July 23, 1915] 



SCIENCE 



125 



For the same data and for the potential func- 

 tion used by Dr. Woodward/^ 



TC'^.OOIO cm. 

 Therefore 



UC =UT + TC' = — 2.94. 



This result agrees very well with the value 

 f = — 3.03 obtained by Dr. Woodward for his 

 originally defined meridional deviation. Thus 

 I have shown that for the meridional devia- 

 tion implied by the statement that the sphe- 

 roid instead of the geoid is the surface of 

 reference, it is possible to find a formula, 

 namely 

 (6) UC' = — (0 — 0„) sin 2<t>-h, 



without integrating the equations of motion, 

 and that, for the data given by equations (2) 

 and (3) of Dr. Woodward's note, this formula 

 yields values for the deviation UO' which do 

 not differ much from those obtained by Dr. 

 Woodward for his originally defined merid- 

 ional deviation. 



3. We have just seen that the expression 

 (formula 6) for the newly defined meridional 

 deviation UC begins with the first power of /;.. 

 Let us now show, with the aid of Fig. 1, that 

 the originally defined meridional deviation 



1^ The quantity TC is the negative of the quan- 

 tity which I denoted by Vi in my first paper (Trans- 

 actions of the American Mathematical Society, Vol. 

 XII., No. 3, pp. 335-53). It is the quantity which 

 Comte De Sparre used for his meridional deviation 

 of a falling body. I have shown this quantity to 

 be expressible by the formula 



PjC begins with the second power of h. For 

 this purpose let us think of a series of level 

 surfaces between the geoid OM and the level 

 surface of P^. The locus of the feet of the 

 perpendiculars from P„ to these level surfaces 

 is a curve d passing, necessarily, through the 

 points P„ and P^ and tangent at P„ to the 

 vertical PJ" of P„ (see dotted curve in Fig. 1). 

 Since the curve d is tangent to P^T at P„, we 

 have for a reason given above, 



(7) F^T = l {1/piW + higher powers of h, 



where p^ is the radius of curvature of the curve 

 d at the point P„. It is further evident from 

 Fig. 1, that 



(8) F^C' = P^T -\-TC', 



the positive sense of each of these quantities 

 being taken toward the equator. By relations 

 (3), (7) and (8) 



(9) PiC =\{— + -\lv' + higher powers of /j" 



Hence we see that while the originally defined 

 meridional deviation P^C begins with the sec- 

 ond power of h, the newly, implicitly, defined 

 meridional deviation UC begins with the first 

 power of h. 



4. In commenting on my work. Dr. Wood- 

 ward, after speaking of a certain assumption, 



13 It ia not diflfieult to show that 



A' 

 So' 



where the terms have the same meaning as in the 

 preceding foot-note. Consequently 



Pi 



--(I). 



rC' = J2</ sin 200-1- (^) 1^, f /dg\ ) 



I ^ \dUo}6go' PiC'=Pi?'-l-TC'= j 2«2sin20o-5f ^j V 



where h and 0o have the meanings given above, w is 

 the angular velocity of the earth's rotation, and 

 ffa and (3ff/a?)o, are the values which the accelera- 

 tion g due to weight and the derivative of g with 

 respect to i have at the point Pa, i representing 

 distance measured to the south. For the potential 

 function used by Dr. Woodward (Astronomical 

 Journal, Nos. 651-52), (dg/8i)o = — 8.14X10-" 

 sin 2 00 and hence, since w^c= 5.3173 X 10"' we 

 have for this potential function 



TC = 2.49 X 10-8 sm 200 . " 

 6?o 



which for the data (5) yields 



2'C=-1-.0010 cm. 



This formula I proved for the first time in the 

 Transactions of the American Mathematical So- 

 ciety, Vol. XII., No. 3, pp. 335-53. See also Vol. 

 XIII., pp. 469-90, Astronomical Journal, Nos. 

 670-72 and Bulletin of the American Mathematical 

 Society, 2d aeries, Vol. XXI., No. 9, pp. 444-62. 

 For the potential function used by Dr. Woodward, 



(a3/3?)o = — 8.14 X 10-° sin 20<„ 

 whence, for that potential function 



P^C = 51.33 X 10-» sin 20„ h'/Gg^, 

 which for the data (5) gives 



Pfi' = -I- .021 cm. 



