REPORT ON CERTAIN BRANCHES OF ANALYSIS. 339 



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

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

 and m' a" 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), f (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 oi f{x) alone changes from 



+ to — , or conversely. 



fix) fix) fix) 



+ + - 



(2){ 



(3){ 

 (4){ 



+ + + 



a — — + 



b _ _ _ 



+ - + 



+ - - 



a - + — 



— + + 



In the first two cases, the formulae of approximation are 



b — >, ,,, and a — ^4-Wj and commence therefore with the su- 

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

 tion are a — 4rW and b — ^-7-Kjandcommence therefore with 



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

 be selected which gives the same sign to /" (x) andy (^), 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 « by the chord m N n, which cuts the axis of x in the point N, we shall 



cjr^TvT f{a){b — a) , f{h){h—a) ... . 



find O N= a — •' .)' ^ ., { = 6 — %)' „/ ( » "^^^^^ g'ves a new ap- 



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 I'Academie Royale de Bruxelles for 1 826, there is a 

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

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



z 2 



