494 



SCIENCE 



[N. S. Vol. XLI. No. 1057 



To put this statement in a clearer form for 

 the mathematical reader, let V denote the 

 gravitational potential per unit mass at a 

 point outside, or on, the earth, and let r and 

 ij/ denote, respectively, the radius vector and 

 the geocentric latitude of that point. Then, if 

 tt) denote the angular velocity of the earth and 

 if the point (r, ij/) is attached to and rotates 

 with the earth, the expression 

 V -\- i a'r' eos' ^ 

 is the potential per unit mass at that point due 

 to the attraction and to the rotation of the 

 earth. Calling this expression U, 



U=r + ior'r'aoa'iy=^cms't (1) 



specifies a family of equipotential surfaces 

 about the earth. Thus, for example, U = con- 

 siant specifies the sea surface, provided V, r, \1/ 

 have appropriate values, and this surface, 

 which may be imagined to extend through the 

 ■continents, is called the geoid. Similarly, 

 corresponding surfaces above and below the 

 sea surface are geoidal and may be used, like 

 the sea level, as surfaces of reference. 



Adopting for the moment the simpler hy- 

 pothesis that the shape of the geoid does not 

 depend on longitude, the divergence from 

 parallelism of the geoid (1) and the spheroid 

 {a, h) may be defined in the following manner. 

 Since the linear acceleration components along 

 and perpendicular to the radius vector r at 

 the point (r, ij/) of the geoid U = constant are, 

 respectively, 



dU , dU 

 -r- and — - , 

 dr rdxj/ 



the tangent of the angle between r and the 

 normal to the geoid at the same point is 

 given by the quotient of the second by the 

 first of these partial derivatives.^ 



The angle thus derived is the difference 

 between the astronomical latitude, ^„ say, and 

 the geocentric latitude i/r of the point (r, tf). 



could figure sensibly in the orbits of falling bod- 

 ies when my first investigation of these orbits waa 

 published. 



5 To terms of the order of w° inclusive this tan- 

 gent, using the notation of my paper cited above, is 



■(-+¥) 



sin 2;/' 



| + |^(l-3siiiV)-'>'Vcos=^ 



Using the data for Y and r adopted in my 

 paper cited above, it is found that the general 

 value of this difference is to the first order of 

 approximation, and in seconds of arc, 



00 — lA = 688" sin 20„. (2) 



On the other hand, the difference between the 

 geodetic latitudes <^, say (determined by the 

 normal to the spheroid (a, Z))), and the geo- 

 centric latitude of the same point, is to the 

 same order of approximation 



— 1/' = 700" sin 20. (3) 



There is thug a systematic difference between 

 these two quantities, since the residuals 

 (<^o — ^)> or the so-called plumb-line deflec- 

 tion in the meridian, are assumed to be of 

 compensating plus and minus magnitudes in 

 determining the spheroid {a, h). Otherwise 

 expressed, this systematic difference is such 

 as to make the value of the meridional devia- 

 tion of the falling body vanish to terms of the 

 order of co^ inclusive, adopted in my investi- 

 gation, if reference is made to the geoid in- 

 stead of to the spheroid; and to this order of 

 approximation the discrepancy is completely 

 accounted for. 



It is evident that we may not discard either 

 in favor of the other, of the two surfaces of 

 reference giving rise to this discrepancy, since 

 their departure from coincidence is an index 

 of our ignorance of the geoid especially and to 

 a less extent also of the spheroid used. The 

 geoid specified by equation (1) is obviously 

 less well known than the spheroid, since an 

 assumption must be made concerning the dis- 

 tribution of density in the earth before the 

 moments of inertia which determine the geoid 

 can be computed. Thus the relation (2) is 

 known with less precision than the relation 

 (3) ; but it is now clear that a complete treat- 

 ment of the problem in question requires that 

 both of these relations be taken into account 

 along with the additional relations ((^„ — <f) 

 and (A„ — A), say, or the plumb-line deflections 

 in latitude and longitude, respectively, at the 

 point {r, \p, X). That considerable uncertainty 

 attaches still to the relation (3) is indicated 

 by the range in the following values for the 

 coefficient of sin 2<^ derived by some earlier and 

 by some more recent writers in geodesy. 



