ox THE AFTER-SnOCK'S OF EAT^TIIQUAKES. l\J 



§ 10. To deduce theoretically tlie time-relation of the activity (or 

 frequency) of after-shocks, it will he assumed, firstly, that the activity 

 at any moment is proportional to the magnitude of disturhance in the 

 o-eotechtonic conditi(m then existinü' at or near the (^rii^in of the initial 

 e:u'th(|uake; and, secondly, that the reduction of this magnitude 

 depends on the corresponding activity of after-shocks ; that is, it is 

 assumed that the after-sh<x-ks remove so many points of instahility or 

 Aveakness at the (origin. It must also be supposed that each of the 

 after-shocks has its own series of secondary after-shocks depending 

 on its magnitude. 



I^et // = the aetivity of after-shocks at any instant of time ;r : 

 ?)?, = the corresponding magnitude of distin-hance at or near the origin 

 of the initial earthtpiake. expressed in any arhitrnry measure; and /,', /.' 

 be constants. We then obtain the following two ecpiations : — 



y = l'.in; 

 — dm. = h'.y. (It — l/'.y.dx, 



whence, l)v integration, sup])osing /.:" to be constant, 



y = n.h-\ («) 



II and /) being constants. 



Tlie logarithmic time-decrement of the activity of after-shocks 

 seems to l)e a likely one, when considered from the analogy of certain 

 physical phenomena, but the result obtained by applying equation (<i) 

 to the records of after-shocks of the three recent great earthquakes is 

 not yery satisfactory. 



A formula wliich gives nearly satisfactory results is the f )llow- 



ing, — 



A- 



in wliich // is the frequeney (or activity) at time .r, Ic and h being 



