REPORT ON CERTAIN BRANCHES OF ANALYSIS. 329 



succession of signs of the function X and its derivatives, upon 

 the substitution of different values of x. The conclusions 

 which have resulted from this examination, which we shall now 

 proceed to state, have completely succeeded in effecting the 

 practical solution of this most difficult and important problem, 

 as far, at least, as real roots are concerned. 

 If we suppose 



X = .r™ + a, x""-' + a.2 x'"-^ + ...«» = 0, 

 and if we write X and its derivatives in the following order, 

 XM, X(— \ XC"-^), . . . X", X', X, 



then the substitution of -^ and — -^, will give two series of re- 

 sults, the tei'ms of the first series being all of them positive, 

 and those of the second being alternately positive and negative. 



The same will be the case if, in the place of -^, we put any 



limit (a) greater than the greatest root of the equation X = 0, and 



if in the place of — ^ we substitute any negative value of 



X {— ^) (to be determined by trial or otherwise) which will 

 make the first terms of X, X', X", &c., considered with regard 

 to numerical value only, severally greater than the sum of all 

 those which follow them. In the course of the substitution of 

 values of x intermediate to those extreme values — /3 and «, all 

 the m changes of sign of X and its derivatives, from + to — 

 and from — to + , will disappear, in conformity with the fol- 

 lowing theorems, which are capable of strict demonstration. 



1st. If, upon the substitution of any value of x, one or more 

 changes of signs disappear, those changes are not recoverable 

 by the substitution of any greater value of x. 



2nd. If upon the substitution of two values a and b of x, 

 one change of signs disappears, there is one real root and no 

 more included between a and b. If under the same circum- 

 stances an odd number 2 p + I oi changes of sign have disap- 

 peared, there must be at least one, and there may be 2^/ + 1 

 (where ^j' is not greater than ])) real roots between a and b ; 

 but if an even number 2^ of signs have disappeared in the in- 

 terval, there mai/ be 2p — 2p' real roots, and p' pairs of ima- 

 ginary roots corresponding to it, where j/ is not greater than p. 

 If no change of sign disappears, upon the successive substi- 

 tution of a and b, then no root whatever of the equation X = 

 can be found between the limits a and b. 



3rd. If the substitution of a value a of x makes X = 0, then 

 a is a root of the equation. If the substitution of the same 

 value of .r makes at the same time X = and X' = 0, then 



