85 
385] 
ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. 
13 
IS 
hence, if (fi, v) be the characteristics of the system of conics (4<Z), 
the number of 
the 
conics through P is =/i; each of these has with the given curve 1 intersection 
at 
tal 
P, and consequently k = ¡i. Moreover, each of the conics besides 
meets the curve 
in 
,he 
(2to — 1) points, and consequently i = a' = fi (2m - 1). Hence the 
formula gives 
the 
mh 
number of united points 
irs 
= 2g (2to — 1) + ¡1 (n — 2to + 2), 
en 
= fi (n + 2 to) ; 
fiz. 
or, as this may be written, 
its 
)er 
= gn + vm + to (2/i — v). 
But the system of conics (4>Z) contains (2g — v) point-pairs (coniques infiniment aplaties), 
each of which, regarded as a pair of coincident lines, meets the given curve in to 
pairs of coincident points; that is, the point-pair is to be considered as a conic 
th touching the given curve in to points; and there is on this account a reduction 
P = to (2/i — v) in the number of the united points; whence, finally, the number of the 
conics (4fZ) (1) is = gn + vm. It is hardly necessary to remark that it is assumed 
that the conditions (4Z) are conditions having no special relation to the given curve, 
of 
3r _ 11. As a final example, suppose that the point P on a given curve of the order 
m, and the point Q on a given curve of the order to', have an (a, a') correspondence, 
and let it be required to find the class of the curve enveloped by the line PQ. 
Take an arbitrary point 0, join OQ, and let this meet the curve to in P', then 
(P, P') are points on the curve in, having a (mi, ini') correspondence; in fact, to a 
given position of P there correspond cl positions of Q, and to each of these to 
positions of P', that is, to each position of P there correspond mi' positions of P'; 
and similarly to each position of P' there correspond mi positions of P. The curve 
0 is the system of the lines drawn from each of the cl positions of Q to the 
point 0, hence the curve 0 does not pass through P, and we have k = 0. Hence the 
number of the united points (P, P'), that is, the number of the lines PQ which pass 
through the point 0, is = mi' 4- nil, or this is the class of the curve enveloped by PQ. 
12. It may be remarked, that if the two curves are curves in space (plane or of 
double curvature), then the like reasoning shows that the number of the lines PQ 
which meet a given line 0 is =ma' +m'i, that is, the order of the scroll generated 
he by the line PQ is = ma! + rn'i. 
ve 
ve 
ny 
ics 
gh 
ics 
P;