335 

 to be found, and to this end I consider the pencil of quartic curves 



These curves have the right lines i:,,§,, ^,, §, as bitangents and 

 the eight points of contact are obviously the points A„ A,, A^, A,, 

 B^, B,, Bt, B^. Now the product KK^ is a quartic curve that passes 

 through 14 of the 16 fixed points that are common to the curves 

 of the pencil. For K touches the bitangents at A„ A^, A^, A^ and 

 iT, has contact with §i,f,,§8 at the points B„ B,, B^. Hence the 

 curve KKi belongs to the pencil, /v, touches §, at the point B^, 

 and there is a certain value Aj of X such that 



It is readily seen that X^ = l, that we must have fi z=i J , and from 



§, §, è, s^ ^ KK, - H- 

 we find 

 ^4 = (?3— <?i) iqx-q^) ^1 + {qx—q,){g,-gt) 'ê, + {^,— ^,)(^.-?i)ê, = o. (4) 



Putting 



^4 = f 1 êi + f*, ^, + M« ^1. 

 we write down that ^i (0,1,1) and ^1/(0, — N^,M) are conjugate 

 points with respect to the conic i^^^^. Thus we find the relation 

 f^, Ih 



— ZM-\ (i/iV + NL + LM) — . 3 iV 4- — {MN + iVZ + LM) 



and a similar relation is obtained by means of the conjugate points 

 A^ and A^. In this way it appears that the right line tp^ will be 

 denoted by 



,»,,=:-37'-f-(^|- + |+|)0W+ A^Z + LJ/). . . (5) 



In the preceding we took for granted that it was possible to 

 represent one and the same conic A' in two different systems of 

 coordinates by the two equations 



K=aMp^^ -f hM?,^ + ctf>,» + 2/ip,V., + 2^i|,',ip. + 2Ai^,ip, =^ 0, . (6) 

 ^= §,' + §,' + §,'- 2|,§, - 2§-.§,- 21 J, = 0, . . (7) 

 the i|>coordinates depending on the ^-coordinates as is indicated by 

 the equations (2), and we have now to examine if these two equa- 

 tions are really consistent. 



Here it is noteworthy that, after introducing certain constants 

 /i>^u^*i» the lefthand side of equation (6) becomes 

 ^i[«n>i + (^-^i)ip, + ig^^Mt^ + tp,[(^ + ^>, -f H, + (ƒ— OV'.] + 



and that the lefthand side of equation (7) can be written 



