OF INTERPOLATION AND EXTERMINATION. 339 



expression affords us, by expanding its terms, u^^zu-^—^u 



-T^ ^'"+ 1.2.3 ^ " + 1:5:34— 



,Au A% , 2A'tt 6A% \ , /A=« 3A»i/ 



"="+ (t-i:2 + i:3 ~2r4-)" + (r2-i:e 



11A*« X 2 ,_ /A'« 6A*« ^ ^^i^i^/A-^ \^ 



by equating the terms containing'.the same powers of n (277), 



- dw' /Am A^m \ T dM* 



we have w=m, nn -r- = (-:r- — ^t"^*" ) ^ a^" t- aziAm — 

 dx ^1 1.2 / dar 



1- ..,, = . . ,; and the respective series may be con- 



tinued to any number of terms by the actual developement 

 of the different products. 



Scholium 1. It may be observed that the coefficients 

 of the different terms of the first series agree with those of 

 the developement of the quantity hi (1 + A), and that in fact 

 the whole may be represented to the eye by the expression 



— A=hl (1 + A) M. It was also remarked by Laplace, that 

 the powers of this equation will afford us, with equal accu- 

 racy, the values of the higher fluxions ; thus -r-^- h^= j hi 



(l + A)!^^: but this mode of finding the coefficients is 



little more useful, in common cases, than the original com- 

 putation of Euler. ^ 



Scholium 2. This theorem may very often be of use 

 in deriving formulae from the results of observation, but it 

 is necessary that the observations should be extremely ac- 

 curate, since very minute errors will affect the higher or- 

 ders of differences in a material degree. 



