( 2) 



nally; of this hyperbola the line common to its plane and the plane 

 of the circles is the imaginary axis, the intersection of this line 

 with P Q is the centre, and the normal in this point to the plane 

 of the circles is the real axis. Or, put more generally: the circles 

 of a pencil are represented by a rectangular hyperbola, whose real 

 or imaginary axis is normal to the plane of the circles according 

 to the points common to the circles being real or imaginary, and 

 which breaks up into two straight lines, if these points coincide. And 

 the circles of a general net arc represented by a rectangular hyper- 

 boloid with one or two sheets, according to the circle cutting the 

 circles of the net orthogonally being real or imaginary. 



In different eases the Fiedlerian theory can give a clear and 

 concise idea of the position of ranges of circles. So a. o. the circles 

 having double contact with a given ellipse e. If this ellipse f be 



represented by "— + — = 1 , ^ =x 0, the ellipse — + 1- = 1 y = 



and the hyperbola ^-f-^=:l, j=0 correspond to the two 



a^ — b" a~ 



ranges of circles having double contact with s. As is easily explained 

 these curves are transformed into the focal conies of f by inverting 

 the sign of z""'- 



In the following lines we study the surface that forms the image 

 of the twofold infinite system of Joachimsthal's circles of f. 



2. Through any point F of the plane of s can be drawn four 

 normals to f. The footpoints A^ J5, C, D of these normals may be 

 called „conormal". If the point diametrically opposite to ^ on « is 

 indicated by A\ the known theorem of Joachimsthal says that 

 A\ B, C, D are concyclic, if vt, J5, C, D are conormal. 



This non-reversible theorem has been completed by Laguerre 

 in remarking that the circle A' B C D meets the tangent ^a. in A' 

 to Ê for the second time in the projection 0»' of the centre of g 

 on ta'. In other words: 



If P describes the normal «« in A to f, the corresponding circles 

 A' B C D form a pencil, as all these circles pass through ^1' and 

 Oa- . This pencil being represented by a rectangular hyperbola, the 

 imao-e in question is the locus of a simply infinite number of rec- 

 tangular hyperbolae. However, before we proceed to the deduction 

 of this surface, we investigate somewhat more closely the corres- 

 pondence between the points P of the normal »/„ and the centres 

 M on the line /„' bisecting orthogonally the segment A O'a. 



3. The relation between the points P and .1/ on «« and /«■ is a (1,1) 

 correspondence, i.e. these points describe piojective ranges. If P 



