40 KEPOBT — 1881. 



In the hands of Sylvester, of Kempe, and others, this principle haa 

 been developed into a general theory of link- work, on which the last word 

 has not yet been said. 



If. time permitted, I might point out how the study of particular 

 geometric figures, such as curves and surfaces, has been in many instances 

 replaced by that of systems of figures infinite in number, and indeed of 

 various degrees of infinitude. Such, for instance, are Pliicker's com- 

 plexes and congruencies. I might describe also how Riemann taught us 

 that surfaces need not present simple extension without thickness ; but 

 that, without losing their essential geometric character, they may consist 

 of manifold sheets ; and thus our conception of space, and our power 

 of interpreting otherwise perplexing algebraical expressions, become 

 immensely enlarged. 



Other generalisations might be mentioned, such as the principle of 

 continuity, the use of imaginary quantities, the extension of the number 

 of the dimensions of space, the recognition of systems in which the 

 axioms of Euclid have no place. But as these were discussed in a recent 

 address, I need not now do more than remind you that the germs of the- 

 great calculus of Quaternions were first announced by their author, the 

 late Sir W. R. Hamilton, at one of our meetings. 



Passing from geometry proper to the other great branch of mathe- 

 matical machinery, viz. algebra, it is not too much to say that Avithin 

 the period now in review there has grown up a modern algebra which 

 to our founders would have appeared like a confused dream, and whose 

 very language and terminology would be as an unknown tongue. 



Into this subject I do not propose to lead you far. But, as the 

 progress which has been made in this direction is certainly not less than 

 that made in geometry, I will ask your attention to one or two points- 

 "which stand notably prominent. 



In algebra we use ordinary equations involving one unknown quan- 

 tity ; in the application of algebra to geometry we meet with equations, 

 representing curves or surfaces, and involving two, or three, unknown 

 quantities respectively ; in the theory of probabilities, and in other branches 

 of research, we employ still more general expressions. Now the modern 

 algebra, originating with Cayley and Sylvester, regards all these diverse 

 expressions as belonging to one and the same family, and comprises them 

 all under the same general term ' quantics.' Studied fj-om this point of 

 view, they all alike give rise to a class of derivative forms, previously 

 unnoticed, but now known as invariants, covariants, canonical forms, etc. 

 By means of these, mathematicians have arrived not only at many pro- 

 perties of the quantics themselves, but also, at their application to physical 

 problems. It would be a long and perhaps invidious task to enumerate 

 the many workers in this fertile field of research, especially in the schools 

 of Germany and of Italy ; but it is perhaps the less necessary to do so, 

 because Sylvester, aided by a young and vigoi-ons staff at Baltimore, 

 is welding many of these results into a homogeneous mass in the 



