J. H. Alexander on a Neia Formula for Interpolations. 19 



It is evident upon this, that an equation of the second degree 

 would be much below the divergence of the series ; and that one 

 of the fourth is still more above it. We are, therefore, warrant- 

 ed in regarding it as cubic, and in interpolating accordingly. 



This method applies more emphatically to cases where the or- 

 dinates are not equidistant. For instance, taking a series from 

 experiment on the resistance of still water ; in which the ordi- 

 nates represent miles per hour in the speed of the vessel, and the 

 abscissae are the resistances measured in pounds respectively j we 

 have 



itiHfc- mite*. miles. miles. 



ordinate*; 1-925 2-222 2-628 4045 



lb. II). lb. lb. 



resistance-series: 11 13*08 18-10 47*26 



Transforming these, in dividing each by its first term, we pro- 

 duce 



ordinates: 1 1-154 1-365 2-101 



squares: 1 1*33 186 441 



cubes: 1 154 254 9*49 



resistances: 1 1-19 1*64 4*30 



It is very plain from this, that the resistances increase more 

 rapidly than the simple ordinates, but do not transcend the ratio 

 of the squares. The series of cubes has been given only for 

 abundant illustration. We are limited, then, in this instance to 

 an equation of the second degree. 



These methods will cover the most of cases occurring in prac- 

 tice J nearly all which already, in point of fact, do not transcend a 

 series of the second order. Cases may occur, however, in which 

 for various reasons they could not be applied safely ; and where 

 it is desirable to use all the terms given by observation. To 

 such as these, the formulas I propose are, I believe, peculiarly 

 applicable. These formulae I shall now at length present. 



Calling a, 6, c, rf, . . . . &c, the first, second, third, fourth, 

 &c, (equidistant) terms in a series ; and /, any term whose place 

 in the series, or rather the number of whose place, is n J we have 

 generally 



I. When first differences in the given series are constant, 



/=/* -1 .b-n-2.a; 



II. When second differences are constant, 



A 



.a; 



7i-l .n-2 tC , n ,j .n-3.H-n-2.n-:* 

 1.2 1.2 



III. When third differences are constant, 



1.2.3 1.2.1 ' 1.2.1 



n - 2 . n - 3 . n - 4 



1.8.3 ** ; 



