251' Prof. Sylvester's Theory of certain Multiple Boots. 



To wit, on selecting any r roots out of the entire number, 

 either these r will all be found again in those that are left, or 

 those that are left will contain inter se, one repetition at least ; 

 so that except on the latter supposition any (? — 1) may be ab- 

 solutely sunk out of those thatare left, and there will still be 

 o7ie root common to the [n—2 r+\) remaining, and to the r 

 originally selected to be left out. 



Wherefore calling the roots e, e^ ... e„, and giving fi any 

 value whatever, we have 



s(/^ (.,.,... e,) X (siisiz)x2(;;,::;;; ::...)} =«• 



Hence the simplest distinctive equations indicative of the 

 existence of r pairs of equal roots are to be found by putting 

 ju, equal in succession to all values from up to (r — 1). 



For instance, if we require that an equation of the seventh 

 deo-ree shall have three {)airs of equal roots, we need only to 

 call the seven roots respectively abed ejg, and then our 

 type equation becomes 



From this it appears that the r distinctive equations for 

 r pairs of equal roots are of different dimensions from the r 

 ffeneral or overlying ones corresponding to the multiples 

 r, anyhow distributed; the lowest of the latter being of 

 (^n — r+1) (?2—?-)> the lowest of the former of {n — y).{ri — r— 1) 

 + 2 r (?j — 2 r+ 1) /. ^. of n. (« — 1) — 3 r (« — 1) dimensions. 

 In general we shall find that the more unequally distributed 

 the multiplicity may be the lower are the dimensions of the 

 distinctive equations, and are accordingly lowest when the 

 multiplicity is absolutely undistributed*. 



22, Doughty Street, Mecklenburgh Square. 



* It must not, however, be overlooked, that the equations above given, 

 although decisive as to the existence of /• pairs of equal roots ivhen the 

 multiplicity is known to be not greater than r, do not enable us to affirm 

 with certainty their existence when this limitation is absent : for should 

 the multiplicity exceed ;•, then inevitably (no matter how it may 

 be distributed) (e,i^\ <?r+2 ... <■« ) is always zero, and consequently 

 nullifies each term of every one of the equations in question. In fact 

 (repugnant as it may appear to be to the ordinary assumptions of analy- 

 tical reasoning), it is not possible to express with absolute unambiguity the 

 conditions of there being a niultiiilicity (?•) distributed in any assigned 

 manner by means of r uffi-rmat'we equations alone. 



[To be continued.] 



