e 



324 EEPORT— 1880. 



These are far more useful working formula for large logarithmic tables 

 than any depending upon differences.* 



This process receives an evident simplification when the function 

 tabulated is a simple integral. In that case (px is replaced hj fxxdx, (p'n 

 by x^i ^^d so o^- I*' ^^ therefore of useful application to the direct tables 

 of elliptic integrals, like Legendre's ; but it will not apply to the tables 

 recently printed by the Association, because in these the functions 

 tabulated are not mere integrals, having simple differential coefficients, 

 bat are complicated functions, of which the differential coefficients are 

 still more complicated. Nevertheless, whenever A = V (1 — sin^6» sin^^) 

 is known or discoverable, the formulas of this article may still be of use 

 for interpolating to F^ and E^. 



These, and other methods of interpolation derived from the properties 

 of the function itself, are of especial advantage at those parts of a table 

 where the rate of change of value of the function differs widely from that 

 of the argument. Immediate examples of this are afforded by the tables 

 of logarithmic sines and tangents for small angles. In these and many 

 similar cases the general methods of interpolation, dependent upon finite 

 differences, are practically useless. It should not be forgotten that there 

 are two kinds of difficulty met with in certain of those cases — one in 

 which a very small change in the argument corresponds to a very great 

 change in the entry, which introduces actual indeterminateness into the 

 attempt to interpolate to the latter — and another where the amounts of 

 change fairly correspond, but, the argument varying uniformly, the rate 

 of change of the entry varies rapidly. Each case has its converse. An 

 example of the former is to be found in the attempt to determine a small 

 angle from its cosine — in which case accuracy is impossible ; an example 

 of the second is to be found in the problem of finding the logarithmic sine 

 or tangent of a small angle, in which the only difficulty is the arithmetical 

 one arising out of the particular system of tabulation in common use ; 

 that is to say, a difficulty arising from our having selected, for reasons of 

 a general character, a plan of tabulation not suited to the work to be done 

 in the particular case.f 



III. — Genekal Considerations Relating to the Application of Finite 



DiFFEKENCES TO INTERPOLATION AND QUADRATURE. 



When all that is known of a fanction is, that it takes certain definite 

 values for corresponding definite values of the independent variable, 

 separated by finite intervals, the function itself, and consequently all its 



* Avery full account of this method, with copious examples, is given by Legendre 

 under the title ' Methodes diverses pour faciliter I'lnterpolation des grandes Tables 

 trigonomt^triques ' in the Cunnaissance des 'Iemj)S for 1817, p. 302. The formulfe for 

 logarithms were first given, to one term only, by Dodson, in his ' Antilogarithmic 

 Canon,' dated 1742. The second is also given by Legendre, ' Fonctions Elliptiques,' 

 vol. ii. p. 13. Several other examples will be found in the author's memoir on 

 ' ElUptic and Ultra-elliptic Integrals,' mil. Trans, vol. 152 (1862) pp. 421-427. See 

 also Legendre, op. cit. pp. 34, 61, 62. 



t The difficulty in the former example is inherent, and therefore insuperable. In 

 the latter example it is met by using a table of natural sines or tangents, and finding 

 the logarithm of the interpolated value, or else by an artifice such as the formation 



of special tables of log ^' or of log-^^" ^. These are called Delambre's tables ; but 



sin x X 



they were given long before Delambre's time by John Newton, in his THgonotnetHa 

 Britanica (jsic) dated 1658 ; in the folio edition p. 41, § 6. 



