164 



GRAVITATIONAL METHODS 



[Chap. 7 



tical and horizontal components. For P the vertical component is then 

 proportional to cos {<p + x) and for C, proportional to cos (p. The hori- 

 zontal component is proportional sin (9? + x) for P and proportional sin 

 (P for C. The resultant deflection of the vertical, A^, is equal to the dif- 

 ference of the horizontal components divided by gravity, so that 



A^. = -feM p^^^^ + ^ J - ^ 

 L e2 r^ J 



kM 



R 



— cos (p 



[[- 



2 - cos <p + 

 r 



©'] 



rT| + COS <p 



L 6 ^ . 



kM 



sm <p 



\\- 



R 



COS v? + I - 



©J 



sm ^ 



A^ = - 



g 



kM 



sm <p 



\[ 



1 — 2 - COS v? + ( - 



01 



3 — sm if 



(7-45o) 



As the earth's radius, R, is 6.37 -lO' km and r for the moon = 3.8-10* 

 km and for the sun = 1.5-10^ km, it follows that R/r is 1.7-10"^ for the 

 moon and 4.3-10" for the sun (and may, therefore, be neglected). Then 



SkMR / „ . r 



A^z = - 



A^A = 



2r3 



SkMR 



2r3 



cos 2v? + ^ j milligals 



sin 2(p milligals, and 



A\P = - 



ZkMR 



2gr^ 



sin 2(p 



(7-4bb) 



The coefficient 3kMR/2/ is 0.0824 milligals for the moon and 0.0376 for 

 the sun, and the coefficient dkMR/2gr^ is 0.0173 arc-sec. for the moon and 

 0.0079 arc-sec. for the sun. 



Tabulations of lunar and solar tide-components have been published 

 by K. Jung.^^ R. D. Wyckoff" calculated the total amplitude for the two 

 principal lunar and solar semidiurnal, the principal lunar and solar di- 

 urnal, and the lunisolar diurnal tides for Pittsburgh, and found 0.167 



»»Handb. Exper. Phys. 26(2), 322 (1931). 



•7 Trans. Amer. Geophys. Union, 17th Ann. Meet., pt. I, 46-52, July, 1936. 



