E. G. Brown. — Standard Functions in Interpolation. 425 



POSTSCEIPT. 



A more extended table of the standard terms, which are 

 equivalent to the Lagrange interpolation, has now been com- 

 puted, and is given herewith. The form differs slightly from 

 that of the shorter table in the paper, the common factors 

 in each line being separated. These common factors are 

 connected by the ratio VltI when n, the order of the factor 



J 2?i 



sought, is even, 2L_ or \ when n is odd. Thus the table can 



be extended. Hence the common ratio is about ■§■. The co- 

 efficients of the standard functions are also best extended by 

 a similar process, but it is too complicated, and, moreover, 

 obvious, to be here given. It will be necessary to construct 

 a similar table for the Newton differences if it is required to 

 treat functions of which the values cannot be given on each 

 side of any interval. 



It is to be noticed that the results of these operations are 

 simply identical with the results of the orthodox methods, and 

 where the latter fail for want of convergence these operations 

 fail also. The advantage which the reductions possess is 

 merely the shortening of the formulae. 



The practical use of this reduction lies, I think, in the 

 power it gives in stating the values of functions in tabular 

 form, where space forbids the use of more than a 2- or 

 3-figure argument. An example will now be given of 

 the power of the standard method in this respect. First, 

 we may notice the fact, which is evident from the table, 

 that the Lagrange terms are, all of them, nearly pure stan- 

 dard parabolse. That is to say, their maxima are nearly 

 the same as the value when x = \. Consequently, the 

 value or number represented by any of the Lagrange dif- 

 ferences may amount to \ times the common factor for that 

 difference. 



The example taken is that of the log. gamma function, 



which was tabulated by Legendre for the unit interval 1 . . 2 



in a 3-figure argument, 12-figure table, the interpolation being 



indicated by third differences, which were tabulated thus 



(twelve decimals understood) : — 



T n A 1 A 2 A 8 A 4 



Log. Gamma. 



(1) • 119 



• 120 974 783 415 092 171 440 853 605 919 768 2 



• 121 



The last difference does not seem to have been given, and 

 it is not necessary for Lagrange's method, and twelve places, 

 but would have been needed for thirteen piaces, since it 

 reaches the value 6 or -^ • £ = 0T4 units in the twelfth place. 



