46 Prof. R. Clausius on the 



time to mention the very interesting portion of the Topologie 

 which Listing worked out in detail. You will find a brief 

 synopsis of a part of it prefixed to Clerk-MaxwelPs i Elec- 

 tricity and Magnetism/ and Cayley has contributed an ele- 

 mentary statement of its contents to the c Messenger of Mathe- 

 matics ' for 1873; but there can be no doubt that so important 

 a paper as the Census raumlicher Complete ought to be trans- 

 lated into English. 



To give an exceedingly simple notion of its contents I may 

 merely say that Listing explains and generalizes the so-called 

 Theorem of JEuler about Polyhedra (which all of us, whose 

 reading dates some twenty years back or more, remember in 

 Snowball's or Hymers' ' Trigonometry '), viz. that "if S be 

 the number of solid angles of a polyhedron, F the number of 

 its faces, and E the number of its edges, then 



S + F=E + 2." 



The mysterious 2 in this formula is shown by Listing to 

 be the number of spaces involved ; L e. the content of the 

 polyhedron, and the Amplexum, the rest of infinite space. 



And he establishes a perfectly general relation of the 

 form 



V-S + L-P=0, 



where V is the number of spaces, S of surfaces, L of lines, 

 and P of points in any complex ; these numbers having pre- 

 viously been purged in accordance with the amount of Cyclosis 

 in the arrangement studied. But to make even the elements 

 of this intelligible I should require to devote at least one 

 whole lecture to them. 



Meanwhile I hope I have succeeded in showing to you 

 how very important is our subject, loose and intangible as it 

 may have at first appeared to you; and in proving, if only by 

 special examples, that there are profound difficulties (of a kind 

 different altogether from those usually attacked) which are to be 

 met with even on the very threshold of the Science of Situation. 



V. On the Theory of Dynamo-electrical Machines, 

 By Ii. Clausius*. 



THE practical development of dynamo-electrical machines, 

 like that of the steam-engine in its time, has been in 

 advance of its theoretical treatment, and it is only recently 



* Delivered at the Annual Meeting of the Schweizer. Naturforschende 

 Gesellschaft, held at Zurich, August 8, 1883. Translated from a separate 

 impression from the Annalen der Thysik und Cliemle ) Band xx. ? 1883, 

 communicated by the Author. 



