and Mathematics to Seismology. 193 



instance as each the combined product of surface harmonic 

 and arbitrary coefficient — for a perfect sphere acted on by 

 bodily forces. Thus for terms in V/ in the displacements, 

 we have only to take the results (36), (37), and (38), I, c. 

 pp. 264-265, and in them replace V* by Y/<7i and Si by zero. 

 Doing so, we find for the components of displacement, 

 measured respectively in the directions of the fundamental 

 polar elements dr, r dd, r sin 6 8$, the following results : — 



gpr 



10a(m 



aiCTi 



; f g a g (5m + 7i) \ 



+ n) \ 3m — n J 



+ ^ %(2?; , 1) D X ^'^ 10 '"^-1) 



-mre(4; 4 + 4«' 3 + 34i 2 + 29/ + 10) + « 2 (8i 3 + 8i' 2 + 13i-2)} 

 - r^a-t+H { 10mH(i + 2) - mn (i? + 1? - 2t + 1) - n\ii - 3) }] 



+ j^±!0 gggg? [aV-,- m(, ' t tf;- w _^>{m(,-+ !)-»}] , (48) 



-i*** '"=is-4^ J • • • (49) 



where 



J) i =5(m + n){m(2i 2 + U + 3)-n{2i+l)}, . . (50) 



^=2^Slir^ + ^{10m»(.-- l)P + 8) 



-mn(^' 3 + 12i 2 -6? + 17)-n 2 (8z 2 -17)} 

 -^- 1 6t- i + 2 {10m 2 /(i + 2)-mn(4i 3 + ^ 2 -2i + l)-w 2 (42-3)}] 



+ g^ L \-_i r'+^m^ + SJ-^J. (51) 



§ 26. Before utilizing these results we must determine a» in 

 terms of V/, which is easily done as follows : — The surface 

 being supposed originally spherical, the terms ^aia 1 in (44) 

 arose solely from the action of the disturbing forces, and so 

 must be identical with the variable terms in the surface-value 

 of u. 



Thus writing a 4- 2oj<j? for r in the principal terms in (48). 

 and a for r in the subsidiary, then equating the separate 

 harmonic terms to the corresponding ones in Stf*^ and re- 

 ducing, we find 



_ pa i+l Y/i\(2i + l)m-n\ + {2{i-l)n((2i 2 + ±i + 3)m-(2i+l)n)} 

 1 gpg 15i(2i+ l)m 2 -(8i* + t)i 2 -2i-y)mn + (4i* -2P -di-3)n 2 ' 

 n 5(2i+ l)(3m — n){ {2p4-4d + 3)w»— (2i + l)n}. 



(52) 



