ON QUADRATURES AND INTERPOLATION. 359 



If the second difference is small, the correction for the alternate difference 

 is very easy. A particular case of this formula was suggested by Sir 

 John Leslie * for continuing either Briggs' or Vlacq's logarithm tables. 

 For the difference between the logarithms of (say 11000 and 11002 is the 

 same as between those of 6500 and 5501, and the formula given above 

 enables us to find from it the difference between the logarithms of 10099 

 and 11001, and so on, so that the odd series can thus be quickly calculated, 

 and the extension of the table to double its former range effected with 

 very little more than mere copying. ' Mere copying,' however, when 

 applied to an extensive table of logarithms, is so laborious, and such 

 a fruitful source of eiTor, that this application of the method has never 

 been made. Nevertheless, it is a very convenient process for interpolating 

 tables of physical or other observations, where the second difference is not 

 very considerable. 



Section 8. — Definite m- Tdlnilar Intei-polatioii, 



The problem of definite or tabular interpolation is this : given a table, 

 or a set of differences, corresponding to a given equal interval ; to con- 

 struct from it a table corresponding to some other equal interval, usually 

 a sub-multiple of the former. For example, suppose a function to be 

 tahulated (or given by differences) for every ten minutes, and that it is 

 required to find the means of tabulating it to every minute ; it would be 

 possible to interpolate separately to every intermediate value ; but this 

 ■would be unreasonably laborious, and what is usually needed is to find a 

 set of differences corresponding to the reduced interval, from which the 

 table may be set up, either by arithmetical summation, or by a difference 

 engine. 



Let A be the symbol of differencing for the wider interval, and o for 

 the smaller interval, and let 



E = l + A,e = l + cZ 

 then the fundamental relation between the two scales is E = e", and the 

 analytical problem is simply to express a selected function of e in terms 

 either of E, or of a selected system of functions of it. The equation 

 E = e™ arises simply from a comparison of the original and interpolated 

 series, namely, 



original series Uq .....Ui.. ......... U2 . • . • • 



interpolated series, Mq'^i ^2 • • • '^m-i %n'"'m+i '^,,1+2 • • • ^2m-i """im • • • • 

 Where Uq = tto, E Uq = Uj = U„, = e™«o, 

 E^Uo = E Ui = U2 = U2;„ = e"'((,„ = e^"- «o, &c. 

 All the remainder of the work consists, firstly, of algebraical _ trans- 

 formation ; and, secondly, of the actual arithmetic. The kind of 

 transformation needed turns, firstly, upon .,how the E's are expressed ; 

 secondly, upon how it is desired to express the e's. Thus the E's m.ay be 



expressed either by a mere tabulated series Uq, Ui, U2, U3 , that 



is, by powders of E ; or by Uq and its ordinary differences, that is, by • 

 ascending jiowers of A or E — 1 ; or again by symmetrical differences, 

 that is, by powers of Z = A^ : (1 -f- A) ; or even by ascending differences, 

 that is to say, by powers of A : (1 + A). So it may be desired to express 

 the interpolation by powers of e, oi d or e — 1, of ^ = cl"^ : (1 + (^) or of 



* See the article ' Logarithms,' in the recent editions of the Bncycl. Bi'itannica. 

 The article originally appeared in the supplement to the fourth edition. 



