RELATION OF VARIABLES. 117 



AC" = - [5 X 1.99 + 1 (- 0.73)] = 1.024 (62) 



which are second approximations to the quantities that must be added 

 to A', B*, and C', respectively, to equal A, B, and C. Substituting 

 the original averages A\ B\ and C' plus the quantities AA", AB", 

 and AC'* respectively into equations (56) and (57) gives second ap- 

 proximations (a" and c") to a and c. That is 



a" = (12.47 + 0.199) - (12.38 + 0.037) = 



ai -f (AB'' - A A") = a^ + A^a" = 0.09 + 0.162 = 0.252 (63) 



c" = (12.47 + 0.199) - (8.93 + 1.024) = 



ci + (AB^' - AC") = c^ + A^c"' = 2.715 (64) 



Substituting a" and c" for a and c in equations (53), (54), and (55) 

 gives 



AD" = ^ [2 X 0.252 + 5 X 2.715] = 1.564 (65) 



AE" = - [2 X 0.252 + 3 X 2.715] = 0.961 (66) 



AT" = ^ [5 X 0.252 + IX 2.715] = 0.442 (67) 



y 



which are second approximations to the quantities that must be 

 added to D\ E\ and F'' to give D, E, and F. Substituting the original 

 averages D^ E*, and F^ plus the quantities AD", AE", and AF" into 

 equations (58) and (59) gives second approximations {d'^ and /") to d 

 and /. That is 



d" = d^ + (AE" - AD'O = d^ + AW"' = 1.99 - 0.603 = 1.387 (68) 

 /" = /' + (AE" - AF'O = /'• + Aif' = - 0.73 + 0.519 = - 0.211 (69) 



If this process be continued, the successive approximations will 

 converge to the required corrections AA, AB, AC, AD, AE, and AF, 

 and to the required diflferences a, c, d, and /. But, unless two com- 

 puters are available to duplicate each other's work, a second process 

 of computation is needed to check the results obtained by the first 

 process. Beginning with the third approximation, this is accom- 

 plished by computing directly the differences A^A"', A^B'", A^C'" 

 etc., that must be added to A A", AB", AC", etc., to obtain A A"', 



