135 



trom which it appetirs that F^ contains three pairs of the involution ; 

 the latter is consequently of the tklrd cla.ss. 



2. We shall now suppose that the two nets have a common base 

 point .1; they produce then a triple uwokUlon o'l {\\c tliinl chis.s. We 

 choose the base point A for vertex O^ of a triangle of co-ordinates. 



Through O^ pass oc^ conies of the first net, which are touched 

 there by the corresponding conies. For we have the conditions 



^P.ttjj = T^ P.Z>j3 and JS'Aa^., =z T ^ 6,3, 



3 3 3 3 ' 



so that the parameters A, /', )." are connected by the relation 



2a^J. 



3 



3 

 3 



z=0. 



(3) 



Now we find from (1) 







etc. 



If we substitute these formulae P., a', a" in (3), an equation of the 

 eighth order will arise. The locus of the pairs X' , X" of the triple 

 involution (X') associated to O^^^A is therefore a curve of the 

 eighth order, which we shall indicate by h^ ; A. is a singular point 

 of order eight. 



By (3) two projective systems with index two are separated from 

 the two nets, which systems produce the curve «'. Their intersec- 

 tions with the arbitrary straight line r, are the coincidences of the 

 (4,4), which the two systems determine on r. If r is laid through 

 A, tiie free points of intersection are connected by a (2,2) ; one of 

 the 4 coincidences of this con-espondence lies in A, because two 

 homologous conies touch each other and r in A. Hence it appears 

 that the .singular curve ci^ has a quintuple point in A. This corre- 

 sponds to the fact that (A'^) must be of the third class; the three 

 pairs on a straight line r laid through A are formed by A with the 

 three points in which r is moreover cut by «\ The line .v=X'X" 

 envelops a curve of the fifth class ; for of the system {x) only the 

 lines which touch ci^ in .1 pass through A. 



3. A is not the only singular point of {X^). The homologous 

 conies intersecting in a point Y are determined by 



:Eia,^ 







and 



2).b,; 







3 3 



If these equations are dependent, Y becomes a singular point. 



10* 



