( 4B3 ) 



times that the two points F and F' through which a curve of each 

 of the pencils is possible coincide. 



7. With the help of this result the class of the envelope of the 

 lines connecting P and P' can easily be determined. To this end 

 we have to count how many lines PP' pass through an arbitrary 

 point >S. We find this number by regarding the correspondence 

 between the rays SP and aS'P', which we call I and /'. This is an 

 involutory [ii, /i)-correspondence where n represents the order of the 

 locus L of the points P and P' ; for on an arbitrary ray / (or /') 

 lie n points P (or P'), to each of whicii one point P' (or P) cor- 

 responds. So there are 2/i rays of coincidence which can be furnished 

 either on account of PP' passing through >S or of Pand P' coinciding. 



So for the number of rays of coincidence where PP' passes 

 through aS' we find : 



2 \^rst + 1) _ 2 (r -f .. 4- ^) — {ar + /?« + yt)\ — |4(«« + <r + rs) — 

 — 6(r -f « -f + Ö — 2 (« -}- /? + y -f ff)j = Qrst _ 4 (s« + «r + rs) + 

 + 2 (r + s + - 2« {r _ 1) _ 2/? (.s - 1) - 2y (« - 1) + 26. 



These rays of coincidence however coincide in pairs. For if the 

 line connecting tlie corresponding points P^ and P/ passes through 

 S, then to PxP^ regarded as line / correspond n lines /', two of 

 which coincide with P^P^, for if point P of / is taken in P^ or in 

 P/ the corresponding point P' lies in P/ resp. Pj. Likewise to P^P^ 

 regarded as line /' correspond n lines /, of which also two coincide 

 with P^Pi, from which ensues that P^P^' is a double ray of 

 coincidence^). So to find the number of the lines PP' passing 

 through S, thus the class of the envelope, the above found number 

 must still be divided by 2, so that we get : 



1) One can easily convince oneself of the accuracy of this conclusion by a 

 representation of the correspondence between the rays SP and SP'. To this 

 end we regard the parameters of the lines SP and SP' as rectangular Cartesian 

 coordinates x and y of a point which is the representation of those two lines. 

 The curve of representation (which is symmetrical M'ith respect to the line y = X 

 on account of the correspondence being involutory) indicates by its points of 

 intersection with the line y = x the rays of coincidence. If B is the point of 

 representation of the rays I and /' coinciding in PiPi', the curve of representation 

 is cut in two coinciding points B by a line parallel to the ?/-axis as well as by 

 a line parallel to the .c-axis, on account of PjPi' regarded as / or /' corresponding 

 twice to itself regarded as /' resp. I. So B is doublepoint of the curve of repre- 

 sentation, so that the lino y = .r furnishes two points of intersection coinciding 

 with B. 



