general Theory of Multiple Roots. 251 



To make this manifest, suppose r = 2. Then in order 

 tliat an equation may have two pairs of equal roots, we must 

 have by the above formula 



S( gggg ... g„ ) = S (gi fggga — g» )} =0. 



But if instead of there being two perfect square factors 

 there be one perfect a^^g^factor mfx, it may be shown by the 

 same reasoning as above, that the very same two equations 

 apply. In fact, it may be shown in general that no such equa- 

 tions as those given above can be affirmed in consequence of 

 there being an amount r of multiplicity consisting of unit 

 parts which may not be affirmed with equal truth as necessary 

 consequences of the same amount distributed in any other 

 manner whatever. How to obtain affirmative equations suffi- 

 cient as well as necessary (under certain limitations) will ap- 

 pear at the close of this present paper. 



It is worthy of being remarked, that if we make f^i denote 

 the sum of the products of the quantities to which it is pre- 

 fixed, taken fA and jw. together, the equations of affirmation be- 

 come identical with those obtained by eliminating between 



/.*and /^*. 

 "^ dnd 



It can scarcely be doubted that the illustrious Lagrange, 

 had he chosen to perfect the incomplete theory of equal roots 

 given in the Resolution Numerique, by applying to it his 

 own favourite engine of symmetric functions, could scarcely 

 have failed of stumbling by a back passage upon Sturm's me- 

 morable theorem. 



Let us now proceed to show how a polynomical known to 

 contain one or more perfect square factors may be decom- 

 posed. 



Let us begin with supposing that it contains but one such 

 factor ; so thaty^r = <^ x . (x — a)^ 



I shall show how to obtain the equations C {x — a) = 0, 

 D 4; j: . {x-a) = 0, E {x~af = 0, F . (^j x) = 0, each in its 

 lowest terms. 



1. To form the equation L jr + M = 0, where x ~ a, it is 

 easy to see that if we write down in general the expression 



{x—e{) (cc^e^ ... g„) this will become zero whenever the root 



g, left out is not one of the equal roots (a) : so that in fact 

 (calling the two equal roots g, gg respectively) 



• Sec^my Note on Sturm's Theorem, Phil. Mag., December, 1839. [L. and 

 Ji. Phil. Mag. vol. xv. p. 434.] 



