( 538 ) 



For the new origin A\ the pair of tangents out of A\ to the conic through 

 the other basepoints. < >r, if one likes, just as ./'' -f- //'-' is with the exception 

 of a numerical factor, the fourth transformation ("Ueberschiebung")ofthe 

 first member of the equation Q = of the lines connecting A' t to 

 the remaining- basepoints, so a?" -\- 5y 2 represents, likewise with the 

 exception of 'a numerical factor, the fourth transformation of the first 



Py 



member of the equation - - = 0, which indicates with respect to the 



,/■ 



new origin A\ the live lines connecting A\ to the remaining base- 

 points. 



Finally the equation of c" is 

 ()V _7 () « + 20^-7^ + l) + 22V-V-3^^1)-f4QV-2)=0, (1) 

 or entirely in polar coordinate- (o, <p) 



4(^-2)9 B C o S 5y=(^+l)(^ 4 -49'+l)±(^-l)V(9 , -l)(49 1 H ' " ""/-!)• (2) 

 It is easy to show that this curve admits of no real points differing 



from the six basepoints A'i and the ten points of intersection of the 



triplets of connecting lines. If for brevity we write (2) in the form 



L cos o(p = M ± \/\. 



then we find 



— V shS h(f — (.I/ 2 -f N — L'-) db 2 M \/N ... (3) 



and 



M' -fA T — IJ = 2 (o 2 — l) 2 (2o 2 -l)(f/ -6 ? a -fl4 ? 4 -f 2o 2 — 1), 



(M'+N-L*)* - W N = 4q*{q*—1) 4 ( (/ »._2) s ((> 4 -7p , 4-l) 1 ! 



If now wc moreover nolice that N is negative and therefore 



1 

 <;is nip complex when o 2 lies between — and 1, the following is 



O 



immediately evident : 



a. The lirst member of the second equation (4) tends to zero, 



when q* assumes one of the values 0, 1, 2, — (7 ± 3e); it is positive 



for all other values of q 2 . 



b. If VN is real and q* dilfers from unity the second member 

 of the lirst equation (4) is positive; for the equation 



has, as is evident when the roots ^ 2 are diminished by 1 — , besides 



Li 



1 



one real negative root only one real positive one between — andl. 







c. If 9 s dilfers from 0, 1, 2, — (7 =b 3e) the second member of 



• (4) 



