and Mathematics to Seismology. 195 



this critical value of n has the very ordinary value 21 X 10 7 

 grammes wt. per sq. cm. 



Thus if such a value as m/n-=.2 were admissible the con- 

 tingency of a,i/Yi becoming enormously large for a high 

 value of i would be quite a possible one. Unless, however, 

 as I have previously pointed out, n/m be very small,, the 

 term in u, independent of the angular coordinates, would in 

 a body of the earth's mass be enormously greater than is con- 

 sistent with the mathematical theory of elasticity. Therefore, 

 so long as the present calculation is justifiable, the denomi- 

 nator in the value of a,-/V/ can differ but little from that 

 occurring in (54), and we are thus thoroughly justified in 

 neglecting all the higher harmonic terms in the potential 

 relative to the term containing the second harmonic. 



§ 28. As a small departure of n/m from would exercise 

 but little influence on numerical values, it will be best, as we 

 are dealing with data so uncertain, to neglect n/m altogether. 



Thus, putting 



i=2, cr 2 = P 2 , n/m = 0, 



V 2 '=^(M/E)(«7R 3 ), (57) 



we have for the displaced surface 



r=.a + a 2 ¥ 2l ...... (58) 



where 



ad a ■ 



= j| (gpa/*) (M/E) (a/R)*+ { 1 + A (gpa/n) } . (59) 



An equivalent result is given in Thomson and Taitfs i Natural 

 Philosophy/ art. 840. The result will also be found, along 

 with that answering to i = 2, m = 2n, in Mr. Love's ' Treatise 

 on Elasticity/ vol. i. pp. 302, 303. 



The corresponding surface displacements are 



u = ~a¥ 2 (gpa/n)(M/E)(a/Ry/{l + 2gpa/l9n}, (60)* 



v a = - ^a sin0 cos 6 \gpa/n) (M/E) (a H) 8 / {I + 2g pa/ 19 n\.(6l) : 



The term in u independent of the angular coordinates abso- 

 lutely vanishes for n/m = 0, and both components of the 

 surface displacement, and so the resultant displacement itself, 

 are reduced owing to the self-gravitation in the common 



ratio l:l + 2gpa/(19n) (62)* 



* The material being as here incompressible, it may be proved that 

 for any value of i in (45) the displacements are everywhere the same as 

 in a sphere of radius a, over whose surface act purelv normal tractions 

 eo[ual to paia-fVi'^- { 1 + (</pa/n)i/(2i 2 +4Z+3) } . 



