108 



IVom A and Bk two more points E', E" in common ; the lines 

 C\ E', Cj E" touch 7i in C^ and are apparently the onlv possible 

 lines s passing through 6\ ; hence 6\ is a node on the curve {s)^. 



The curve %^ belonging to C'l is represented by the figure com- 

 posed of y^^ and a curve *)\\ which has nodes in Bjc- This may 

 be found independently of what is mentioned above. The trans- 

 formation replacing a point X by the corresponding points X', X", 

 transforms a straight line / into a curve y", consequently the curve 

 Yi^ into a figure of order 44. It consists of 7/ itself (foi' this curve 

 bears go^ pairs A^ A^'), twice y^", the curves ti\ ^k^ y^ and twice the 

 locus of A^"; the latter is therefore of order four. 



Tf E is brought into the centre M^ of the /Mying on /j', f" passes 

 into Yi" and a curve ftj" with node M^. Of the latter 6 tangents 

 pass through M^, whereas this point lies on 2 tangents of y/; from 

 this it ensues anew that the lines d envelop a curve of the eighth 

 class. As yi" apart from A and B\ has with jtj^ four points in 

 common, which must form two pairs of the /^ and so determine 

 two lines s, M^ too is a node of the curve (.s-)^. 



If E lies in A, e" consists apparently of a^ and the three lines 

 h]i\ whereas ^J is the figure com|)Osed of an «^ and the three lines 

 hmi' F'oi' ^ i" ^' ^^ is replaced by the figure formed by tv" and a 

 curve T^, also passing through T and liaving with a^ besides the 

 four points ^4, B^ two more pairs collinear with T\ consequently 

 T is also a node of (.s')4- 



For Bk f" consists of /:?^.' and the line BkA; e J of |ii' and B,J}n- 



7. Passing on to the consideration of the involutive correspondence 

 (A^ Y) we cause A^ to describe the straight line /, and we try to 

 find the locus of the corresponding points Y. On each line X' X" 

 lies a second pair Y', Y" ; the curves <f^ and f/^ which intersect 

 in the points Y' , Y" we shall associate to each other. In order to 

 determine the characteristic numbers of this correspondence, we 

 consider the involutions 7 ■^ which are formed on a curve y' oi' ^f^ 

 by groups of (A^). 



The sides of the L described in a 7' envelop a conic; among 

 the 12 tangents, which this curve has in common with the curve 

 of involution {óc)g belonging to /" must be reckoned the two lines 

 A', A", for which A' is one of the intersections of / and r/'-. The 

 remaining 10 contain each a pair Y', Y" ; consequently each cf'^ is 

 in the said correspondence associated to 10 curves r/^ 



The involution T^ on a g^ possesses a curve of involution of the 

 third class; for B^ bears in the first place the line èj, which contains 



