174 



INTERPOLATION. 



Application terval, then in all cases h=12, and at the 1 1 Id hour 



of the r 1 



Theory. x= s therefore, in this case, - r = ; and by the for- 



*- * ft '1 



Example 

 4th. 



mula put down in last example, 

 1 3 



In like manner for the Vlth hour, we have -j- = y' 

 and 



J_ 



16 



x 3 

 And for the IXth hour, -j- = , and 



It V 



128 



Now, to find the differences, 



3d Difference. 



+ 21" 



Hence, we have &y = 22411", tfy = 340, tfy =21, 

 and these being substituted in the formulae, we have 

 for the 1 1 Id hour ...... y = 54 53' 8" 



In like manner for the Vlth . . y = 56 26 20 

 and for the IXth ...... y = 57 59 53 



In this way, tables of any kind may be constructed, 

 by computing accurately the values of the ordinates at 

 equal intervals, and interpolating any number of inter- 

 mediate ordinates. 



Ex. 4. Given a correct distance from the sun on the 

 four following days at 12 at night, to find its distance 

 Dec. 20. 



Dec. 12. Distance .... 301 



21 ........ 620 



24 ........ 715 



26. ....... 772 



This case, in which the intervals between the given 

 successive values are unequal, may serve to exemplify 

 formula (2). By employing the usual notation, and 

 reckoning x the time from Dec. 12. we have 

 .r s = 0, #,= 301, 



*,== 9, y,= 620, 



x,= 12, ^ 2 =715, 



x,= H, ^,= 772, 



* = 8, y is required. 



From the values of x we get 



* x a =8, x x,= 1, x x 2 4, x , = 6, 

 *o *, = 9, x x, = 12, X Q *,= 14, 

 a, x a = 9, JT, x 2 = 3, x, x, = 5, 



* a *<) = 12, x 2 x , = 3, x 2 x rs 2, 



These differences, and the values of y being substituted 

 in the expression 



_ Jf.--.) (* *.) ( * , ) 

 "(.-*.) (x-*,) (* -x,) yv 

 (x x ) (x j,) (x x t ) 



J. ~ " 

 r 



(x *) (x x,) (a- . 

 T 7Z - \ (-, ~ / - 



\ X t *o) ( X ) *tj V.' T 3 



It becomes after abbreviation, x 



1 6* 2 



Applic:itio 

 of the 



Tlinny. 



8 

 3l X 



772, 



or y = 586.3, the distance of the comet from the sun, 

 Dec. 20. 



Ex. 5. In the year 1500, Bernard Walther at Nu- Example 

 remberg observed the chord of the sun's distance from 5th. 

 the zenith, by a large parallactic instrument of Ptole- 

 my, as follows. 



June 2. O. S. . 45467 June 12. O. S. . 44883 



8 44975 16. . . .*. 44990 



9 44934 



From these observations, it is required to find the true 

 instant of the summer solstice. 



In this example, we might make the given chords 

 the ordinates of a curve, and the corresponding times, 

 reckoned from some fixed instant, the abcissx ; we 

 would then have five ordinates, and their abcissce, to 

 find the abcissa corresponding to the least ordinate ; 

 so that the problem requires for its solution, besides 

 the theory of Interpolation, that of maxima et minima; 

 which is given, FLUXIONS, Art. 61 66'. 



If we employ the five given ordinates at once, the 

 function to be made a minimum will be of the fourth 

 degree, in respect of the variable x ; and the problem, 

 in general, will lead to the solution of a cubic equation. 

 To avoid this, it will be better to take only three of the 

 five ; and then the curve will be a common parabola, 

 having its axis perpendicular to the abcissa x ; in fact, 

 the problem, from its nature, admits of this simplifica- 

 tion, for it is evident, that the values of y must be 

 equal at equal distances from its least value, which is a 

 property of the parabola. As there are five given or- 

 dinates, there may be various sets of three formed from 

 them, and those may be selected which are most suit- 

 able, and a mean taken among the results. 



Let us now suppose that x and y are any indefinite 

 values of the co-ordinates. Then, the given values of 

 the abcissa: being a: , x f , a 2 , and of the ordinates 

 y> #2> #?> the general formula, in the case of three or- 

 dinates, gives 



_ (xx,)(xx 2 ) (x J 



(x a x 2 ) y ^ (*,-* 

 (xx ) (xx t ) 



which, by taking away the denominators, and develop- 

 ing the terms, so that they may be arranged according 

 to the powers of x, becomes 



(*.-b) (*,-*,) (.*,) 



** 



+ C. 



And here, A and C denote constant quantities made up 

 of the particular values of x and y. 



Now as y, and consequently A y, is to be a mini- 

 mum', the fluxion of the second member of this equa- 

 tion, after the terms are divided by the fluxion of x, 

 must be =0. (FLUXIONS, Art. 63.) Hence, taking the 

 fluxions, (Rule A, Art. 26,) and transposing, &c. we 

 get, 



(*l*l ) . Vo (x* -xl ) .y, + (*;-*; ) y, 



