general Thewy of Multiple Roots. 253 



Let us begin with supposing that /a: has one pair only of 

 equal roots; to form the simplest quadratic equation contain- 

 ing this pair, write down 



{x-e{){x-e^)x (7^e^ ... g„) x ( ll^l^.r)' 



Now if ^1 and e„ are the two equal roots in question the 

 multipliers of [x — e^) {x—Cc^, neither of them vanish. 



If Cj and ^2 are neither of them equal roots ( e^e^^... e„ ') = 0. 



If one of the two only belong to the pair of equal roots 



r^ '^ ] = 0. 



Hence it is clear that 



2 ((^-.,) {x-e,) X (J^^T;^^ X (^> 'I ^,, ^J ) = 



is the equation desired. 



In like manner if there be (;) pairs of equal roots the equa- 

 tion of the (2 r)th degree which contains them all may be 

 written 



xUx-e,) [x-e,)...{x-e,;) x ( ^.,.4.i...^0 x('' e^,,.e,>^-\^ 



The coefficient of x^*" in this equation is clearly of (?j— 2r) 

 (?j— 2r— 1) +4.?- (?z — 2?-) i.e. of(?« + 2r— 1) (w— 2 r) di- 

 mensions. The coefficient of a;'' in the equation which con- 

 tains the r equal roots unyoked too'ether is of (?z — r) {n—r— 1) 

 dimensions, and consequently the coefficient of x^^ in the 

 square of this equation would be of 2 {ii — r) {n—r — 1) dimen- 

 sions, i. e. would be ii-+ 6 r^ — (4'?' + l) 7i dimensions higher 

 than needful. 



Finally, to obtain an equation clear of simple as well as 

 double appearances of the equal roots, we have only to write 

 the complementary form 



t-[{x—e^r+i) {x-eir+2) {x-e„) xC ^2,+i — g» ) 



X (^1 ^^ ••• ^-M1 = 0. 



Let us, now that we are more familiarized with the notation 

 essential to this method, revert to the question with which we 

 set out and endeavour to obtain r sucli equations as shall 

 imply unambiguously the existence of;- pairs of equal roots. 



The existence of r such pairs enables us to assert the fol- 

 lowing disjunctive proposition, which cannot be asserted when 

 the same amount o[ multiplicity is distributed in any other way. 



0. 



