( 423 ) 



at in this way we shall call the locus proper, to distinguish it from 

 the total locus to be arrived at by allowing one of the points Pand 

 P' to be a fixed point of intersection of two of the pencils. 



Suppose the pencils (C\) and (C/) show « fixed points of inter- 

 section and that this number amounts to j? for the pencils (G) and 

 {Cf) and to y for the pencils (C,) and (6s). 



The degree n of L is determined from its points of intersection 

 with an arbitrai'y straight line /. On /we take an arbitrary point Q,.,. and 

 through Qrs we let a C,- and a Cs pass, which cut each other besides 

 in the basepoints and in Qrs still in rs — y — 1 points. Through 

 each of these points we let a curve Ct pass. These rs — y — 1 curves 

 Ct cut / in t{rs — y — 1) points Qt, which we make to correspond 

 to the point Q,.s • To find reversely how many points Qrs correspond 

 to a given point Qt of / we take on / an arbitrary point Qr through 

 which we allow a 6'. to pass cutting the Ct through Qt in rt — /? 

 points differing from the basepoints. Through each of those points 

 we allow a Cs to pass, of which the points of intersection with / 

 shall be called Qs. To a point Qr now correspond 6^ {rt — /?) points 

 Qs and to a point Qs correspond r {st — a) points Qr. The Irst — ar — ^s 

 coincidences QrQs are the t points of intersection of / with the Ct passing 

 through Qt and the points Qrs corresponding to Qt, whose number 

 therefore amounts to 2 rst — «r — ^s — t. 



So between the points Qrs and Qt of / we have a {rst — yt — t, 

 2 rst — (xr — ^s ■ — ^)-correspondence. The 3 rst — ar — ^s — yt - — 2t 

 coincidences are the points of intersection of / with L and the points 

 of intersection of / with the curve of contact of the pencils (6r) and 

 (6s), i. e. the locus of the points of contact of the curves Cr and Cs 

 touching each other. If there are two systems of curves (fti, v^) and 

 (Ma' ^2) ^)j tli6 order of that curve of contact is 



^i^v^ + ftjVj + n^ii^ ^). 



1) A system of curves (^^4, v) is a simply infinite system of curves, of which 

 fz pass through an arbiu'arily given point and v touch an arbitrarily given straight line. 



~) This order is found by counting the points of intersection with an arbitrary 

 hne J. To this end we consider the envelope of the tangents of the curves of the 

 system (,«1, v^) in its points of intersection with /; this envelope is of class juj+vi, 

 the tangents of that envelope passing through an arbitrary point Q of I being 

 the tangents in Q to the ^wj curves of the system through Q and the line I 

 counting v^ times. In like manner does the system (|:/2> "2) gi^^ an envelope of class 

 fji2 + vo. The («1 + vi) (uo + V2) common tangents of both envelopes are the 

 line I counting vjvj times and ,m-i,"2 4" p-r-'2 "^Ps'-'i other lines whose points of 

 intersection with I indicate the points of intersection of I with the curve of contact. 

 For a deduction with the aid of the symbolism of conditions see Schubert, "Kalkiil . 

 der abzahlenden Geometrie", p. 51 — 52. 



