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courses have this year (1897) been published for the first time by 

 the London Mathematical Society; they have attracted considerable 

 attention, and have already led to a remarkable paper by Mr. G. B. 

 Matthews, F.R.S. 



In these researches Sylvester, standing upon the shoulders of 

 Cauchy, showed how to form an algebraical expression, involving 

 the imaginary roots of unity of different orders for the general co- 

 efficient in the associated generating function. It was a piece of 

 analytical skill that could only have proceeded from a rnind endowed 

 with imagination of the highest order. 



In 1864 appeared in the 'Philosophical Transactions of the Royal! 

 Society ' a paper which will perhaps be considered his greatest 

 achievement. The title is " Algebraical Researches : containing a 

 Disquisition on Newton's Rule for the Discovery of Imaginary Roots, , 

 and an allied Rule applicable to a particular class of Equations, 

 together with a complete Invariantive Determination of the Character 

 of the Roots of the General Equation of the Fifth Degree, &c.' r 

 Newton had given in the * Arithmetica Universalis ' a rule for dis- 

 covering an inferior limit to the number of imaginary roots in an 

 equation of any degree, but without proof or indication of method or 

 marshalling of evidence. Maclaurin, Campbell, Euler, and Waring 

 had also treated the question, but either failed to obtain a solution 

 or had fallen into serious error in the attempt. Sylvester's memoir, 

 described by him as a Trilogy, falls into three parts ; in the first he 

 establishes Newton's rule in regard to algebraical equations as far- 

 as the fifth degree inclusive ; in the second he obtains a rule appli- 

 cable to equations of the form 



m being any positive integer, and a, b real coefficients ; in the third 

 he determines the absolute invariantive criteria for ascertaining the 

 exact number of real and imaginary roots appertaining to an equa- 

 tion of the fifth degree. Here, as in his treatment of the Partitions 

 of Numbers, he has frequently resorted to geometrical intuition. In 

 the present investigation every superlinear function is conceived to 

 be in association with a pencil of rays constructed in a definite 

 manner, and much of the argument is given in the language of the 

 geometry of pencils. During a conversation with the writer in the 

 last weeks of his life, Sylvester remarked as curious that notwith- 

 standing he had always considered the bent of his mind to be 

 rather analytical than geometrical, he found in nearly every case 

 that the solution of an analytical problem turned upon some quite 

 simple geometrical notion, and that he was never satisfied until he 

 could present the argument in geometrical language. 



During these years he continually wrote upon the theory of inva - 



