( 426 ) 



If we take for the systems the two pencils (G) and (Cs) then 

 (n^ = jLt, =r 1 and (as ensues imnriediately from the principle of corre- 

 spondence) Vi = 2 (r — 1), r, = 2 (s— 1). So the order of the curve of 

 contact is 



2r-\- 28—3. 



For the number of points of intersection of / with L remains 

 3rst-ar-^s-yt—2t—{2r+2s—3) = 3{rst + l)-2{r-\-s-\-t)-{ar-\-^s-ir7t). 



So we find: 



The locus L of the pairs consisting of two movable points by which 

 a curve of each of the pencils is possible is of order 



n = 3 {rst + 1) — 2 (r + s + — («^ + i^s + tO 5 

 here a is the number of fixed points of intersection of the pencils 

 [Cs) and (Ct), ^ that of the pencils (Ct) and (Cr) arid y that of 

 (Q and (Cs). 



2. Whilst the preceding considerations remain accurate when of the 

 basepoints of one and the same pencil some coincide, we shall suppose 

 in the following that the pencils {€].), {Cs) and (G) have respectively 

 r% 6'' and f different basepoints, so that we can only allow the 

 basepoints of one pencil to coincide in part with those of an other 

 pencil. Then « is the number of common basepoints of the pencils 

 (Cs) and (G) (which can however also belong to (G)), etc. If the 

 pencils have no common basepoints (« = /? = y = 0), the order of 

 the locus becomes 



3{rst + 1) — 2 (r + s -1- t). 



This is also in the case of common basepoints the order of the 

 total locus as long as that is definite, i. e. as long as there are no 

 basepoints common to the three pencils. If there is such a point, this 

 furnishes together with an entirely arbitrary point a pair of points PP 

 through which a curve of each of the pencils is possible; of this 

 pair of points however only one is movable. The locus proper however 

 is still definite then. 



A basepoint of the pencil (Cr) only we call Ar, a common base- 

 point of the pencils (CI) and (CO which is not a basepoint of the pencil 

 (6r) we call Ast and a common basepoint of the three pencils we 

 call Arst- If' <^ is the number of points Am tlien the number of 

 points Ast amounts to «' =: « — cf, that of the points A,t to ^' = ^ — d 

 and that of the points Ars to y' = y — rf, whilst the number of points 

 Ar is equal to /•' — ^' — y' — d, etc. By introduction of «', ^', y' and 

 d the order n of this locus proper becomes 



