32 W. P. G. Bartletl on Interpolation in Physics and Chemistry. 



, i^.M 



^3A< 1 ^^t 



y^AH"^ 



_€A^_ m^-t■^ 1 























+ 11885 





-2-0 [1]= +4-316 



2. 















2J=- 666-8 





nVs 



24738 



276-7 



+ 2655 



+ 20-6 



"S-0 



t3]=-206 





158-8 



18758 











[2']=- 10125 



6. 







178-3 



3589 



1 



0-8 



L3'j=-1019 



I: 



1183 



^6619 



'^9-1 



31 



+ii 



[3"]=-m 



lo' 



+ 47-7 



+ 5640 



- 66-3 



11700 



90-7 



-0-1 



A= 4686-6 







16871 



198-3 





906 





B=: - 15-539 























46329 



544-5 





+ ^2-9 







u. 



+ 68 20 



+ 68753 ' - 808-0 



+ 28297 





+ 5-2 





In this example, to which Cauchy's method is applied, the 

 numbers in the column y express the solubility* of anhydrous 

 sulphate of soda in 10,000 parts of water at 14 different tem- 

 peratures.— column t contains the corresponding temperatures 

 less 50° (for convenience of development). The work is earned 

 as far as the determination of four terms in the development 

 according to powers of (temp. —50°); but a comparison of the 

 values of y'^ with those of y'" shows that nothing is gained by 

 the addition of the fourth term. Since the values of A, B, &c., 

 are obtained by using (6) it is not necessary to obtain the coeffi- 

 cients a, b, &c., at all. In taking the sum :s' the signs are to be 

 changed in the first nine cases ; therefore ■2"(+) means the alge- 

 braic sura of the last five cases, and ^'{--) that of the first nine, 

 ^•"(-f-) is the algebraic sum of the 1st, 2d, 3d, 4th, 13th and 14th 

 cases, and ^"{-) of the rest. S"'{+) includes the 4th, 5th, 6tb, 

 7th, 8th, 9th, and 14th, and ^"'(-) the rest. The values of 

 ^"{+) and -i"'(-) are written on the same horizontal line oppo- 

 site the argument 2"'; similarly with -S""(+) and ^"'{-) for y": 



M. Bienayme has shownf that if each case of the equations, 

 2/z=A-i-Bi+C«2-f&c. 



were multiplied by the proper least-square factor (different for 

 each case and for each of these equations) before taking the sums 

 X, 2', &c., the process would become merely another form for 

 the expression of the elimination in least-squares given by Gauss+ 

 and others. 



The advantages of Cauchy's method are its simplicity, the ease 



» According to Lowel, Annales de Chimie and Physiqwe, ^''^^ P^^^^'Jj^ j^^^^^, 

 matiq^fx7uC^Sig^'299^^ Se^lso Cauchy's Note, Compter, Rendui, xxxvi, 27 Juin, 



X in the Disquisitio de elements PedladU ; and translated into French on pag;e 

 137 of tha Miihode des Moindres Carris. 



