9 1; Prof. Encke on Intet'polation. 



I{t is exactly = i, or if x is exactly the mean between q and 

 r, it is the same as to accuracy whether the interpolation pro- 

 ceed forward from q or backward from r. The first form 

 would be by (IV)* 



X = Q+ 1 aq + i^ A^Q+ -;;;^ 3"' a=q 



and the second by (III)* 



X = R-iAQ+ i^7^;A^R-^J^A^Q 



In adding them together all odd differences disappear ; and 

 if the respective sums of the even differences which are on the 

 same horizontal line with Q and R are denoted by k', k", &c. 

 or if we put 



Q + R = k; A*Q + A^R=/c'; A^Q+A^R = F&c. 

 the foi'mula will be 



^^ A— 2^: 2.4 2 "^ 2-4.6.8 2 2.4:6.8.10.12 2 *" 



- tF 8 i'^ 16 r" 24 r M 5 ^ 



In using this formula we may again dispense with the re- 

 gard to the signs by another consideration. If we designate 

 the two differences which form any /-, by /3 and (3', and the next 

 preceding and following ones by '/3 and |3", and form this ar- 

 rangement : 



"-" /3'-2/3+'/3 



/3"-2/3'+ ^ 



we shall have if T = /3 + /3' r+^ = /3" - /3' - /3 + '/3 

 = /3"+ '/3 - k'\ or ^" + ^„+i = |3" +'/3. But the correction 

 has always this form k^—ak"''' where « is positive and < 1. 

 The correction applied to /c" or to /3 + /3' will consequently 

 always have the effect of rendering the sum /3 + /3' more di- 

 stant from the sum of the preceding and following differences, 

 excepting the case in which k"^'' has changed its sign by a 

 former correction, which is easily observed in making several 

 successive interpolations, and will never lead to errors. The 

 last formula (V)* is so accurate, and at the same time so con- 

 venient, that in calculating a table it will be best to calculate 

 the accurate values for intervals which are distant by a whole 

 power of 2, and to find the intermediate values by this formula. 

 As an example, we will take the longitude of the moon for 



the 



