516 
ON POLYZOMAL CURVES. 
[414 
106. A circular point at infinity 7 or J, may be an ordinary or a singular 
point on the curve, and the tangent at this point then counts, or, in the case of 
a multiple point, the tangents at this point count a certain number of times, say 
q times, among the tangents which can be drawn to the curve from the point ; the 
number of the remaining tangents is thus =n — q. In particular, if the circular point 
at infinity be an ordinary point, then the tangent counts twice, or we have q = 2 ; if 
it be a node, each of the tangents counts twice, or q = 4 ; if it be a cusp, the tangent 
counts three times, or q — 3. Similarly, if the other circular point an infinity be an 
ordinary or a singular point on the curve, the tangent or tangents there count a certain 
number of times, say q' times, among the tangents to the curve from this point ; 
the number of the remaining tangents is thus = n — q\ And if as usual we disregard 
the tangents at the two points 7, J respectively, and attend only to the remaining 
tangents, the number of the foci is = (n — q) (n — q'). 
107. Among the tangents from the point 7 or J there may be a tangent which, 
either from its being a multiple tangent (that is, a tangent having ordinary contact 
at two or more distinct points), or from being an osculating tangent at one or more 
points, counts a certain number of times, say r, among the tangents from the point 
in question. Similarly, if among the tangents from the other point J or 7, there is 
a tangent which counts r' times, then the foci are made up as follows, viz. we have 
Intersections of the two singular tangents counting as 
Intersections of the first singular tangent with each of 
the ordinary tangents from the other circular point at 
infinity, as 
Do. for second singular tangent, ..... 
Intersections of the ordinary tangents . . . . 
r'r foci. 
{n-q - r') r „ 
(n-q -r)r' „ 
(n-q-r) (n -q- r) „ 
Giving together the (n — q) (n — q') foci: 
and the like observation applies to the more general case where the tangents from 
each of the points 7, J include more than one singular tangent. 
108. There is yet another case to be considered; the line infinity may be an 
ordinary or a singular tangent to the curve: assuming that it counts s times among 
the tangents from either of the circular points at infinity, the numbers of the 
remaining tangents are n — q — s, n — q —s from the two points 7, J respectively, and 
the number of foci is =(n — q—s)(n — q' — s). 
109. In the case of a real curve the two points 7, J are related in the same 
manner to the curve, and we have therefore q = q'; the singular tangents (if any) 
from the two points respectively being the same as well in character as in number. 
Writing n — q — s — n— q' — s, =p, and not for the present attending to the case of 
singular tangents, I shall assume that the number of tangents to the curve from each 
of the two points is =p; the number of foci is thus =p 2 ; and to each focus there 
corresponds a directrix, viz., this is the line through the points of contact of the 
tangents from the focus to the two points 7, J respectively.