103 



onl.v. With au arbilrary </•', /i, lias moreover iii conunoii the three 

 points which form a quadruple with 7i, ; consequently 27 points in 

 all. So the triplets of {P') belonging- to B, lie on a curve of order 

 nine, which passes three times through each of the remaining base-points. 

 We found that i>\ and H, belong to three quadruples; the three 

 paij'S, which those quadruples cojitain besides, belong to the singular 

 curves /i/ and ^^\ Thej have moreover in the seven remaining 

 points Bk, 63 points in common ; the remaining 12 common points 

 are found in the singular points 1)/^. 



3. The locus of the points of inflection / of {rp^) has triple poijits 

 in Bjc, has therefore with an arbitrary (f>\ 9 X 3 -|- 9 = 36 points 

 in common ; it is consequently a curve of order twelve, t'". On a 

 curve d' lie only 3 points of inflection ; we conclude from this, tliat 

 1.^- has nodes in the twelve points U/, ; in each of those points t'^ 

 and ff' have the same tangents. 



The points P' , P", P", which have / as tangential point, lie in a 

 straight line, the harmonic polar line h of 1. So t^'^ is the locus of 

 the points, which in {P'^) are associated to linear triplets. 



The curves /ii" and t^" have in the singular points B and D 

 8 X 3' 4- 12 X 2 = 96 points in common ; on fi^' lie therefore J 2 

 points I, so that B^ belongs to 12 linear triplets. From this it ensues 

 by the way, that the involution [B^) lying on /i/ has a curve of 

 involution {p) of class twelve; for the line p = P' P" will only pass 

 through i?i if P'" is a point of inflection, while P lies in B^. As 

 B^ is point of inflection of three (f\ (P^) has three linear triplets, 

 consequently [p)-^^ three triple tangents. 



The locus ;. of the linear triplets has, as was shown, 9 dodecuple 

 points B ; as (f* bears nine points of inflection, therefore 9 linear 

 triplets, it has with A 9 X 12 -|- 9 X 3 = 135 points in common. 



Consequently the linear triplets lie on a curve /^\ 



4. We shall now consider the curve o, into which a straight 

 line /* is transformed, if a point P of ;• is replaced by the points 

 P' , which form a quadruple with P ; for the sake of brevity we 

 shall speak of the ti-ansformation {P, P'). If we pay attention to the 

 intersections of r with ,'?^.'' and with rf/j', we ariive at the conclusion 

 that Q has nonuple points in Bk and triple points in I)h. It has 

 therefore with a ((^ in Bk 81 points in common ; further these 

 curves cut moreover in the three triplets which correspond with the 

 intersections of / ^ and r. Consequently ^ is a curve of order thirty. 



