W. P, G. Bartlett on Interpolation in Physics and Chemistry. 29 



absolute sum of the special values of u.^ To show more dis- 

 tinctly how the numerical application is to be made, we shall 

 here arrange some formulae for computation, changing for this 

 purpose some of Cauchy's notation and giving the development 

 an entirely different form. 



Let it be assumed, as usual, that the observed quantity y, con- 

 verges when developed in the form 



(1) y— A-fBi+Ci2-f &c. 



If this assumption does not give, on trial, a convenient formula, 

 the logarithm or any other function of y may be tried in the 

 place of y, and its development made in precisely the same way. 

 It will be easy to see, moreover, that any variables we please 

 may be substituted for the different powers of t, provided only 

 the series is convergent. The function will first be developed in 

 the form 



(2) y=^+3$zf<-|-aC^2i2^jO^=i3^ &c. 



in which ^^, J^t^^ &c., are functions of the form a+ht+ct'+kc. 

 and are respectively of the first, second, &c., degrees in t. The 

 numerical values of ^, B, &c., and the expressions for ^t, &c., 

 bemg found, the series (2) is immediately reducible to the form 

 (1). Let s be the number of observations given to determine 

 A, B, &c. ; then the formulae required in practice are 



St 2t^ Si^ sr 



^^~ 2'^t' ^^~ 2'Jt' ^'^i 



^^t2=zJt2-8^jt, J2t^=^t^-^^Jt, . . . J^tr=Jr-Mt 



^3=^1,^212' ^""^"^2^3 



Jtii3--J2i3_Y^J2t2 . . .■J3ir—J2(n_^^JSi2 



2"^2t2 



which is quite simple. 



•"YS'uai mnographed memoir pubUsnea m loso, or lo ira repuoiicauon m i 

 ■LiouyUle's Jaurjial d« Mathematiques, tome ii, page 193. The same thing i 

 appended as a note to the first Tolume of Moigno's Caleul Diffh-entiel, pag« 

 «w a partial translation of it in the U. 8. Coatt Survey Report for 1860, p. 3 



