484 Mr. J. Prescott on the 



density above it appears that the actual earth must yield 

 appreciably less than a sphere o£ uniform density would 

 yield. 



Let the position of a particle of the earth be indicated by 

 polar coordinates, the earth's centre being the pole and the 

 radius vector through the tide-producing body being the 

 initial line. Assuming the layers of uniform density to be 

 spherical in the unstrained state everything will be sym- 

 metrical about the initial line or axis of reference as we 

 may call it. Let r, 6 be the coordinates of a particle in the 

 unstrained state; r-fe, 6 + rj its coordinates when strained 

 by the tide-producing force. Let i, /denote the strains in 

 the directions of e and 77, and g the strain at right angles to 

 both of these. Let 8 denote the cubical dilatation, co the 

 rotation in the plane of r and 6. 



Then 



.= £-■, (1) 



Or 



<r + 9(><H a , 



V 8e)-rS0 



= ^tj 4- - approximately, (2) 



_ 27r{(r-fe)sin (fl + 77) — r sin 0} 

 9 ~ 2th- sin 



rri cos 6 + e sin 6 . , 



= — : — 7; , approximately, 



r sin 6 ll J 



= fl/cot0+,- 9 .......... (3) 



h = i+f+g 



Oi' r 0(3 ' 

 = - 2 ^~ (r 2 €) + — ^ J^ ( v sin 6) . . . . . (4) 

 Also it is not difficult to prove that 



