1 36 Prof. Sylvester on the Amount and Distribution 



only result was that it became yellow when heated. I next tried 

 it with borax, in which it dissolved very slowly, and did not 

 colour the glass. I was still at a loss to say what it was, but 

 suspecting that it contained some metal, which was indicated 

 by the white sublimate, 1 tried it with carbonate of soda on 

 charcoal, and it speedily yielded brilliant metallic globules, 

 which tarnished rapidly after cooling ; they were malleable, 

 and when flattened out presented the appearance of tin. 



No doubt now remained as to its nature; and I have only 

 to add, in corroboration of the assertion of Messrs. Mills, 

 King, and Weaver, that " native oxide of tin " exists in the 

 county Wicklow. 



XXIX. Introduction to an Essay on the Amount aiul Distribu- 

 tion oftheMidtiplicitij of the Roots of' an Algebraic Equation. 

 By J. J. Sylvester, F.R.S. Sfc., Professor of Natural Phi- 

 losophy in University College, London*. 



T USE the word multiplicity to denote a number, and di- 

 stinguish between the total and partial multiplicities of the 

 roots of an algebraic equation. 



There may be /• different roots repeated respectively //j h^ 

 ... A*" times. 



r is the index of distribution. 



^j ^2 ... //,. are the partial mutiplicities, and if /i = //i+ //g 

 + ... +h, 



h is the total multiplicity. 



The total multiplicity it is clear may be defined as the dif- 

 ference between the index of the equation and the number of 

 its roots distinguishable from one another. 



In this Introduction, I propose merely to consider how ex- 

 isting methods may be applied to determine the amount and 

 distribution of multiplicity in a given equation, and conversely, 

 how equations of condition can be formed which shall imply 

 a given distribution and amount. 



Let the greatest common factor betweeny.r (the argument 



dfx 

 of the proposed equation) and -j-^ be called,/, a?. 



And in like manner, let the greatest common factor of/i^ 



and - ^ ^ be called fa x . and so on, till in the end we come to 

 d X 



. , df. X 

 /I . .r, which has no common factor with —^ — . 

 J" ^ ax 



Let k^ l„ ... kr denote the degrees in ,r of fv f x ...f . x 



respectively. 



• Communicated by the Author. 



