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10] ON THE THEORY OF ALGEBRAIC CURVES. 
Suppose 
(1) 17=0, V=0 intersect in three points, we must have a= A, or the curve 
17=0 must have one of its asymptotes parallel to one of the asymptotes of V=0. 
(2) The curves intersect in two points. We must have a = A, a' = A', or else 
cl = A, ¡3 = B; i.e. 17 = 0 must have one of its asymptotes coincident with one of the 
asymptotes of the curve V= 0, or else it must have its two asymptotes parallel to 
those of V = 0. The latter case is that of similar and similarly situated curves. 
(3) Suppose the curves intersect in a single point only. Then either a = A, 
cl = A', cl" = A", which it is easy to see gives 
U=(y-Ax-A')(<j-i3x-0) + K^—l 
or else cl= A, cl = A', (3 = B, which is the case of onfe of the asymptotes of the curve 
17= 0, coinciding with one of those of the curve V = 0, and the remaining asymptotes 
parallel. As for the first case, if a, aj are the transverse axes, 6, 6 X the inclinations 
of the two asymptotes to each other, these four quantities are connected by the equation 
a 2 _ tan 6 cos 2 \ 6 
a* tan cos 2 ^0 1 ’ 
and besides, one of the asymptotes of the first curve is coincident with one of the 
asymptotes of the second. This is a remarkable case ; it may be as well to verify 
that 17=0, F=0 do intersect in a single point only. Multiplying the first equation 
by y — Bx — B', the second by y — fix - ¡3', and subtracting, the result is 
reducible to 
(A-/3)(y-Bx-B')-(A-B)(y-/3x-f3')= 0, 
„ A (B' - /T) + B/3' - B'/3 
y~ A * = . 
i.e. y — Ax — (7=0. 
Combining this with V = 0, we have an equation of the form y — Bx — D = 0. And 
from this and y — Ax — G = 0, x, y are determined by means of a simple equation. 
(4) Lastly, when the curves do not intersect at all. Here a = A, a'= A', /3 = B, 
¡3' = B', or the asymptotes of U = 0 coincide with those of V= 0 ; i. e. the curves are 
similar, similarly situated, and concentric: or else cl = A, a! = A', cl” = A", /3 = 7?; here 
U=(y — Ax — A') (y-Bx- /3') + K, 
or the required curve has one of its asymptotes coincident with one of those of the 
proposed curve; the remaining two asymptotes are parallel, and the magnitudes of the 
curves are equal. 
In general, if two curves of the orders m and n, respectively, are such that r 
asymptotes of the first are parallel to as many of the second, s out of these asymptotes