Rigidity of ike Earth. 497 



For when we know the po.-ition of the tide-producing body 

 and the direction of the plumb-line, we can calculate the 

 height of the tide on the equipotential surface relative to 

 the solid earth, except of course when the tide-produeino- 

 body is at the zenith or nadir of the point of observation, lor 

 then the deviation of the plumb-line is zero, since the crest 

 of one of the tides is at the point of observation. 



In order to compare calculated results with observations, 

 it will be necessary now to find the ellipticity of the equi- 

 potential surface near the earth's surface and to subtract 

 from this the ellipticity of the solid earth. This difference 

 may be called the ellipticity of the equipotential surface 

 relative to the earth. 



Now the potential of the solid earth in the strained state 

 at an external point (r' 3 6) is (see Ronth's Statics, vol. ii. 

 Art. 196) 



T-^^+^friW. . 07) 



Since the second term in V 7 contains e and is therefore 

 small, we may put a for r in this term in the neighbourhood 

 of the earth's surface. Also 



47T7 j pr 2 dr=ga? (G8) 



Jo 



Hence, near the earth's surface, 



y/_^ , 47ryP 2 C a d 5 

 v — ~r H ^^— 1 p-j-h^e)dr. . 



But when 



p= w (l0-7-55) • • • 



we have found that 



«=A,+A 1 y + A,£ + (71) 



Consequently 



I p j-(r 5 e)dr = pr°e— I r 5 e~^dr 



r. . 15 10 (' . 7 

 =pr b e+ — g- 1 r*edr 



• (69) 



. (70) 



