Manchester Memoirs, Vol. xlvz. {igo2), No. 15. 7 



With these in the compound wave, we are to combine 

 the relations, from (5), 



k = qco^\\fi, /; = ^/coshy. 



We have to consider two special cases of simple waves, 



which consist of solutions possible with F functions only, 



or with / functions only. These arise out of the a's. only. 



The first case is that of a wave of pure strain, often 



called the sound wave. In this case 



rti = «2 = <^s = 0, 



and we have the condition from (5) 



2/^^cosh73 = k"-, 



giving the rate of propagation J2h. 



The second case is a rotational wave without conden- 

 sation, and it requires a = 0, or r=i) '\n (6.). 

 We therefore have 



Ci =/>cota, 

 ci2= — ^tana, 



as = 2/cot2a (9). 



The displacement in this case is in the wave-front and 

 parallel to the Earth's surface, and we have no limitation 

 of the rate of propagation, except that it must be less 

 than /i. These waves will be referred to later. 



We remain with two kinds of compound waves, that 

 given by the //s in (7), and that given by the a's, after we 

 have removed from (8) the terms in / which have just 

 been considered. These conditions then give 



tanh^y 

 I + tanh-y 



tanh^y , , 



a = - 4ai- — — — TTT-v,, (10), 



^ (1 + tanh-y)^' ^ ^' 



which should be compared with the d relations in (7;. 

 Each of these indicates a wave which satisfies the surface- 

 conditions and which may travel at any rate less than //. 



