INTERPOLATION. 



which expreffion will then reprerent the correfponding ordi- 

 nate to the abfcifs z ; and confequently the interpolation oi 

 the feries may be made general. Thus in the example 

 propofed. 



64 



37 



Therefore, confidering y as the correfponding ordinate to 

 the abfcifs z, we have 



■^^ (•tzJ.)4. tL^ll: 



0(- 



By a little attention it will be feen that the feries is pro ^ 

 duced from the fucceflive multiplication of the terms of thi- 

 3 5 7. 

 4' 6' 8' 



manifeft, and the interpolation of this, except for the term 

 between i and I, is evident ; the other interpolated terms 



being -, -, - , &c. : let then the term between i and i 

 be X, then the feries after interpolation will be 



' "" 2 3' 4' 5' 6' 7' 

 and as the propofed feries is generated from the continued 



feries — , 



in which the order of formation 



&c. 



as it ought to be, from the nature of the feries. If, there- 

 fore, now we had to interpolate a term between each of the 

 given ones, thefe terms would be (lA)', (25)') iih) > 

 (4i)^ &c. 



And the fame method is applicable in all cafes, but it 

 fometimes admits of abbreviation, if the law of the feries can 

 be determined, and therefore this {hoiild be the firll confi- 

 dcration when the law is not evident. 



Let there now be propofed the feries 



wTiere the law is not obvious, the interpolation of which 

 is required. 



Series I 



id difference — ; 



ad difference 



3d difference 



4th difference 



5th difference 



affuming y for any ordinate correfponding to any 



multiplication of the feries i, — , 



&;c. ; fo will the 



1 i 



2 4 6' 



interpolated terms required, be generated from the con- 

 tinued multiphcation of the interpolated terms of this lall 

 feries, which will therefore be 



2 2.4 2.4.6 



X, - X, ~- X, ^ X, &c. 



3 3-5 3-S-7 



All, therefore, that is required, is to find the value of .v, 

 from which any interpolated term may be determined ; and 

 it may be further oblerved, that the term x of the above 

 generating feries mull neceffarily be the fecond term of 

 the required feries after interpolation, or that term which 

 falls between i and i. Let therefore the differences be taken 

 as follows, "ulz. 



6i_ 



2j6 



t- &c. 



136 



He „. . ^ 



propofed abfcifs z, the equation of the ( 

 the fummits of the ordinate!, will be 



^ = I - i = + g . — - 



paffmg through 



I) 



«(» 



0(^-^1 



(^- I) (z-2)(. 



But as the objea of our 



(<.- I) r^-2) (^-3:1 (^-4) 



3-4-5 



quiry is more particularly 



direded to the finding of the firft interpolated term 

 prefcnted above by x, we mult make % = i; whence the 



from what has been before obferved, the interpolation of 

 the propofed feries is now complete, being as follows : 

 J _., 1 % Lli i_4 ^ ^ -5-5 lii_6 . 

 ' • ' ^' 3 ' 2 . 4' 3 . 5 • ' 2 . 4 . 6' 3 . 5 . 7 ^' '^'^^ 

 If it had been required to interpolate two terms between 

 each of the given terms, it would have only been neceffarv 

 to have affumed a = i, and j, as is evident ; and therefore 

 needs no illuftration. 



Hence we fee that the general equation of any curve, 

 made to pafs through the fummits of any number of equi- 

 j-a._. __j: _ 1 - d,e, f, &c. wiU be 



C 4- 



iS 



above is reduced to 



'=-4+T6^^36"'64 

 in which feries the order of formation of the co-efficients 

 is obvious, A,. B, C, D, &c. being reprefentatives of the 

 preceding terms ; and hence, by the fummation.of the feries, 

 the value of * is found equal to 0.64052, and cojifcquently 



diftant ordinates. 



D &c. 



{<{ 



3 ' 



+ {c 



") 



2b + a) 



( = 



(' - ^) 



&c. 



which is Newton's formifla in his " Methodus Differentialis." 

 The limits of our article will not admit of a farther de- 

 velopcment of the theory of interpolations, we muft there- 

 fore refer the rcaik-r, who wiflics for more complete in- 

 formation. 



