and Mathematics to Seismology. 181 



hence 



where ?i and r 2 are the inferior and superior limits of r in 

 the integration with respect to a/ m 



Suppose the origin vertically over the point where the 

 slope is to be found, or x=y = 0, and draw the axes of x and y 

 parallel to the sides of the loaded rectangle. Take for the 

 coordinates of the corners of the rectangle — 



®i> 2/i 5 #2) Vi 5 #2, 2/2 5 x 1} y 2 ; 

 and suppose 



^2>^i ? y a >yi. 



The following result is then easily obtained from (13) : 



4™ /^y=y=o =2 ^ fa + *V + *a 2 + z*) (y x ± *Jy? + x? + z*) 



p \dx)*=* V) g (y x + V yi * + ^ 2 + ^ 2 ) (y 2 + vV + ^ 2 2 + * T ) 



+ ^2 f _j_ / ^ _ .3/1 \ 



+ ^ a +^\"vyT«+^+^" ^p+fep)}* ■ (14) 



This combined with the corresponding expression for divjdy, 

 which can be written down by symmetry, supplies complete 

 information as to the slope at all depths. By putting z = 

 in (14), or by direct calculation, we get 



dw 



(at origin) ^ fa^Vg b*± *frj!+5 g ) (y»+ V.Vi* + * . 



. _2> 



-7— litli UlJ.ii.lIJl = — 1U& _ 2__ ^ • ~ * / t-i k\ 



* 2 ™ (yi+ VyT+^P) (y 2 + VsF+^J' ' ' (15) 



where y 1% y 2 must be treated algebraically. Thus ABOD 

 representing the loaded rectangle, DM, CN perpendiculars 

 on Ox, we have, in the case shown in fig. 1 : — 



^ fat 01_ (l-q)/> log (OP + DM)(OB + BN) 

 dJ atV >~ ^r^ lo g (UA+AM)(00 + ON) ; • ( 16<l ) 



in the case shown in fig. 2 : — 



d ™ ht0) _ a-v)P hc (OD + DM)(OB-BN ) 



dx (at Uj ~ ^^T l0 « (OA-AM)(OC + CN)- ( 16? ')* 



* An equivalent but somewhat longer form for the logarithm 

 (leading directly, however, to [17]) is given in ecm. (7) artfsiS of 

 Thomson and Tait's ' Natural Philosophy.' 



