124 



SCIENCE 



[N. S. Vol. XLII. No. 1073 



2. For the sake of simplicity let us assume 

 (as Dr. Woodward did before he got very far 

 into his solution) that the distribution of the 

 earth's gravitating matter is such as to make 

 the potential function independent of the 

 longitude (.». e., correspond to a distribution of 

 revolution). Let Pg denote the point (fixed 

 with respect to the earth) from which the body 

 falls. In Fig. 1 the plane of the drawing is 

 assumed to be the meridian plane of ?„. This 

 plane contains the axis of rotation OZ, and 

 cuts from the geoid and the spheroid (both of 

 which are surfaces of revolution of axis OZ) 

 the meridian curves GH and AB, respectively, 

 GH drawn fuU and AB dashed. The point P^ 

 is the foot of the perpendicular from Pg to the 

 geoid GH. The straight line P^Po is then the 

 vertical of PA and the angle (f>, which it makes 

 with the equatorial plane (perpendicular to 

 the axis OZ) is the astronomic latitude of P^. 

 The straight line P^T (not coincident with 

 PpPj) is the vertical of ?„ (i. e., the normal 

 at Pf, of the level surface through P^). The 

 angle <^(, which it makes with the equatorial 

 plane is the astronomic latitude of P^.' The 

 path (with respect to the earth) of the falling 

 body is a curve c which does not lie in the 

 meridian plane of P^, but is tangent at P„ to 

 the vertical P„T of Pp. This curve c pierces 

 the horizontal plane of P^ (i. e., the plane 

 through Pj perpendicular to P^P^) in a point 

 0. Let us denote by c' and C the orthographic 

 projections of c and 0, respectively, on the 

 meridian plane of P^. Then c' is also tangent 

 to P^T at P(,. According to the definitions 

 originally adopted by Dr. Woodward, 



8 The difference "between 0o and <t>i is given by 

 the formula 



do — <^i = ~ ^ ^ h + higher powers in h, 

 Si 



where h is the distance of Po above Pi, gi is the 

 value of the acceleration due to weight at Pi, and 

 (.dg/dOi is the value, at Pi, of the derivative of g 

 with respect to i, where I represents distance meas- 

 ured to the south at Pi. For the potential func- 

 tion used by Dr. Woodward {Astronomical Journal, 

 Nos. 651-52), 



-(ag/a^)i ^g3 ^10-12 gin 2.»i. 

 ffi 



C'C is the easterly deviation of the falling body, 

 PiC is the meridional deviation of the falling 

 body. 



He now says, however, that he referred the 

 motion of the falling body to the spheroid 

 (AB, Fig. 1). By this he must mean that he 

 measures the deviation of the falling body 

 from the foot U of the normal drawn from P^ 

 to the spheroid. The angle <p which this 

 normal (shown in Fig. 1 by the dashed line 

 PqU) makes with the equatorial plane is called 

 the geodetic latitude of U. In other words, 

 the statement that the spheroid is his surface 

 of reference implies that UO' is the merid- 

 ional deviation of the falling body. That this 

 is the implication is also borne out by the fact 

 that the value of this deviation agrees with the 

 value which Dr. Woodward actually found. 

 In order to show this let us first observe that 



(1) UC' = UT + TC', 



the positive sense of each of these quantities 

 being taken toward the equator. If (p and 

 <^(, be expressed in radians, 



(2) UT= {<!>„ — <p)h, 



where h^P^P^ is the height of fall. Since 

 the curve c' is tangent to P^T at P^, and has 

 no cusp there,^ 



(3) rC'=:J(l/p„)7f2 + Mgher powers of V 



where p^ is the radius of curvature of c at Pg. 

 By equations (2) and (3) of Dr. Woodward's 

 note,ii (p — (^(, =; 12" sin 2<^, and hence in 

 circular measure 



(4) — 00 = .00006 sin 20. 

 Hence for the data 



(5) ;» = 49024 cm., = 45°, 



assumed in his example in the Astronomical 

 Journal (Nos. 651-652) 



UT = — 2.94: cm. 



» The curve c has a cusp at Po as has also its pro- 

 jection on a plane perpendicular to the meridian 

 plane of Po. 



10 See "Introduction to Infinite Series," by 

 Osgood, p. 39. 



11 Science, No. 1057. 



