and Mathematics to Seismology. 189 



Luni-Solar Effects*. 



§ 21. Another possible cause affecting the indications of 

 pendulums and spirit-levels is the gravitational action of the 

 heavenly bodies, especially the sun and moon. If we regard 

 the earth as a sphere of mass E and radius a, and suppose the 

 moon's mass to be M and its distance from the earth R, there 

 exists in the earth a system of bodily forces of which the 

 principal come from a potential 



^(M/E)(a/R) 3 (r 2 /a)(3cos 2 6'-l)/2, . . . (39) 



where g is " gravity " at the earth's surface, neglecting 

 " centrifugal force." The moon is supposed to lie in the line 

 = 0, the earth's centre being origin, and r, ordinary polar 

 coordinates. As explained in art. 812 of Thomson and 

 Tait's ' Natural Philosophy,' there results at the earth's surface 

 a component force radially outwards 



#(M/E)(a/R) 3 (3cos 2 0-1), 



and a component along the tangent 



T' = 3#(M/EXa/R) 3 sin0cos0, .... (40) 



directed towards the point under the moon (6 = 0). 



Both components being small compared to g, the direction 

 of gravity is, owing to the direct attraction alone, deflected 

 through the angle 



ty' = tan-43(M/E)(a/R) 3 sin0cos0} . . (41) 



from the vertical. The angle being very small may be re- 

 placed by its tangent. 



Thomson and Tait suppose 



(M/E)(>/R) 3 = 10- 5 /182, .... (42) 



and thence draw the following conclusion: — "the plummet is 

 deflected towards the point of the horizon under either moon 

 (0 = 0) or antimoon (0 = tt), by an amount which reaches its 

 maximum value . . . G^'017 when the altitude is 45°." They 

 add — "The corresponding effects of solar influence are of 

 nearly half these amounts." According to this conclusion 

 direct luni-solar influence should make itself felt in any system 

 of pendulum or spirit-level observations in which the accuracy 

 is of the order 0""02. 



§ 22. The data on which the above calculation is based are 

 pretty accurately known, which constitutes a reason for 



* Strictly the problem is a dynamical one ; as yet only an u equili- 

 brium " solution is available. 



