ON QUADBATUBES AND INTEEPOLATION. 323 



II. — Interpolation et Known Properties of the Particular Fitnction. 



Use of Ta\lor's Theorem. 



The calculus of finite differences is of such general and easy appli- 

 cation that its nse has sometimes superseded other methods which are 

 preferable in particular cases. This is especially true of the ordinary 

 logarithmic tables, including those of circular functions. "Where the 

 first difference is constant, or nearly so, it is sufficient to use proportional 

 parts ; but when the second and third differences have to be taken into 

 account, it is frequently prefei-able to use the properties dependent on the 

 form of 1;he function. The advantage of this is very marked in the in- 

 verse use of the table, where the argument has to be found from the 

 entry. 



The formulpe required may in some cases be obtained by algebraical 

 transfonnation of the function ; but a more general method is afforded 

 by the nse of Taylor's theorem in the following form : — 



Let y ^= fxhe the nearest tabular entry, and let 



y ± I = (j> (x ■+ h) 



be the interpolation required : / and h, or, often preferably, log I and log h, 

 have to be determined in terms of one another and of (p. 

 The direct application of Taylor's theorem gives 



±1= ±h<p'x + ^^ <p"x ± ^-^f'x + . . . . 



orZ = 7e.^'. (l+l^- + ^i^ ^^ ± . , . . "I 

 ^1-2 (j^'x ^ 6 <i,"x • • • • / 



whence 



logl = \ogh + logf'x ± "^^-^ 



4 \((>'x) T f.f ^^^^y 



where m is the modulus of the logarithms : writing x = ;//y, gives similar 

 formulae for h and log h in terms of I and \\jy. The differential co- 

 efficients of \^x may be determined either directly, or by the common 



formulae which give ^^, J| &c. in terms of % ^ &c. There is no 

 cly dy^ dx dx^ 



advantage in setting out the general formulae, because it is easier to obtain 

 the formula suited to each case by direct differentiation, than by substi- 

 tution. 



In the case of common logarithms, making log (x±1i) = log x±Jc, 

 log Jc=\og(^2^)+iV^ nearly 



log 7t=log (Mxk) ±^1 nearly. 



Where m is the modulus of common logarithms and M its reciprocal, so 

 that 



10 + log 7)1=9-63778 43113 00537 

 221 



Y 2 



log lf=0-36221 56886 99463. 



