BY MARTIN GARDINER, C.E. 71 



PROCESS OF INVESTIGATION. 



1. If we assume any three points a lt b^ c^ in the line 

 L , and that we draw the straight lines x o^ 6 X o^ c^ o l to 

 cut L 2 in the points 2 , b 2 , c 2 ; then 



Similarly, if we draw a o. 6 o , c a o,, to cut L in points 



8 * * B Z o * 



a *> b 9 . c 9 ; and that we draw a o_, 6_ o,, C Q o,, to cut L in 



OOO OOOoOd ^ 



4 , 6 , c^ ; and that we proceed thus until a n o n , b n o , c o n , cut 

 L X in a n 1? 6 n 1? C M x ; then evidently we have the following 

 relations (each one of which is similar to the above) : 



n+1 n+l 



C n+l 



And from these n equations we at once derive the equation : 

 ^- : ~ = : * +1 : (!) 



n+l n+l 



From this we learn that p l is known (by the problem of 

 " determinate section " of Apollonius, or because it is a double 

 point of known homographic divisions on L X ) ; and therefore also 



Moreover, we learn that in the non-porismatic state of the 

 data, there are two and but two positions for p v both real or 

 both imaginary. 



