984 



For the ftirther iiivesligation tliere remains coiiseqiientl}' the com- 

 bination of one singular point of llie W^^ order, and four singular 

 points of the 2"^^ order : we shall indicate them by Cand Bl (^•=1,2,3,4)- 

 In addition to this we have moreover tlwen singular points of the 

 first order Ak- 



Then there are further three singular lines of the l^t order, ak, 

 four singular lines of the second order and one singular line of the 

 third order. 



The curve {Cf l)elonging to (J has in C a node, which is at the 

 same time node of the curve of coincidence y\ The two curves 

 iiave in (' six points in common ; so also six points outside C; to 

 them belong the two coincidences of the /'^ lying on {Cy ; the 

 remaining four can only lie in the points />. 



As {Cy forms part of tlie curve fM§ •^). belonging to C, a singular 

 line a, passes through 6'. With y\ a^ has in common the coincidences 

 of the [- lying on it, and the two coincidences lying in C ; conse- 

 (juently a.^ cainiot contain any of the points Z^. By the transformation 

 [P,P') it is transformed now into a figure of the G''^ order, of which 

 [Cy and a^ itself form a part ; so the ligure consists further of the 

 singular lines n., and r/,, belonging to two singular points A^, A^ 

 lying on a^. 



The singular line a^ is ti-ansformcd by {/\P') into ^/,^, and a figure 

 of the 5''' order, arising from singular points on that line. As a, 

 does not pass through C and as it must contain, besides the 

 coincidences of the /\ situated on it, (wo more coincidences wiiicli 

 can only lie in points B. we conclude that it bears two points 

 7ii, B^ and the point .4^. From this ensues at once, that n^ too 

 passes through A^, and contains the points B^, B^. 



We consider C, B^, B.^, A^ as base-points of a pencil (qp-) of conies ; 

 C, B^, B^, A^ as base-points of a second pencil (i(''). If each (p" is 

 made to intersect with each ip\ a {P^} will arise, having singular 

 jioints in C, Bk, Ak (see § 12). If to each (f' is associated the \p\ 

 which touches it in C, then the pencils rendered projective by it, 

 generate the ligure {Cy -{^ a^; from this it is evident that [Cy does 

 not only contain the points Bk< but also the singular point A^ = 

 = {B,B,, B,B,). 



It is easy to see now, that A^B,, A,B^, A^B„ and A^B, are the 

 singular lines of the 2"^^ order. For the ff' formed by .43^1 and 

 CB., is cut by (tp') in a /' on A^B^ and a series of points (P) on 

 CB.,; so CB^ is the axis of the involution (//,//') belonging to Jj^j. 



As the axes of the involutions {^p , p"), determined by the four 

 singular lines of the 2"'^ order pass through one point C, the centres 



