334 



THIRD REPORT — 1833, 



+ 



/(A) 



fi (h\^^ ~ "j when no regard is paid to the sign of/' (a) 



and f {b). In this case new limits must be taken successively, 

 intermediate to a and b, until f («') and f {b') one or both of 

 them change their sign. 



In the second case, if there be two imaginary roots cor- 

 responding to the interval 

 between a and b, then the 

 curve whose equation is y=X 

 though similar in its other ge- 

 ometrical properties to fig. 1, 

 will not cut the axis between 

 a and b. In this case the sum of the subtangents a n' and 

 h in' will either exceed the interval a b, or will ultimately ex- 

 ceed it, when the interval a 6 is sufficiently diminished. The 



corresponding analytical character will be that "^rjjl + frrA 



is either greater than b — a, ox that it may ultimately be made 

 to exceed it *. 



Thus, in the example referred to above, p. 332, write down 

 the following scheme : 



Xiv Yi" Y'i Yi Y 



and place above and below the indices 1 and 2, in the succes- 

 sion of indices 0, 1,2, the values of X"' and X" respectively, 

 without regard to sign, corresponding to a: = and x = 1; then 



Q S 1 4( 



we shall find -^ 7 1 and, a fortiori, therefore 7^ + Toj also 



greater than 1, which is the interval between which the roots 

 required are to be sought for : it consequently follows that two 

 of the roots corresponding to this interval are imaginary, and 

 there remains, therefore, only one real root between and 1. 

 If we suppose 



X=x^ + a^ + ar*-2x'^ + 2a;~-l = 0, 



* The new values a' and b' of a and b may be made a + "{,)■-( and b — \, ,,{ , 



/ («) f (0) 



which are «' and m' respectively : a second trial will generally succeed. 



