ON PRESENT PROBLEMS OF ALGEBRA AND ANALYSIS 



BY HEINRICH MASCHKE 



[Heinrich Maschke, Associate Professor of Mathematics, University of 

 Chicago, b. Breslau, Germany, October 24, 1853. A.B. Magdalenen Gym- 

 nasium, Breslau, 1872; Ph.D. Gottingen, 1880. Post-graduate Heidelberg, 

 Breslau, Berlin, and Gottingen. Professor Mathematics Lvisenstadt. Gym- 

 nasium, Berlin, 1880-90; Electric Engineer at Weston Electric Company, 

 Newark, New Jersey, 1890-92; Assistant Professor of Mathematics, Uni- 

 versity of Chicago, 1892-96.] 



As set forth by the Committee directing the affairs of this Interna- 

 tional Congress, the address which I have the distinguished privilege 

 of delivering to-day shall be on " Present Problems in Algebra and 

 Analysis/' - - but it is not provided by the Committee how many 

 of these problems shall be treated. 



The different branches of algebra and analysis which have been 

 investigated are so numerous that it would be quite impossible to 

 give an approximately exhaustive representation even only of the 

 most important problems, within the limits of the time allowed to 

 me. I, therefore, have confined myself to the minimum admissible 

 number, namely one, or rather one group of problems. 



Of this one problem, however, this Section of Algebra and Analysis 

 has the right to expect that it is neither purely algebraic nor purely 

 analytic, but one which touches both fields; and at least in this 

 respect I hope that my selection has been fortunate. 



I purpose to speak to-day on the Theory of Invariants of Quad- 

 ratic Differential Quantics. Invariants suggest at once algebra, 

 differential quantics: analysis. At the same time the subject also 

 leads into geometry, - - it contains, for instance, a great part of 

 differential geometry and of geometry of hyperspace. But is there, 

 indeed, any algebraic or analytic problem which does not allow 

 geometrical interpretation in some way or other? And when it comes 

 to geometry of hyperspace, - - it is then only geometrical language 

 that we are using, --what we are actually considering are analytic 

 or algebraic forms. Moreover, rigorous definitions and discussions 

 of geometrical propositions of an invariant character in particular 

 can only be given by tracing them back to their analytic origin. 



In the following exposition I shall first speak on the various in- 

 variant expressions of differential quadratics as they occur in geo- 

 metry of two and more dimensions, and then take up the purely 

 analytic representation in the second part of the paper. 



This corresponds also to the historical development of the sub- 



