CURVES WHICH SATISFY GIVEN CONDITIONS. 
149 
or substituting for 2D its value expressed in the form n — 2m-\-2-\-x, we have 
t—3n — 3 m+«, 
which is right. 
102. For the purpose of the next example it is necessary to present the fundamental 
equation under a more general form. The curve 0 may intersect the given curve in a 
system of points P', each times, a system of points Q', each # 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, u, a'), (b, (3, (S'), 
&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 
(f 3 , /3') correspondence, and the number of united points is =b, and so on. The theo- 
rem then is 
j?(a— a— oi')-j-g'(b— j3— /3')+ &c. + Supp. = 2&D, 
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' cotangen- 
tials 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 r be the number of double tangents, then 
the number of united points is =2 r. The curve 0 is the system of the n— 2 tangents 
from P to the curve ; each tangent has with the curve a single intersection at P, that is, 
Jc—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 (2 r, (3 , /3') to the 
points (P, P'), we have 
2{a— a — a'}-f{2r— (3— Supp. =2(n— 2)D. 
From the foregoing example the value of a — a — a' is =4D — ». In the case where the 
point P is at a cusp, then the n— 2 tangents become the n— 3 tangents from the cusp, 
and the tangent at the cusp; hence the curve © meets the given curve in 2(n— 3)-f3, 
=2n— 3 points, that is, (n— 2)-\-(n— 1) points; this does not prove (ante, No. 96), but 
the fact is, that the cusp counts in the Supplement (n—l) times, and the expression of 
the Supplement is =(n— l)z. It is clear that we have (3=j3’=(n—2)(m—3), so that 
the equation is 
8D — 2k + 2r — 2(n - 2)(m -3)+(n-l)z=(n- 2)2D, 
that is, 
2r=2(n-2)(m-3)+(n-6)2D+(-n+3)z; 
or substituting for 2D its value =n — 2m-\-2-\-x and reducing, this is 
2r=w 2 +8m— lOn— 3%, 
which is right. 
104. As another 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 in! have an (a, a!) correspondence, and let 
t 2 
