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try ; and the undivided opinion as to their value and importance 

 held hy those who are best qualified to judge of them, sufficiently 

 establishes Mr. Cayley's claim to be regarded as one of the most 

 eminent and profound mathematicians of the age in which we live. 



Mr. Cayley is among the foremost of those who are successfully 

 developing what may be called the organic part of algebra into a 

 new branch of science, as much above ordinary algebra in generality 

 as ordinary algebra is itself above arithmetic. The effect is a vast 

 augmentation of our power over the comparison and transformation 

 of algebraical forms, and greatly increased facility of geometrical in- 

 terpretation. 



To give any full account of Mr. Cayley's labours would be im- 

 possible, from mere want of space ; and such account, were it given, 

 would be intelligible to none but the highest order of mathemati- 

 cians ; moreover, you are well aware, it could not come from my 

 own knowledge of the subject. I have, however, considered it my 

 duty to lay something before you, in the most general terms of 

 description, about these very remarkable papers, obtained from 

 those who are competent to describe them. 



Mr. Cayley's memoirs relate almost exclusively to pure mathe- 

 matics ; and a considerable proportion of them belong to the subject 

 Quantics, defined by him to denote the entire subject of rational and 

 integral functions, and of the equations and loci to which these give 

 rise ; in particular the memoirs upon linear transformations and co- 

 variants, and many of the memoirs upon geometrical subjects, belong 

 to this head. Among the memoirs upon other subjects may be 

 mentioned Mr. Cayley's earliest memoir (1841) in the Cambridge 

 Mathematical Journal, " On a Theorem in the Geometry of Position," 

 which contains the solution in a compendious form, by means of a 

 determinant, of Carnot's problem of the relation between the distances 

 of five points in space ; the memoir in the same Journal, " On the 

 Properties of a certain Symbolical Expression/' which is the first of a 

 series of memoirs upon the attraction of ellipsoids, and the multiple 

 integrals connected therewith ; a memoir in Liouville's Journal, 

 which contains the extension of the theory of Laplace's functions to 

 any number of variables ; and the memoirs in the same two Journals, 

 on the inverse elliptic integrals or doubly periodic functions. The 

 earliest of the memoirs upon linear transformations was published 



