INTERPOLATION. 



173 



Theory of The law of this series is evident, its terms being ma- 

 InterpoU- nitestly the first of the successive orders of differences 

 _, multiplied by the co-efficients of a binomial raised to 



the power - r . 



n 



There is another form under which this expression 

 may be put, which is at once concise and elegant. The 

 iymbol Ay? having been used to denote^, y , the 

 difference between two successive values of'^, let us 

 now put A.V, to denote y y,, which is the whole in- 

 crement that the function y receives by changing from 

 , its first particular value, to its present magnitude. 

 The foregoing expression may now be put under the 

 ' 



Fx i:: I '.' 

 lit. 



We have aUo JT = 3.1415926546 

 ii=. 0.001595(^,36 

 x x,= O.C084073M>4 

 x j = <) i n 84073164 



*! *,.=* 



Pics* value* being substituted in the formula* 



' - -i! 

 iL A3 . 



)(' 



Theory. 



.... (5) 



observing that in the developement of the second mem- 

 ber by the binomial theorem, we must transfer the ex- 

 ponent* of the powers of the quantity Ay to charac- 

 teristic A ; that is, we must write z-y, for (ty )', and 

 A'jTe for { A y,)'i and so on. 



APPLICATION OF THE THEORY. 



We shall give some examples of the application of the 

 theory of interpolation. 



Ex. I . Let it be required to find the common loga- 

 rithm of .'5.1415926536, l>y means of a table containing 

 the logarithms of numbers from 1 to 1000 to 10 deci- 

 mals. 



In this case, we may regard the numbers as the suc- 

 cessive value* of the abcuna x, and their logarithms, a* 

 the corresponding value* of the ordinate y. If we take 



.=3.14, jr,=3.15, x,=3.16, *,=3.17, * t =3.\S, 

 we shall have the following conrCMponding value* of v, 

 and their difference* : 



tdDiC 3dD*C 4th HiC 



jr=log. 3.14152926536=0.4971498726. 

 This is not the easiest way of finding the logarithm of 

 a number consisting of a great many figures ; but it '""V 

 serves very well as an example of the interpolation of a 

 term in a series. 



Ex. 2. Let it be required to find the moon's longi- Example 

 tude, Nov. 18. 1817, at 16 1 "- 22 m - 16% having given, 2tL 

 from the Nautical Almanac, 



1817. 

 Nov. 18, Noon 



Midnight 

 19, Noon 

 Midnight 



Questions of this nature continually occur in practi- 

 cal astronomy. The tables give the calculated places. 

 of a phenomenon at equal intervals, and from these its 

 place at some intermediate instant is to be found. From 

 our general expression, ( 4, ) the rule for four calculated 

 places is this : 



Call y a , y ,, y ,, y t , four succeeding given places of 

 the phenomenon at equal intervals of time, and h the 

 common interval ; put x for any indefinite intermedi- 

 ate time, reckoned from the instant for which the first 

 place is given, and y for the corresponding place of the 

 phenomenon; then 



it will be found, after executing the arithmetical ope- 

 ntion^that 



Moon's Longitude. 



]! 26 52' 0" 

 2 57 11 

 8 59 32 

 14 59 28 



V =O.W83I05538 43769 



l.rr.-.j-., 



43402 

 13721796 +274 



43218 

 13678578 

 ^,=0.5024871200 



Here it appears, that the differences of the sucoenive 



order* are alternately pocitive and negative ; and that 



& , = + 0.001 3809057 



= O.OOOOJH3T69 



= +0 0000000277 



= 0.0000000003 



In the 

 22* 1&-. 



it example, A=12 hours, and ar=l6 h - 



4= 1.364*6, ^-1= .36486, 4- -2=. 63574. 

 A A n 



The particular values ofy and their differences, are 



SdLhff. 



25 



Hence it appears, that 



Ay -+21911, 'jo= HO, **y= 

 The first term of the value of y, is !! 26* 52' 0* 

 the second, +29892* 



the third. 42.4'' 

 the fourth, 1.3 



44" 



29S48 = 8 17 28 



Moon'* Long. Nov. 18 4 - 16" 1 - 22- Ifr i 5 9 28 



Ex. 3. The calculated distances of the moon's centre 

 from the star Aldebaran, are 



1817. Nov. 1. Noon 53 20* 16" 



Midnight /!<) 33 47 



Nov. 2. Noon 65 52 58 



Midnight 72 18 10 



Find, by interpolation, the distances at the Hid, Vlth, 

 and IXth hour of the first interval. 



A* in the last example, let y , y,, y,, y,, denote the 

 given distances, and y the required distance at each in- 



I-'vinn-i 

 3d. 



