Rigidity of the Earth. 491 



value for n. Since it is generally asserted that the rigidity 

 of the earth is about the same as that of steel, I shall give n 

 the value 8xl0 8 grams per square centimetre, which is 

 nearly the value for steel. 



In order that the results may have any worth, the 

 expression for p should be chosen so as to satisfy, as nearly 

 as possible, all that we know about the earth's density. 

 Thus we must make the mean density about 5'5iu and the 

 density at the surface about 2'5w, where iv denotes the 

 density of water. It will be found that both these conditions 

 are satisfied by the equation 



p = wllO-7-5 



( W -T 5 h) < 3i > 



With this expression for the density, the differential 

 equation (21) can be satisfied by taking 



^•2 ^,4 „S 



« = A +A 2 - s +A 4 - 4+ A e - + .... 



= 2A„,- (32( 



where m may be zero or any positive even integer. 



That the boundary conditions can also be satisfied by this 

 series will be seen in the course of the work. 



T 



Let x be written for -. Then on substituting: for p and e 

 in equation (21) we get, after multiplying through by a, 



L ' {5\ m + 2/ 3 5 (m + 2)(/» + 7) J 



-" (m-2)m(m + 3)(m + tyx™- 1 ! +7'5wha\v 3 = 0. (33) 



By equating the coefficient of x z in this to zero, we get 

 2.4.7.9 A . _ . .C_/ 3 m \ . 10 



nA i + 607r 7 aW{2(2-| -5L) A.-| A j 



6 



+ l'Dicha 2 = 0, (34) 



Also, when m is zero or any positive even integer, we 

 find, by equating the coefficient of x m + 5 to zero, 



en 2 3 f 10 . 3 m 2 +9m + 4 . ) 



-- 6 (m-f 4)(m + 6)(m + 9)(m + ll)A m+6 = 0. (35) 



Now 



4 



7,7ryrt(5"5w) = g dynes. 



