﻿the Rigidity of the Earth . 221 



where n is the coefficient of rigidity, 



m „ „ incompressibility, 



V is the potential of attraction, 



$=^[x <2 {b* + c <2 )+yXa* + c' 2 )+z' 2 {a' 2 + b' 2 )--2xyab-2xzac-~2yzbcl 

 p means the mean density, 



dx dy dz ' 



V 2 is the known Laplacian symbol. 



V may now be divided in two parts, the first corresponding 

 to the potential of the unstrained body, the other containing 

 all the terms which arise in consequence of the deformation : 

 so we may write 



v=v 1 +v a . 



On the other hand, with the help of the formula 



a 2 + 6 2 + c 2 =l, 



we may write for 



6 2 + c 2 , 1-a 2 , 



a 2 + c 2 , l-£ 2 , 



a 2 + & 2 , l-c\ 



Now if the Z axis of coordinates coincides with the polar 

 axis of the earth corresponding to the unstrained state, the 

 k potential 



V 1 + pO) 2 (<27 2 + ?/ 2 ) 



exerts no deformation, so that in equations (I.) V reduces to 

 V 2 and <£ becomes 



<£= -^- [a? 2 a 2 + ?/ 2 6 2 + 2 2 (c 2 - 1) + 2xy ab + 2xz ac + 2yz be]. 



We see that <£ is a solid harmonic of the second degree. 

 But the disturbing potential is 



To find it, we proceed after the method of Prof. Darwin*, 

 and obtain readily 



w - v « + *-I5h^-*- • • • (IL 



* " On Bodily Tides," Phil. Trans, for 1879, p. 9. The calculus is 

 u literatim " the same. 



