Analysis of Micro seismograms. 467 



corrections to each v by solving equation (15) alone subject 

 to the minimum condition. 

 Taking then (15) subject to 



V 2 r+ 3 + V 2 r+2 -\- V 2 r + l +,t ,2 /-+ V 2 r -I+V 2 r -2 



being a minimum, we obtain 



r r +3= — ^-2 = ^, v r+2 = — v r -i = qi\, 1 



V r +1 = - V r ~ q 2 \ h • • (16) 



where \=R r /2(l + q l * + q 2 2 ). J 



On repeating with the u's substituted for the corresponding 

 ?/'s the original series of operations, a new series of resi- 

 duals was obtained whose average value was double the 

 original R's. The values of y were therefore corrected by 

 subtracting "from them half the total values of the w's as 

 given by the above formulae. These corrected values of y are 

 given in column (7) and the resulting residuals in column (8). 

 It is seen that all large residuals have been removed, and 

 this has been done without altering any y by more than 

 two units, an amount well within the probable limits of error 

 in reading the curve. 



Seeing that the values of q i and q 2 are about unity, it is 

 clear that an alteration of a y by a single unit may give rise 

 to a unit residual ; hence a complete removal of the residuals 

 is difficult : but the fact that of the residuals only two are 

 equal to 4, six are equal to 3, and the remainder are 1 or 0, 

 shows that all the equations of conditions are now well 

 satisfied. 



6. We now proceed to determine the amplitudes and 

 arguments oE the component terms. With the corrected 

 values y we form fresh values of a and Ea (Table II.}. 

 This table also contains the values of P 1 r = Ea r — z 2 a r (col. 4). 

 Column (5) contains Si (PV-i — PV) where Si — 1/(2— z x ) 

 X fa — z 2 ) =0*71, and column (6) (^(PV-i +P 1 ,) where Ci = 

 1/2 sin ^ 1 (^ 1 — r 2 ) = 1*20. In column (7) the corresponding 

 arguments \t + a 1 = <f>i are tabulated, and in column (8) the 

 amplitudes c x . Column (10) contains the corresponding- 

 arguments 4> 2 for the terms involving # 2 , and column (11) 

 the values of c 2 . These values nre calculated with S 2 = 

 }/(2-z 2 ){z 2 -z 1 ) = -'61 and c 2 = V2 sin <9 2 (r 2 -^)= -P53. 



The last column gives the values of c , which are obtained 

 by subtracting From y' r the sum of the values of Si(P 1 r -i — P 1 ?-) 

 and S 2 (P 2 r- 1~ P 2 r)- T ne constancy of the values so obtained 

 affords a check on the accuracy of the work and a further 

 justification of the assumptions made. 



2H2 



