CYCLOGRAPHIC TRANSFORMATION OF 

 ORDINARY SPACE 



Arnold Emch 



1. Intkoduction. 



In my foregoing paper on general congruences of curves in space, 

 I have referred to a particular case of congruences of straight lines 

 which result from the partial differential equation. 



[ox 



A-2 + 1 



when X, y, z are the co-ordinates of a point describing a curve in 

 space.^ 



These congruences have the closest connection with the geom- 

 etry of the circle in a plane, and with the method of cyclography as 

 it has been established by Steiner, Fiedler and others. This method 

 admits of beautiful applications, and as it seems to be but little 

 known, I shall attempt to present its principal features in an ele- 

 mentary manner,^ and show its relations with more advanced theories. 



^ S. Lie, loc. cit. 



^ In 1826, J. Steiner announced that he had a manuscript, "iiber 

 das Schneiden (mit Einschluss der Beriihrung) der Kreise in der 

 Ebene, das Schneiden der Kugeln im Raume und das Schneiden der 

 Kreise auf der Kugelfliiche," ready for print. As W. Fiedler 

 remarks, this paper is not in his collected works and must, therefore, 

 have been lost. The foundations of this method were laid by 

 Cousinery, who, in 1828, (Paris) published his Geometric jpers;pective. 

 He introduced the cercle a distance, which plays such an important 

 part as Distanzkreis in Fiedler's early investigations, and solved by 

 this means Apollonius' problem. The establishment of this method 

 as an independent geometrical branch, however, is due to Professor 

 W. Fiedler, of Zurich, who, in 1880-82 published two memoirs 



