426 



Transactions. — Chemistry and Physics. 



De Morgan (Diff. Calc, p. 587) gives an abridgment of 

 this table in a 2-figure argument, to use which it is neces- 

 sary to reconstruct one decad of the original table before 

 an interpolation can be made, which obviously is a tedious 

 process. 



On differencing out the first portion of the 2-figure table, 

 which appears to be the least convergent portion, I find that 

 A 5 has a maximum of 1050 units, and A 6 of 50 units. 



Hence, on the Lagrange system, A 5 represents -^ x 1 °/ ° , 

 or 12 units, and A 6 ¥ |g x --£-, or 0-24 units. 



Now, reducing to the standard system, A 6 will be re- 

 presented by o-fg x g 5 o°o, or 0001 (VI.), with a maximum 

 value of about 0000,1 units, and A 5 by &• -\ -^ -, or 0-3 (V.), 

 with a maximum of 003 units, or 3 units in the thirteenth 

 place. 



Summing up, we need for a 12-figure table — 3-figure argu- 

 ment, three differences ; 2-figure argument, five differences, 

 or four standard terms : and for a 13-figure table — 3-figure 

 argument, four differences ; 2-figure argument, six differences, 

 or four standard terms. 



The following is a specimen line of a 2-figure table in 

 terms of Lagrange differences and in standard terms : — 



x Log. Gamma. 



(1) • 11 



12 974783415092 1687233927 60670390 761542 18304 605 34 



13 ... . 



Standard Terms. 



A 1 

 (-) 



1687233927 



P 



(-) 



30143123 



C 

 (-) 



63456 



IV. 



(-) 



190 



x Log. Gamma. 



(1) • 11 ... . 



• 12 974783415092 



• 13 ... . 



These are the actual numbers. If any one wishes to 

 check them it should be noticed that; the differences in the 

 original table are Newton's or leading differences, so that the 

 interpolation must be made with the binomial coefficients, and 

 not with Lagrange's or Bessel's functions. 



Finally, it may be noticed that if it is preferred not to use 

 the standard functions their terms might be translated back 

 into Lagrange or Bessel differences, so that existing tables 

 may be used for interpolating without losing the advantage of 

 the reduction. 



