338 Mr. R. F. Muirhead on 



Let a be the vertex for any point P, and P' the corre- 

 spondino- new point, and let a' be the new point for a. The 

 line a P P^ looked at as an old line has its new line passing- 

 through a^ and P^ Thus P^ is the intersection of the old 

 line a P and its new line. 



Hence C P^, the new line whose old line has a as vertex, i& 

 the locus of the intersections of all old lines through a with 

 their corresponding new lines. C P^ we shall call the base- 

 line for a. 



Thus we have the Proposition : 



All the old lines through an old point on the axis intersect 

 their corresponding new lines in a fixed line through C, the 

 base-line of a. 



By means of this property, we have the following simple 

 construction to find the new point corresponding to any old 

 point R, having given two points a ^ on the double-line, their 

 new points a^ and ff\ and their respective base-lines, C A^ and 



Draw RaP^ R/5Q' meeting C A' in P^, CB^ in Q^ 



Then R^, the intersection of a^ P^ and yS^ Q^ is the new point 

 for R. 



Double-Points. 



In our general transformation it appears in virtue of the 

 vertex-property that the anharmonic ratio of points in any 

 line through C is unaltered by the transformation, and this 

 can be shown to be true for other lines as well. 



For if P Q R S be four collinear old points, and these be 

 joined to a point 0, we get a pencil cutting any line L 

 through C in pgrs^a range equi-harmonic with P Q R S. 

 Now transform the whole figure, and we get two equi- 

 anharmonic ranges f' q' r' s' and P^Q^R^S^, of which the 

 former is equi-anharmonic with p q r s. Hence {PQRS} 



= {P'Q'K''!S;}. _ ::P^-=: 



Now the line «/3 corresponds to itself. Hence it must 

 have two double-points Dj and Dg? I'^^l ^^ imaginary. 



Thus C Dj, C D2, and D^ Dg are double-lines (lines which 

 transform into themselves) and C, Dj, J) 2 are douhle-2:)oints, fe 



A double-point has its vertex in coincidence with itself, so 

 that the construction of new points has to be modified when 

 the given object-points lie in the double-lines. We may 

 proceed thus : — 



Let Pi be an old point on C Dj^ and Pj^ its new point. 

 Through C draw any other line (C D2 would do) cutting 

 Di D2 ill E, and from any arbitrary point in D^ D2 draw 

 Pi, OP/ cutting CE in^j and/. ^ 



