340 THIRD KEPORT — 1833. 



4. In the application of these rules some precautions are oc- 

 casionally necessary. Thus, \^f" {x) a.nA.f{x) have a common 

 measure f {x), and if a root (a) of ip (.r) = be included between 

 a and b, then there is a point of inflection of the parabolic arc 

 between a and b at the point of its intersection with the axis. 

 Under such circumstances, the method of approximation must 

 be applied to the equation f (x) = 0, and not to the primitive 

 equationy(.r) = 0, for the purpose of determining the value of «. 

 Again, if there exists a common measure off (x) andf{x), which 

 becomes equal to zero, for a value of x between a and b, then 

 there are two or more equal roots of f{x) = in that interval, 

 and the final succession of indices is no longer 0, 0, 1. Other 

 precautions connected with the subdivision of the interval b — a 

 are sometimes required, which the limits of this Report will not 

 allow me to notice in detail. 



It remains to add a few remarks upon the rapidity of the ap- 

 proximation, and upon the means by which it may be ascer- 

 tained. If we express the primary and secondary intervals 

 b — a and b' — a' by i and i', it may be very easily proved that 



i -t . 2 fib) ' 



where f (« . . . b) denotes some value whichy" (x) assumes when 

 we substitute for x a quantity between a and b : and if we form 

 the quotient (C) which arises from dividing the greatest value 

 of/" (a) and/" (b)* by the least value of 2f (a) and 2f (b), 

 and suppose k the order of the greatest articulate or subarticu- 



approximation in Newton's method must be made, are established by a com- 

 bination of analytical and geometrical considerations, and in which also the 

 new limits b' and a' are respectively found by what he terms the rule of ian- 

 gents in one case, and by the rule of chords in the other. The first is the 



subtraction of the subtangent h b' or 'i, from b, as involved in the or- 

 dinary Newtonian approximation when the proper limit is selected. The 



second is the determination of the value of O N, or a — „\,{ — 7r-r4, or 



f{b)—f (a) 



fiil^ ~ fi\ > by the method taught at the beginning of this Note. It is 



evident that these conclusions involve all that is important in Fourier's re- 

 searches upon this part of the subject. 



This memoir of M. Dandelin, which contains a very full and a very clear 

 exposition of the whole theory of the Newtonian method of approximation, 

 preceded by five years the publication of M. Fourier's work. 



• Since no root of/'" (x) = is included between a and b, it follows that 

 either/" (a) or f"{b) will be the greatest value of /" {a . . . b) -. the same, 

 remark applies likewise to/' (a) and/' (l). 



