106 



llie conic y/,^ passing through Ch, wiiich cöiiic belongs to (y^); the 

 straight lines x intersect in a point Mk, the centre of the /\ 



In order to find the loens of the pairs, corresponding to B^ , we 

 associate to each r/^ the f/'% which touches it in B^ . The pencils 

 being projective on this account pi'oduce a curve of order Jive, ^^\ 

 which has a triple point in 7>\, nodes in B.^, Bg and passes through 

 A and C),. If the straight line jj^=:X'X" is associated to the straight 

 line, which touches the corresponding curves (f' and (f'^ in B^, a 

 correspondence (J, J) arises between the "carve of involution" enveloped 

 by X and the pencil of rajs B^ ; from this it ensues that {jj) must 

 be a rational curve. As no other lines .v can pass through B^ but 

 the tangents at ^^^ in the triple point />\, (.x') is a rational curve of 

 the third class, has consequently a bitangent ; on it lie two pairs of 

 (A'"). To the tangents of {.v)^ belong the lines AB.^ and ^4^3. 



Thei-e are three singular straight lines hk^=^ ABk; each of them 

 bears a /^ of pairs X', X". The corresponding points X lie on the 

 line b„,„ = B,nBn- 



3. The curve of coincidences (locus of the points Xim X') has 

 triple, points in B]. and passes through A and 6V With the singular 

 curve ■)''\ it has JO intersections in A and Bu; as it touches it in 

 C'l and at the same time contains the coincidences of the involution 

 (A', X") lying on 7%, it is a curve of order seven'^), which will be 

 indicated by d'. It passes through the 12 nodes of (^') and the 3 

 points {l>hbi,n). 



As ff' has six points in common with (f^, apart from Bjc and Ch, 

 the involution 7' of the A inscribed in (f^ possesses ó-ü' coincidences. 

 In the same way it appears that the inv^olutions P lying' on a" and 

 ii].^ possess four coincidences each. 



The supports of the coincidences envelop a curve {d) of class 

 eic/ht ; for through ^4 pass in the tirst place the lines /;/,-, each bearing 

 two coincidences, and which consequently are bitangents of (t/) and 

 further the tangent in .1 at <(^ which will touch (cZ) in A. 



4. To the points X of a straight line / correspond the pairs of 



points A' and A" of a ('urve ^., which has in common with / the 



two [)airs of the (A^-) lying on /, besides the points of intersection 



of / and Ö' ; hence ?. is a curve of order eleven. By paying attention 



to the intersections of / with the singular curves n\ /^)t^ and y/i', 



we see that /" passes three times through A, five times through B]c 



and two times through Ch. 



ly^This corresponds to this well known proposition: the locus of the points where 

 a curve tp"* of a pencil is touched by a curve f" of a second pencil is a curve 

 of order 2(m+w) — 3. 



