834 THIRD REPORT — 1833. 



f (h\ 



+ fufi ^b — a, when no regard is paid to the sign of/' (a) 



and f (b). In this case new Kmits must be taken successively, 

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

 them change their sign. 



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

 responding to the interval p- „ 

 between a and b, then the ^ 

 curve whose equation is 2/=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 nil 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 ^^r^. + ^irrr: 



is either greater than 6 — a, or that it may ultimately be made 

 to exceed it *. 



Thus, in the example refen*ed to above, p. 332, write down 

 the following scheme : 



X", X-, X", X', X, 

 6 8 



(0) + _ + + _ 

 13 2 3 



(1) + + + + + 



18 14 



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 ^=\; then 



8 V . . 8 14 



we shall find -^ 7 1 and, a fortiori, therefore -pr + rn, 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=a;5 + a;4 + a^-2x2-|-2jr-l = 0, 



* The new values a' and 6' of a and h may be made a 4- ., \~{ and b — \^ ,J . 



' f'(a) f{b)' 



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



