6 GWYTHER, Rate of Propagation of an Earth-Tremor. 



I shall proceed with the case of the compound wave ; 

 since we can, from the results, infer the conditions for the 

 simple cases by neglecting (5). 



The general condition (2) requires that 



2 (rtisin'^a + ajcos^o) 



rtg = rti + ^2 - 



I + tanhV 



^' = ^'^ + ''-^ TTt^i^hS ' • • • (6) 



using (5). 



The surface conditions I shall take to be that the 

 surface is free. From ^ = 0, 7"= 0, when 2 = 0, we infer 

 that 



^tanh/3 + /;itanhy = 0, 

 lh = b^. 

 The conditions affecting the ^'s are now complete, 

 and give 



h = bi, 



tanhV 



bs = 2bi- 



I + tanhV 

 /^= -/^icoth/jtanhy (7). 



From the condition that ^ = 0, when ^ = 0, we get 



2/r 

 ( I - -^cosh.^j3)a + 2 («8 - ^1 - aa) sinhV = 0, 



or 



_ 4(gisin^u + a2C0s^a)tanhV 

 ^~ (i+tanhV)' 



To present the conditions affecting the a's in a con- 

 venient form, write 



a, =;r+/cota, 

 ai = r -/>ta.n a ; 



then 



tanh^ 



tanh' 



+ 2/'C0t2a, 



tan by 

 4''(i+tanhV)' ^ ^' 



