4 GwYTHER, Rate of Propagation of an Eajth-Tremor. 



If a solution has the form 



log{(;c - A-')cosa + {y -y)sina + iz\ 

 or 



{ {x - A:')cosa + {y -y)sina + /z}"*, 

 etc., it gives rise to infinities along the line 

 (.r - A-')coso + {y -y)sina = 

 in the free surface. 



Since we are really only concerned with real values of 

 the displacements, I shall consider that (and therefore 

 /^and/) are expressed in the form 



%( U+ iV) + 0( U- iV)'\ + 40'( U+iV)-<p'{U- iV)\li. 

 If we consider now 



log{ (a- - a')cos a + {y -J^'')sin a + /z} 



+ log{(.v - jv')cosa + {y -y)sina - z's}, 



and by integration adapt this for cylindrical polars, we 

 obtain 



l0g{3+ Vr2 + Z"), 



where 



r^^{x-xJ^-{y-)!)\ 



While differentiation with regard to z leads to the form 

 (^- + 2-2)"^, and so on. 



Analogy with M. Boussinesq's conclusions indicate 

 that infinities arising in this manner are not necessarily to 

 be excluded, but that they may only be admitted in 

 special forms. 



The form of wave most likely to be represented in 

 this manner is a form which can travel alone, but if a 

 single pole of any character can be shown to lead to results 

 satisfying the conditions, we can introduce such poles 

 periodically and obtain solutions of the form 

 log tan{(jc: - a:')cosa + {y -y)sina + iz) 



+ log tan{(A: - a;')cosa + {y -y)sin(t - iz). 



