Rigidity of the Earth. 487 



From equation (10) we get 



(2ra> sin £>) = -. j ~ (A) -2re | (2 sin cos 2 0-sin 3 0), 



9 



55' 



.... (14) 

 whence, 



? P { sop (2,,<u sm tf) ) = ? i i? ^ - 6 ' ,(? j sin **• 



.... (15) 

 Also, 



We have yet to express W in terras of r, 6, and e so as to 

 substitute its value in (13). 



Let P 2 denote the spherical surface harmonic (cos 2 # — J). 

 Then 



e=r*P 2 (17) 



And by Routh's Statics, Part II. Art. 176, the potential of 

 the earth's attraction at an internal point whose coordinates 

 are r + e, 0, is 



+ — /- 1 pr 2 dr+ \ ' \ Pl -(r"e)dr 

 = 47ryl prdr+ g — 1 p — rfr 



+ ^ ( i_,p 2 )j%^ + *$&£, j^,, 



.".'... (18) 



where 7 is the constant of gravitation and a the mean 

 radius of the earth. 



If m denotes the mass of the tide-producing body, and 

 d its distance from the earth, the potential of its tidal 

 action is 



3<ymr 2 



2d* 



(co$z0-i) = ihr 2 P 2 say, 



37m 

 where h denotes the constant -~p" 



2K2 



