( 431 ) 



If reversely we have a tri|)U't of {)oiiils PP'P" lying on curves 

 of each of the pencils, then P is a doublepoint of L, for P' as well 

 as P" corresponds to P, and so it must be possible to displace P 

 in such a way that the corresponding point desci'ibes a branch passing 

 through P' and in such a way that a branch passing through P" 

 is described. The curve L has thus two branches PI and P2 passing 

 through P to which the branches PI and P'2 correspond. Through 

 the point P' (which is of course likewise doublepoint of Zr^as well 

 as P") a second branch P'3 passes and through F" a second branch 

 P"3, which branches correspond mutually. If a point Q describes 

 the branch PI the curves C,-, Cs, Ct passing through Q have a 

 second common point describing the branch P'\, whilst a third 

 common point P" appears and again disappears when Q passes the 

 point P. This third common point displaces itself (along the branch 

 P"2) when Q describes the other branch passing through P, whilst 

 then the common point coinciding with P' appears and disappears. 



Triplets of points PP'P", and therefore doublepoints of L 

 ditfering from the basepoints, there will be as a triplet of points 

 depends on 6 parameters and it is a 6-fold condition that a curve of 

 each of the pencils must pass through it. So we have: 



The curve L ha.s doublepoints, differing from the basepomts of the 

 pencils, belonging in triplets together and forming the triplets of points 

 through ivhich a curve of each of the pencils is possible. To one or 

 other branch through a doublepoint of such a triplet corresponds a 

 branch through the second resp. the third doublepoint of this triplet. 

 Moreover L can however have pairs of doublepoints indicating the 

 double corresponding pairs of points. To the two branches through 

 the doublepoint of such a pair correspoiid the branches through the 

 other doid)lepoint of the pair. 



6. The number of coincidences of the correspondence between P 

 and P' can be determined as follows. The points P and P' coincide 

 if the cur\'es C,-, Cs and Ci passing through P have in P the same 

 tangent. Then P must lie on the curve of contact R,.s of the pencils 

 (O and {Cs) as well as on the curve of contact Rri of (C,) and (C,). 

 The number of points of intersection of those curves of contact 

 which are of order '2r -f- 2s — 3 resj). 2r -\- 2t - 3 amounts to 



(2r + 26' — 3) {2r -\- 2t — 3). 



Some of these points of intersection however do not lie on the 

 third curve of contact R^t , and so they must not be counted. The curve 

 A^•.s• namely passes once through a basepoint A,, or A^ and three 

 times through a common basepoint Ays or .4,5^^ in fact in a point of 



