118 MC'EWEN AND MICHAEL. 



AB'", AC'", etc., and so on. For example, the third approximations 

 found by substituting d" and /", given by equations (68) and (69), 

 for d and /in equations (50), (51) and (52) are 



AA"^ = 0.191 (70) 



AB"^ = 0.2.38 (71) 



^Qiii ^ 0.747 (72) 



But by substituting the differences AH" = d" — d^ = — 0.603 and 

 AV" = /" - /'■ = 0.519 for d and / in equations (50), (51) and (52) we 

 obtain 



A^A'" = 0.154 (73) 



A^B"* = 0.039 (74) 



^iQiii = _ 0.277 (75) 



which when added to A A" = 0.037, AB" = 0.199, and AC" = 1.024 

 (equations 60 to 62) give AA"^ = 0.191, AB"^ = 0.238, and AC"'' = 

 0.747, thus checking the results obtained in equations (70), (71), and 

 (72). 



In the same way the values o^",AD"', AE'", and AF"' are obtained 

 directly and checked by first differences A^, and, beginning with the 

 fourth approximation, computation of first differences is checked by 

 that of second differences A^, as indicated in table 2. The values 

 obtained in the second and third approximations computed by the 

 processes just described, the values obtained in the fourth and higher 

 approximations computed by first differences and checked by second 

 differences, the final limiting values obtained in the seventh approxi- 

 mation, and everything required for making each computation is 

 given in table 6, which is for the most part self-explanatory. Suffice 

 it merely to call attention to the fact that the coefficients 2 and 5 

 of equation (50) multiplied respectively by the first approximation to 

 d and / (1.99 and — 0.73), by the second approximation to A^d and 

 Ay (— 0.603 and 0.519), and by the third approximation to A^d and 

 AY (0.533 and — 0.410), and divided by 9, gives the second approxima- 

 tion, 0.037, to AA, the third approximation, 0.154, to A^A, and the 

 fourth approximation, — 0.109, to A^A, and so on. But the coeffi- 

 cients 2 and 5 of equation (53) multiplied respectively by the second 

 approximation to a and c (0.252 and 2.715), the third approximation 

 to A^a and A^c (— 0.115 and 0.316), and the fourth approximation to 



