W. p. G. Bartlett on Interpolation in Physics and Chemistry. 33 



with which the important check (5) ia applied to the work, and 

 _ the fact, which is of great importance in many of its applications 

 ' to physics, that it is not necessary to determine beforehand how 

 ^many coefficients A, B, &c,, are to be eliminated. The objec- 

 tions, which seem fatal to it as a substitute for least squares 

 where the latter is properly applicable, are of very little import- 

 ance in cases where the form of the function is wholly assumed 

 and the formula therefore only to be trusted within the limits of 

 the series of observations; for in these cases the formula which 

 gives a minimum value to the sums of the squares of the differ- 

 ences between the computed and observed quantities, is not ne- 

 cessarily better than many others giving other sets of properly 

 distributed small differences, especially as "the errors of observa- 

 tion " are frequently mixed up with others of the same order of 

 magnitude, arising from errors in the values of the variables, 



If however the series of observations is very extensive (like 

 those, for instance, on the tension of steam) the labor of finding 

 an empirical formula becomes altogether greater than its value, 

 and it is better to tabulate the function without reference to any 

 " interpolation-formula " satisfying the whole or even any great 

 number of the observations. For this purpose the observed 

 quantities must first be reduced to equidistant values of the 

 variable, and then these may easily be interpolated to as frequent 

 intervals as we please by the methods in common use. 



To accomplish the first object there are several methods. The 

 mechanical one of plotting a curve (however valuable in suggest- 

 ing the true physical law of the phenomena) cannot often be used 

 for this purpose witb as much accuracy as computation- In the 

 method of interpolation by ''divided differences''* each determined 

 place depends only on a very few of the adjacent observations, 

 and a series of such places, unless the observations were accurate 

 to the last figure, would not be apt to harmonize. Another way 

 13 as follows : let one of the equidistant values of the variables 

 be ^„, then the observations may be represented, in the vicinity of 

 ^, by the series, 



(7) y^kJrn-'io)+<^i'-'oY+^<^- 



in which A is obviously the required value of y corresponding 

 to «=i^, and may be determined in eacli case from as many ob- 

 servations as we please to use, Cauchy's method applied m this 

 way to some of Kegnauit's observations has been found to gwe 

 an accurate table of vapor-tensions with yQTj little labor. De- 

 terminations of A were made for every 6^ of temperature from 

 ^° to 39°, and ten observations were used for each determination. 



