REPORT ON CERTAIN BRANCHES OF ANALYSIS. 339 



O b' and O a' are the new limits b' and a : and if ordinates b' n' 

 and a' m' be drawn to the curve, and n' b" be di-awn a tangent, 

 and m' «" parallel to n' b", then O b" and O a" will be the new 

 values b" and a" of b' and a'. The progress of the approxima- 

 tion, upon the continued repetition of this process, will now be 

 sufficiently manifest. 



3. If we consider the different arrangements of the signs of 



f {x), /' {x), f{x), in the transition from the inferior limit a to 



the superior limit b, they will be found to be the following, it 



being kept in mind that the sign of f(x) alone changes from 



+ to — , or conversely. 



+ + - 



(3){ 

 (4){ 



+ + 



- + 



+ — + 



b + _ _ 



- + - 



b - + + 



In the first two cases, the formulae of approximation are 

 ^ ~ "^f /7\ and a — - ^.Sii , and commence thei-efore with the su- 

 perior limit. In the last two cases, the formulas of approxima- 

 tion are a — \.,; l and b — ^4-^,andcommence therefore with 



the inferior limit. In other words, that limit must in all cases 

 be selected which gives the same sign to y" (x) and J" {x), whe- 

 ther + or — . The construction of the portions of the corre- 

 sponding parabolic curves included between a and b in these 

 several cases, will at once make manifest the reason of the selec- 

 tion of the superior or inferior limit and likewise the progress 

 of the approximation itself*. 



* If, in the figure p. 338, we join the extremities m and n of the ordinates a m 

 and 6 n by the chord m N n, which cuts the axis of x in the point N, we shall 



proximate inferior limit in the first two cases considered in the text, and a new 

 superior limit in the last two. Other constructions are noticed by Fourier, 

 which give similar results. 



In the M^moires de VAcadimie Royale de Bruxelles for 1826, there is a 

 memoir on the resolution of numerical equations by Dandelin, in which the 

 analytical conditions which must be satisfied bv the limit, towards which the 



z 2 



