407] CURVES WHICH SATISFY GIVEN CONDITIONS. 267 
cusps is (specially) a united point, and counts once, whence the Supplement is = k- 
Hence, writing n for the class, we have n + 2 (m- 1) + k = 2D, or writing for 2D its 
value = w? — 3 m + 2 — 28 — 2k, we have n = m 2 — m — 2S — 3k, which is right. 
101. Investigation of the number of inflexions. Taking the point P' to be a 
tangential of P (that is, an intersection of the curve by the tangent at P), the united 
points are the inflexions; and the number of the united points is equal to the number 
of the inflexions. The curve © is the tangent at P having with the given curve two 
intersections at this point; that is, k — 2; P' is any one of the m — 2 tangentials of 
P, that is, a! = m — 2 ; and P is the point of contact of any one of the n — 2 tangents 
from P' to the curve, that is, a = n — 2. Each cusp is (specially) a united point, and 
counts once, whence the Supplement is = k. Hence, writing i for the number of 
inflexions, we have 
l — (m — 2 ) — {n— 2) + k = 4P; 
or substituting for 2D its value expressed in the form n — 2m + 2 + k, we have 
t = 3n — 3m + k, 
which is right. 
102. For the purpose of the next example it is necessary to present the funda 
mental equation under a more general form. The curve © may intersect the given 
curve in a system of points P', each p times, a system of points Q', each q times, 
&c. in such manner that the points (P, P'), the points (P, Q'), &c. are pairs of points 
corresponding to each other according to distinct laws; and we shall then have the 
numbers (a, a, a), (b, /3, /3'), &c., corresponding to these pairs respectively, viz. (P, P') are 
points having an (a, a') correspondence, and the number of united points is = a; 
(P, Q') are points having a (/3, /3') correspondence, and the number of united points 
is = b, and so on. The theorem then is 
p (a — a — a) + q (b — /3 — /3') + &c. + Supp. = 2JcD, 
being in fact the most general form of the theorem for the correspondence of two 
points on a curve, and that which will be used in all the investigations which follow. 
103. Investigation of the number of double tangents. Take P' an intersection of 
the curve with a tangent from P to the curve (or, what is the same thing, P, P' 
cotangentials of any point of the curve): the united points are here the points of 
contact of the several double tangents of the curve ; or if t be the number of double 
tangents, then the number of united points is = 2r. The curve © is the system of 
the ii — 2 tangents from P to the curve; each tangent has with the curve a single 
intersection at P, that is, k = n— 2; each tangent besides meets the curve in the point 
of contact Q' twice, and in (m — 3) points P'; hence if (a, a, a') refer to the points 
(P, Q'), and (2t, /3, /3 ) to the points (P, P'), we have 
2 {a - a - a'} + (2t - /3 - /3'} + Supp. = 2(n- 2) D. 
From the foregoing example the value of a — a — a' is = 4D — k. In the case where 
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