( 430 ) 



ö{st -\- tr -\- rs — a — ^ — Y — h) 



points of intersection. 



5. The movable points of intersection of L vvitlj (',■ ; these are 

 those points of intersection which displace themselves when we 

 choose another Cr- These are found as the pairs of common points 

 of the simpl}^ infinite linear systems of pointgi'oups intersect on Cr 

 by the pencils {Cs) and (G). The number of these are found from 

 the following theorem : 



If there are on n carve of yenus p two .sinvply infinite linear 

 systems of pohityroups consisting of a and b points, the number of 

 common ijairs of pjoints of those systems is 



(« - 1) (6 - 1) - p. 



In our case a = rs — y, b = rt — |? and (as Cr is an arbitrary curve 

 of the pencil (6';-)) p ^ ^ {r — 1) (r — 2). For the number of pairs of 

 common points we therefore find 



^rs _ y _ 1) (r« _ ^ _ 1) _ 1 (,. _ 1) (r -2), 



and for the number of movable points of intersection of L and C,- • 

 2{rs — y—l){rt — ^—l)~{r— 1) (r - 2). 

 So the total number of points of intersection is: 



r{drst -[- 3 — 2r — 2s — 2« — ar — ^s — y«), 

 in accordance with the formula we have found foi' the order of L. 



5. The pairs of points FF' through which a curve of each of 

 the pencils is possible determine on L an involutory (l,l)-correspon- 

 dence ; in the following we shall indicate F and F' as corresponding 

 points of L. 



If F falls into a doublepoint of L differing from the base- 

 points, then in general two different points Z'" and P" will correspond 

 to F according to our regarding F as point of the one or of the other 

 branch of L passing through F. The curves of the pencils passing 

 through F now have two more common points F' and F", so that 

 we get a triplet of points FF' F", through which a curve of each 

 of the pencils is possible. 



It may however also happen that tiie points F' and F" coincide. 

 In that case correspond to the two branches througii F two branches 

 through F', so that F is likewise doublepoint of L. The curves of 

 the pencils passing through F have now but one other common 

 point P', but now the particularity arises that J* or P' can be 

 displaced in two ways such that the other common point is retained. 

 So FF' is then to be regarded as a double corresponding pair of 

 points. 



