( 3) 



is at infinity on «„, this is also the oas(! with M ou /„-. So the 

 point at infinit}' common to «a and /„' corresponrls to itself, i. e. 

 the projective ranges are in perspective. The centre of perspective 

 la is immediately found as the point common to the joints Pi M^ 



and FiM^ (fig. 1) of the pairs 

 4, of corresponding points (Pi.Mi) 



and (Po, ^h)- This point being 

 found, it is possible to indicate 

 the centre M of the circle of 

 .ToACHiJiöTHAL corresponding 

 to any point P of //„. 



Analytically the obtained 

 results are given back as fol- 

 lows. The coordinates of ^4 

 being acosrp^ i sw (^, the equa- 

 tions of the right Hues "a,Za', 

 P^M^, P2M2 are successively: 



J'l ^h ■ 



cos (f 



ax 



- + 



cos (f sin (p 



sin (f 

 2 by 



c"-, la' ■ ■ 

 = c", P2, i/?, 



— 2 ax 2 by 



V -. = c- . 



cos (p sin (p 



— 2 ax by 



coscp 



sm (p 



So the centre of perspective Ta has the coordinates — — cos (p , 



a 



— sin rp and this point describes the ellipse e', the four vertices of 



b 



which are the four real cusps of the evolute of e. And the joints 



A Ta and A' T^ likewise envelope ellipses, etc. 



4. The simple relation between the lines «« and U- proves 



4 iF" 4 //"^ 

 immediately that /„- is normal to the ellipse — + — ^ = 1 and 



a^ b^ 



that through any point M pass four of these normals /„'. In other 



words, any normal to the plane of the circles meets four of the 



rectangular hyperbolae and so contains eight points of the locus. 



As the point at infinity common to all these normals does not lie 



on one of the rectangular hyperbolae, the locus is a surface of the 



eighth order. We confirm 'his result by the deduction of its equation. 



The cones that form the images of all the circles through ^4' and 



all the circles through 0"' are represented by 



1* 



