Manchester Memoirs, Vol. xivi. (1902), No. 15- 3 



where 



r^ = „r^ + y + s^. 

 These statical elastic solutions are not more easy of 

 mental conception than is the rigid dynamical description 

 of an impact between bodies, and I shall omit their con- 

 sideration in what follows. Also I do not concern myself 

 with standing oscillations which do not depart greatly 

 from a form of possible equilibrium displacement, but 

 confine the investigation to those modes of displacement 

 which are capable of regular progressive transmission. 



Writing 



\ + 2^ = pk^y and IX = p/i^, 

 we notice that any solution of A"^ = () or constant, can be 

 made to give a suitable value for F ov /, by introducing 

 kt or /it in a proper manner. Thus, if 



0{(.v - .r')cosa + {y -y')s,ina + h} 

 is the solution of a^0 = O, and we seek a solution suitable 

 for a wave propagated normally into the earth, we should 

 replace rr by 



scosh/3 -/^/sinh/3, or 2coshy->^/sinhy, 

 it would then be clear that to satisfy the conditions when 

 ^ = 0, we should require 



/&sinh/3 ^y^sinhy, 

 and that we should have two wave-systems travelling with 

 velocities /^tanhjS and ktanhy respectively. 



If, on the other hand, we desire to represent a pro- 

 gressive surface wave, we should write the amplitude 



{x - a:')cosa + (y -y)sina - /^/sech/3 + /ctanh/3, 

 or 



{x - A:')cosa + (y-y')sina- //^sechy + tztanhy, 

 and the waves would, to satisfy the surface-conditions, 

 usually combine into one compound system travelling at 

 the rate 



^ = /^ sech /3 = /i sech y . 



