28 W. P. G. Bartlett on Interpolation in Physics and Chemistry. 



If all the observations are required to be rigorously satisfied, 



is probably as good a way as any of arranging tbe elimination, 

 since it is only necessary to multiply the various factors to ob- 

 tain the function y, developed in powers of the variable t\ y^, 

 y,, &;g., being the observed values corresponding to the values 

 t^, ^,, &c., of f.* Otherwise the determination of the constants 

 may be made either so as to satisfy exactly some of the observa- 

 tions, or so as to satisfy them all within moderate limits — say 

 the probable errors of the observations themselves. The former 

 proceeding is theoretically imperfect, because it makes some of 

 the observations of no account whatever in determining the 

 values of the constants, using them, if at all, only to help the 

 selection by successive trials of the form of the function. The 

 latter is generally impracticable in a direct form, unless the con- 

 stants enter linearly into the equations, in which case the method 

 of least squares will always give good results; but if besides 

 this, the successive terms in the development, either of y, or of 

 any given function of 3/, form a convergent series, it will gene- 

 rally be advantageous to use Cauchy's method, which, notwidi- 

 standing its violation of the law of probable error, is practically 

 sufiacient, and indeed far the best, for almost all the physical 

 formulae that it is worth while to develop at all in an empirical 



This method not being, like least squares, generally accessible 

 in a working form, it is proposed to devote special attention to 

 its operation. Its principle is to neglect at each step all the 

 terms of lower orders, leaving in general a form 



and then of all the values 



which might be given to the constant a, by assigning different 

 sets of values to the ^'s, to select that in which the k's are all so 

 taken (=±1) that the denominator above written becomes the 



are tn+1 observed values of y, there will be only i»n( 



