96 Prof. Encke on Interpolation. 



in the present case, as the sum U' is to be multiplied by 

 2 • 8 • fV ~ 2 j^' ^^^^ have consequently for 

 6'' ... k' = —2. 28,3 ... - 1 X-' = + 18,54 

 IB" ... ^ = -^ 1 . 47,0 ... - i /f = 4- 1 3,38 

 And next : 



April 5. 0" 164°13'46",2 



6 167 12 13,3 



+ 2.58.27,1 

 58. 8,5 

 57.52,9 

 57.39,5 



-18,6 

 15,6 

 13,4 



12 170 10 21,8 

 18 173 8 14,7 



6. 176 5 54,2 

 Interpolating again into the middle, we have for 



9*> ... V = - 34,2 - iA:' = + 4,3. 



April 5. 6^ 167.12. 13,3 



9 168.41.19,7 +^-^^'^'*_4,3 

 12 170.10.21,8 1-29.2,1 



from which by the common formula for interpolation (II)* 

 for 1\ t = ^ 167. 12.13,3 



+ 29 . 42,13 



+ 0,48 



7'' ... 167 . 41 . 55,9 as above. 

 One might have dispensed again with the attention to the 

 signs in inteipolating into the middle, because it appears at the 

 first view that k must be augmented. 



If the interpolation is not to be for entire hours, but for a 

 time which contains single seconds, it will be sufficiently ac- 

 curate, in calculating the factors of correction for the higher 

 differences, to substitute for the true t an approximate fi'action. 

 Thus if it is required to find the longitude for the time of the 

 reappearance of 82 Leonis^ ^^ 24' 16", we shall have proceed- 

 ing from April 5. 12'', t= — -~ , for which the approxi- 

 mate value — will be obtained. 



The factors will therefore be -^ , — , — ; the corrected 



differences 20", 9, 1' ]2",8, 5° 56' 13",20. Applying to the 

 latter the exact value of/, we obtain 7*' 24' 16". ..167° 53' 56", 7, 

 which agrees with the interpolation from the values found for 

 e*', 9'', and 12\ 



XIV. 0/» 



