332 THIRD REPORT — 1833. • 



One change of sign is lost in the transition from — 10 to — 1, 

 and there is therefore one real root between them ; the sign of 

 the last term is therefore necessarily changed from + to — . 



For X = 0, there is a result laetween two similar signs ; 

 there is therefore a pair of imaginary roots cori'esponding, and 

 consequently a loss of two changes of sign. 



There is no root of the equation between and 1 . 



There is a loss of three changes of sign in the transition from 

 1 to 10, and therefore there are three roots corresponding, one 

 or all of which may be real : the application of a subsequent 

 rule will show that two of them are imaginary. 



It is obvious, in a series of derivatives, X^™^, X^"*"'', . . . X'*"' 

 ... X, that X'*"', X^"*"'^ may be considered as the derivatives 

 of the {m — r —ly^ and {m — r — S)* order from X'*"^, as well 

 as the wj"' and {tn — 1)* derivatives from X, and that the same 

 rules may be applied to the separation of the roots of tliese de- 

 rivatives when they become equations, whether they be consi- 

 dered as belonging to the inferior or to the superior order. The 

 substitution, therefore, of a and b successively for x, will show 

 the number of roots of the successive derivative equations which 

 are found in this interval, which will be equal successively to 

 the number of changes of sign which have disappeared in the 

 transition from one value of x to the other. If we now place 

 under the several results of the substitution of a and b, a series 

 of zeros or numbers as indices to signify that no change, or an 

 indicated number of changes of signs, have disappeared, then in 

 passing from the left to the right, we shall find first zero, and sub- 

 sequently, whether immediately or not, the numbers, 1, 2, &c., 

 which will indicate the number of roots which must be sought 

 for, in that interval, in the derivative or other functions, consider- 

 ed as equations, which are severally placed above them. Thus, 

 i{ X. = x'^ — x^ + 4< x^ + X — 4) — 0, then from the scheme 



X'% X"', X", X', X, 

 (-10) + - + _ + 



we infer that there is one root of X = 0, and no root of any of 

 the several derivative equations situated between — 10 and — 1 j 



