. With respect to this equation Pro- 

 fessor Moulton remarks " If the rotating body 

 is a figure of revolution about the axis of 

 rotation whose density does not depend upon 

 the longitude, the function V can be devel- 

 oped as a series of zonal harmonics in the form 



7 = --h;^(l-3sinV)." 



A similar remark with regard to this expres- 

 sion has been quoted above from Professor 

 Eoever, the inference being, apparently, that 

 in some manner the expression (4) limits the 

 distribution of the earth's mass to one of revo- 

 lution. As a matter of fact, however, the ex- 

 pression (4) implies no such restriction; on 

 the contrary, it applies equally to a body of 

 any form and of any distribution of density, 

 the sole requirement being that the point 

 {r, ij/, A) lie at a distance from the centroid of 

 the body equal to or greater than the greatest 

 distance of any element of mass in the body 

 from the same point. The considerations 

 which permit us to assume (B — A) small, or 

 possibly negligible, in this and other problems 

 of geodesy, must depend, unfortunately, on 

 other sources of information than the expres- 

 sion (4). Some attention to these considera- 

 tions was given in each of my papers referred 

 to in the first paragraph of this note. 



Without going further into the subject at 

 this time it may suffice to remark that it now 

 appears illusory except as a mathematical exer- 

 cise to push the solution of the differential 

 equations of motion of a falling body to terms 

 involving the second derivatives of V without 

 including the third term in the right-hand 

 member of (4), without taking account of 

 the known relation between these derivatives, 

 and without taking account of plumb-line de- 

 flections, which often exceed the discrepancy 

 shown by equations (2) and (3). 



E. S. Woodward 



February 22, 1915 



ABTHUB VON AUWEES 

 The problems that confront the astronomer 

 differ from those with which workers in other 

 departments of science are engaged in many 

 important particulars, but in none more than 

 in the magnitude of the data involved. So 

 great is the number of the stars, so vast, both 

 in space and in time, the scale of their motions, 

 that in general it transcends the powers of an 

 individual, or even of a single observatory, to 

 collect, within the span of a lifetime, the mate- 

 rials for comprehensive studies, or to collate 

 and discuss them. Cooperation is probably 



