METHODS OF INTERPOLATION. 307 



The general plan for graduating irregular series of numbers, wliose 

 application to the construction of tables of mortality has now been in- 

 dicated, will undonbtedly be found usefnl in other directions. P^very 

 physical law is a mathematical relation between one or more variables 

 and a function. To ascertain the form of this relation, or the law of 

 the natural phenomenon, we must obtain, by observation or experiment, 

 a number of values of the function corresponding to known values of 

 the variable, and then endeavor to find some analytical formula which 

 will connect and express them all. For a statement of the nature of 

 this general problem, and of the graphical and tentative methods which 

 have been employed for its solution, see the discussion of experiments 

 for ascertaining tlie law of variation of the density of water at different 

 temperatures, given by M. Jamin in the Cours de Fhymjne de VEcole 

 PolyfecJmique, Vol. II, pages 39 to 50. The number of observed values 

 of the function is ordinarily much greater tlmn the number of constants 

 in the desired formula. If there is but one independent variable, and 

 the observed values of the function are ])lotted as ordinates to a curve, 

 the corresponding values of the variable being the abscissas, this curve 

 will be a more or less irregular or wavy line, because the ordinates which 

 fix successive points in it are subject to the errors of observation. In 

 " an exact equation of this line, the number of constants would, in gen- 

 eral, be as large as the number of observations taicen. The i)roblem 

 presented is, to simplify the equation by reducing the number of con- 

 stants, while preserving a form of curve which shall approximate to the 

 original one as closely as possible. Our first method of graduation 

 secures such approxinnition by taking the ordinates of the original curve 

 in groups, and making the arithmetical means of the ordinates in the 

 corresponding groups in the new curve severally equal to those in the 

 original one. The equation of the new curve can only contain as many 

 constants as there have been groups assumed. This plan has obvious 

 advantages over the one usually followed, v/hich is, to selector compute 

 as many normal ordinates to the original curve as there are to be con- 

 stants in the equation of the new one, and then subject the new curve to 

 the condition of passing through the extremities of these ordinates, thus 

 making the a(;curacy of the new curve depend on that of the observa- 

 tions, as represented by the selected ordinates, instead of depending 

 alike on all the observations in each group. 



When it is not convenient to have the observed values of the function 

 corresi)ond to equidistant values of the variable in the first place, they 

 can be reduced to equidistant ones either graidiically, or by ordinary 

 interpolation with Lagrange's formula, or with (32), which is merel}' one 

 form of a special case under it. The irregularities of the series may 

 then l)e diminished by the second method of adjustment, ;nid, fimilly, 

 the first method will give an equation which will exi^ress the law of the 



