( 429 ) 



curves must toucli each other in that basepoint, that this is a contact 

 of the second order, so that from the number of rays of coincidence 

 three must be subtracted to find that of the lines irst wanted. 

 From this ensues that tlie multiplicity of the point Aj-^i amounts to 

 ,t 4- tr + /-.• - i<( + ^? + y) _ 5. 

 So we find : 



A basepoint of tlm pencil (C,) only is a 



{st — « — 1)- 



fold /tomt of the locus proper L. A common basepoint of the 

 pencils {Cs) and {Q) /vhich is not basepoint of (C,) is a 



{rs J,-rt-^-y~ 3)- 



fold point of L and a common basepoint of the three pencils is a 



{^t -\- tr -\- rs — (t — ^ — 7 — 5)- 



fold point of L^). 



4. With the help of the preceding the points of intersection of 

 L with an arbitrary cur\'e of one of the pencils, e.g. a CV, are 

 easy to indicate. These are : 



1. The r^ — ^ — V -h ^ points Ar counting together for 



(,.' - ^- y + d) (..« - « - 1) 



points of intersection. 



2. The i? — Ö points Art counting together for 



{^ — Ö) {sr -j- St — « — 7 — 3) 

 points of intersection. 



3. The 7 — d points Ars , giving 



(y - 6) {tr -f ^s — a — |i — 3) 

 points of intersection. 



4. The Ö points Arsi, giving together 



') If there are no points Arsi (5 = 0) and tlioielbre the total locus is not inde- 

 finite, we can also ask after the multiplicities of the points Ar and Ast as 

 points of the total locus. Now the improper part of the locus consists of x curves 

 Cr , <3 curves Cs and y curves Ct . Of these pass tlnough a point yJr the « curves 

 Cr and through a point Ast the /3 curves C, , the y curves Ct and one of the 

 curves Cr . From this ensues : 



A point Ar is an (st — 1), a point Au an {rs-\-rt - 2)- fold point of the 

 total locus. 



So the multiplicity of Ar as a point of the total locus is not changed by the 

 coincidence of the basepoints, whilst the multiplicity of Asi is equal to the sum 

 of the multiplicities which this point would have if it were only basepoint of the 

 pencil (r>) or (mly basepoint of the pencil (C). 



28* 



