in the Theory of Determinants. 417 



of them unfamiliar, viz. " gedoppelte Verbindungen " and 

 " Producte mit Versetzungen," and turned over the necessary 

 pages to see what exactly these terms were used to denote. 

 To my astonishment, " Producte mit Versetzungen " proved 

 to be nothing more nor less than Determinants ; and I soon 

 saw that what I had been fortunate enough to light on was a 

 very complete treatise on the subject, extending to 113 pages. 

 The historical interest attaching to this work of Schweins 

 would alone render it desirable that some account of the con- 

 tents should be published; but an additional inducement exists 

 in the fact that some of the results are not without a savour 

 of freshness even at the present day. 



2. The Theorie der Producte mit Versetzungen begins at 

 p. 317. There is an introduction of four pages, in which the 

 author refers in the first place to mathematicians who had 

 previously dealt with the subject, and then indicates, in the 

 form almost of a table of contents, the nature and extent of his 

 own investigations. The names he mentions are those of Euler, 

 Cramer, Bezout, Hindenburg, Rothe, Laplace, Desnanot, and 

 Wronski. One of these, Desnanot, it may be remarked, is 

 even now new to us in connexion with the history of Deter- 

 minants. As for their influence upon Schweins, it may be 

 said in a general way that, while he belonged to the Combi- 

 natorial School of Hindenburg, it seems to have been the 

 writings of Wronski that in the main directed and inspired 

 his efforts. 



What is not a little remarkable is the omission of Cauchy's 

 name, the one really great name that could have been referred 

 to in connexion with the subject. Cau dry's memoir (Sur les 

 fonctions &c.) appeared in the year 1812 ; so that, so far as 

 date of publication is concerned, it w r ould seem to have stood 

 as good a chance of being known as either the work of 

 Wronski (1811 &c.) or of Desnanot (1819). Its existence 

 undoubtedly lessens the importance to be attached to Schweins 1 

 work. 



3. The Theorie consists of four sections (Abtheilungen) , 

 subdivided into portions which we may call Chapters. The 

 first Section has five chapters; the second also/rg ; the third, 

 one; and the fourth, four. 



Sect. I. Chap. 1. 

 On the Nature of "Producte mit Versetzungen.'''' 



4. The " elements " from which the products are formed 

 are denoted thus : — 



a, a, 



Aj, A l3 . . . 



