467 



of which the human body is composed, but even assisting us to obtain 

 more accurate notions of those higher qualities, those intellectual and 

 moral qualities, by which man is eminently distinguished from all 

 other animals. In the name of the Royal Society I present to you 

 this Medal as a token of their high appreciation of your labours. 



A Royal Medal has been awarded to Mr. James Joseph Sylvester, 

 F.R.S., for his Memoirs and Researches in Mathematical Science. 



Professor Sylvester's mathematical writings extend over a period 

 commencing in the year 1837 ; separately, and as a whole, they dis- 

 play in an eminent degree the originality and inventive and generali- 

 sing power of their author, and they have very greatly contributed to 

 the advance of pure mathematics, more particularly as regards the 

 Finite Analysis or Algebra, in the widest sense of the word. Several 

 of the earlier papers relate to subjects which are resumed and de- 

 veloped in those of the last ten years ; and on this ground it is right 

 to allude to the researches on the theory of determinants, and 

 the dialytic method of elimination; and also to the remarkable 

 discovery as to Sturm's Theorem. It is well known that the theorem 

 in its original form gave only a process for finding the functions 

 which determine the number and limits of the real roots of an 

 equation ; the determination of the actual expressions of these 

 functions in terms of the roots was an extension and completion of 

 the theory, the merit of which belongs exclusively to Professor Syl- 

 vester. The subject is considered in detail, and various new and 

 valuable results in connexion therewith are obtained, in the elaborate 

 memoir in the 'Philosophical Transactions* for 1853, "On a 

 theory of the Syzygetic relations of two rational and integral functions, 

 comprising an application to the theory of Sturm's functions and 

 that of the greatest algebraical common measure." The same 

 memoir contains also a very original theory of the intercalations or 

 relative interpositions of the real roots of two independent algebraical 

 equations, and a new method of finding superior and inferior limits 

 to the roots of an equation, characterized by the employment of for- 

 mulae involving arbitrary coefficients which may be determined so as 

 to bring the limits into coincidence with the extreme roots. The 

 memoir contains also, in connexion with the subjects to which it 

 primarily relates, valuable researches on the theory of Invariants. 



