Prof. Encke on Interpolation. 95 



the above calculated occultation on April 5. We have from the 

 Ephemeris : 



A A A^ A» A* 



April 4. 0'' 152°15'56",6 



12 158 15 45,5 +^° ^^' ^^"'^ _ w . .„ . 

 5. 164 13 46,2 ^ 5« 0,7 +23",1 



12 170 10 21,8 5 5^3^^^ '''' 21,9-^"'2 



6.0 176 5 54,2 ^^5^2,4 ^ '^ ^^^^ ^^^ 



12 182 42,8 '" 



In order to find the longitude for April 5. 7^ we must proceed 

 from April 5. 12" and apply formula (III)*. The factors x—a, 

 x—b, &c. in V and VI, always divided by the number ex- 

 pressing the order of the difference, are the same as the quan- 

 tities denoted in (III)* by /, ^^, &c. ; we have, therefore, 

 these factors of correction -^j ~, ~, ^ ; and reversing the 



order -12. 11 2_ A 

 '^''^^^' 48' 36' 24' 12"- 



The correction of the third difference by the fourth is 

 2,5 .19 . . "^ 

 4§ — = I3O; and this is to be applied, without regard to the 



sign, in such a manner as to approximate 21,9 to 19,4. The 

 corrected third difference is consequently 20,9. Hence the 

 second will be = 1 . 3,2 + il . 20,9 = 1 .13,07, as the correc- 

 tion is to effect an approximation to 1 . 25,1. The corrected 

 first difference now becomes 5 . 56 . S5,Q — -J^. 73,07 = 

 6.5Q. 14,29 for the same reason. If we take -^^ of this 

 quantity and subtract it from 170 . 10 . 21,8, we have 

 April 5. 7'' ... 167° 41' 55",85. If we had proceeded con- 

 trary to the above rule, from April 5. O'' by formula (IV)*, 



the factors would have been -j^, ^, ^ , ^, and the cor- 

 rected differences successively 22,3 ; 1 . 13,3 ; 5 . 56 . 50,87, by 

 which the same longitude would have been obtained. In 

 taking due notice of the signs, it will be seen that the above- 

 given rule perfectly agrees with the change of signs in this ex- 

 ample. 



In order to perceive the facility of the interpolation towards 

 the middle by fbrmula (V)*, let it be required to find the lon- 

 gitudes for April 5. 6'' and 18". The fourth differences being 

 uncertain, it is not absolutely necessary to take them in ; and 

 their influence will only be sensible if they are greater than 



m 



