10 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi..xiv. 



The values of e±"W380 contained in Tables VIII and IX were computed in the manner just 

 described from the values, £>±W36o fi^Q latter function was evaluated for the first 10 decimals 

 of the exponent by successive multiplication of the appropriate factors taken from Tables 

 IV and VII. The values for the remaining decimals of the exponents were obtained by sub- 

 stitution in the exponential series. The product of the two factors thus obtained is the re- 

 quired result. Checks were applied in the usual manner, also by comparison with the values 

 given by Gauss.' 



Values of the exponential function previously obtained provided an excellent check on the 

 fundamental values needed in the computation of sin x and cos x. These values were com- 

 puted at intervals of 0.1 from 0.0 to 1.6, inclusive, by direct substitution in the series and 

 verified by means of the relation 



i~x^ x^ ~i [~x^ x^ n 



e* = sina; + cosa; + 2 9i+gj + -- +2 37+7T + - - (3)- 



Interpolations were made by means of the formulas 



. , . , . Ax (Ax)' . 

 sin(x + Aa;) =sm x + jy cos x 2^sma; + ,.. 



Ax . (Ax)' 



cos (X + Ax) = cos X — ry Sm X ;j|— COS X + ... 



in which Ax assumes the values 0.1, 0.01, ... according to the interval of interpolation. It 

 will be noted that the two equations together contain terms of the form 



(Ax)" sin x/n! and (Ax)" cos x/n\ 



wherein n assumes successive values of the natural numbers beginning with unity. There are 



thus two series of terms, 



1 . 1 . 1 . 



g-f sm X, ^ sm X, jy sin x, ... 



1 1 1 



■^ cos X, ^ cos X, jy- COS X, ... 



which may be evaluated by dividing the sine or cosine, as the case may be, first by 2, this quo- 

 tient by 3, the. last by 4, and so on, thus avoiding the use of large factors. The computation 

 of both functions is made at the same time, and a complete check is obtained on the tenth inter- 

 polation. The maximum difference between the interpolated values is 10 units, and as there 

 were two interpolations, the maximum error of interpolation is of the order of magnitude of 

 20 units in the twenty-fifth decimal. Table X was computed with the assistance of a com- 

 puting machine by substitution in the trigonometric expansions for sin (x + Ax) and cos (x + Ax), 

 and verified by assigning various values to x and Ax; also by forming the smn of the squares 

 of the sine and cosine for several values of the argument. The values of sin x and cos x con- 

 tained in Tables XIII and XIV were computed by substitution in the respective series and 

 verified by means of equation (3). 



Writers on interpolation emphasize the importance of interpolation by differences while 

 not much attention is given to interpolation by means of derivatives. This procedure does 

 not seem justifiable as the time lost in retabulating and differencing the quantities is some- 

 times much greater than the loss of time due to the possible increased labor and difficvdty of 

 computation by the derivative formula. Furthermore the check provided by the derivative 

 formula is much more rehable than that of the difference formula when both the interval of 

 interpolation and the interpolated values are large. Neither method provides an absolute 

 check, for experience proves that positive and negative errors of equal or approximately equal 

 magnitudes very frequently escape detection. The same is true of the various methods of 

 mechanical quadratures which could be used for the same purpose. 



' Loc. cit. 



