hiination of x, the relation o;' = .i- + 2 + 8' + 9 X 4' ; hetldé .^^15. 



The transformation (A^ X'), which replaces each point by the two 

 points, which (Z') associates to it, transforms therefore a straight 

 line into a curve of order fifteen luith an octuple point and nine 

 quadruple points. 



As / contains three pairs A^A^', which supply six intersections 

 witti V\ the curve of coincidences 6 is of order nine. Apparently 

 d"' has a quintuple point in A and nodes in B^. 



With a\ 6' has 5X6 + 9X4 = 66 intersections in A and^/^; 

 the remaining six are coincidences of the involution of pairs lying 

 on a\ Analogously we find that /- has four coincidences on /?t\ 



The supports d of the coincidences envelop a curve of the tenth 

 class (f/)io, which has a quintuple point in A. 



5. The locus of the pairs X' , X" , which are collinear with a 

 point E, is a curve f^ passing twice through E where it is touched 

 by the lines to the points E' and E" , which form a triangle of 

 involution with E. It is clear that e^ wi]l pass three times through 

 A and twice through each point B; it is consequently of class 30. 



To the 26 taugents of f', passing through E, belong 10 lines d; 

 the remaining ones are represented by 8 bitangents, which are 

 straight lines s. 



If E is brought in A, then e** passes into «^ For a point Bj. t" 

 consists of ih' and a curve f^.\ which passes through A and the 

 points Bi and has a node in B^. The two curves have 14 inter- 

 sections in the singular points; the remaining two are points ^' and 

 E", belonging to E=Bj,. The 6 tangents passing through B/, 

 at a- are supports of coincidences; the curve (r/)^, has JJj. for 

 node. 



The curve e' has with d" 51 intersections in A and Bk; of the 

 remaining common points 10 lie in the coincidences mentioned above, 

 of which the supports d pass through E. Consequently there lie on 

 t" 11 coincidences A^=e A^', of which the supports do not pass 

 through E, whereas A' and A^" are collinear with E. These 11 

 points belong to the curve e^, which cojitains the points X, for 

 which the line x = X'X" passes through E. The curves t' and e^ 

 also have the points E' and E" in common, forming a triangle of 

 involution with E. As E is collinear with 5 pairs of the J' lying 

 on ((' and with 2 pairs of the P lying on Bk, s^ passes five times 

 through A and twice through B^. Consequently e^ and s^ have in 

 all 3 X 5 + 9 X 2 -f 13 = 64 points in common ; the locus of X 

 is therefore a curve 8^'. 



