tlie A.vial Dioptric System. 337 



i'or linos which pass tliroufrh (!, tlic new lines are the loci of 

 the new points corresponding- to the old points on these lines. 



Tluis in the transtbrniation defined here, to every old line 

 or old point there corresponds a new line or new point such 

 that the old lines through the old point have their new lines 

 throuoh the corresponding new point, and the old points in 

 an old line have their new points in the corresponding new 

 line, and vice versa. 



It is obvious that C is its own new i)oint and a f^ its own 

 new line. 



AVe shall next prove that the lines joining the points on 

 any Hue C L through C to their new points, are concurrent 

 in a point lying in oc p. 



Let Lq Pq Q^,, Li Pi Qj, L2 P2 Q2 be three straight lines con- 

 current in E, meeting C A in Pq, P^, Pg and C B in Qq, Qi, Q2 

 and C L in Lo, Li, Lo ; and let Po', P/, P2', Qo', &c. be the 

 corresponding new points. 



Then Po' Q/, P/ Q/, P2' Q2' will be concurrent in W, and 

 will meet the now line for C L in Lq^, L^^, Lo^, where Lq', L^', L2' 

 are the new points for Ly, L^, L2. 



We have 



R'{C L; L/ Lo'} =E/{C Po' P/ P2'} = a{C Po' Pi'P2^} 

 = a{CP,PiPo}=R{CP,,PiP2^-R{CLoL,L2}. 



Thus C Lq' Li' Lo^ and C L^ L^ Lo are equi-harmonic ranges, 

 so that Lq Lq', Li Li^ and L2 L2^ are concurrent in a point \. 



Further, since ayS is a line which corresponds to itself, its 

 intersections wdth C L and C L' are corresponding points, 

 whose join therefore passes through \. Thus \ lies in u ^. 



The point X we shall call the vertex for C L, or for any 

 point in it. It is the point of concurrence of all the lines 

 joining old points in C L to their new points. 



It is now obvious that we should get exactly the same 

 transformation as before if we were to employ C L, C L', and 

 \, and any other old line through C with its new line and its 

 vertex, instead of C A, C A', «, and C B, C B', /3. 



It is easy to see that if C goes to infinity in a direction 

 perpendicular to a/3, then that part of the system which 

 remains at a finite distance from a/3 reduces to the system 

 studied in Section L, the old and new lines becoming in- and 

 out-rays ; and the old and new points becoming object-points 

 and image-pjointSj while u /3 becomes the axis. 



Reverting to the general transformation from old points to 

 new points, we shall next prove the dualistic correlative to 

 the vertex-property to hold good. 



Phil Mag. S. 0. Vol. 6. No. 33. Sept. 1903. Z 



