338 



THIRD REPORT — 1833. 



1. If /(^) = 0, or X = be the equation, /' (x), f" {x\ or 

 X', X'' its first and second derivatives, then the limits a and h 

 of one of the roots will be sufficiently near for the application of 

 this method of approximation, if the three last indices (p. 332) 

 be 0, 0, 1 . If this be not the case, the interval between a and h 

 must be further subdivided until this last condition is satisfied. 



Under such circumstances there will be no root of the equa- 

 tions/' {x) — and/^' {x) = 0, included between a and h : and 

 if we suppose y — f ix) to be the equation of a paraboHc curve 

 CAB, where O a=- a,Oh =^ h,am =f{a), b n ^f{b), then 



there will be no point of inflection between a and b, and no tan- 

 gent parallel to the axis. The analytical conditions above men- 

 tioned would show that /(«) and f{b) must necessarily have 

 difi'erent signs. 



S. If we suppose b to represent the superior limit of the root 

 (a), then the Newtonian approximation gives us the new su- 

 perior limit b' = b — jrjj\ ; a new inferior limit will be found 



to be a = « frnl '• these limits are still superior and inferior 



limits of the root a, and are both of them nearer to it than the 

 primitive limits b and a. 



If the same operation be repeated by replacing b and a in 

 f{b) and/(«) by b' and a', nearer limits will be obtained, and 

 it is obvious that the same process may be repeated as often as 

 may be thought necessary. And in as much as we obtain both 

 the inferior and superior limits corresponding to each operation, 

 the difference between them will always be greater than the 

 error of each approximation. If we refer to the above figure, and 

 suppose n b' to be a tangent to the curve at n, and a' m to be 



drawn parallel to n b', then b b' = f'{by ^^^ « «' = f^Ty 

 since/' {b) = tan w 6' 6 = tan m a a. It follows, therefore, that 



