440 
ON BRIOT AND BOUQUET’S THEORY OF THE 
[860 
the point (0, — 2), and the equation of the tangents is (y + 2) 2 — 27a? 2 = 0. The curve 
besides meets the line a? = 0 in the point y= 1, where the tangent is horizontal, and 
2 
it meets the axis y = 0 in the points a? = ± ^, at each of which the tangent is 
vertical; these two points, as lying on the axis y = 0, are thus each of them 
permissive; and they are the only points where the tangent is vertical (in fact, the 
tangents from the point at infinity on the line x = 0 are the two lines x = + - s 
3 V3’ 
and the line infinity counting twice, as a tangent at an inflexion). 
For the points at infinity, we have y = — 3a# — 1 — ^x~% + &c., which is of the 
form y = a?iP , included in y = x l ~ m P > and the point is thus permissive; 
there is no prohibitive point, and the differential equation has a monotropic solution. 
This is, in fact, the rational solution u = (z — c) — (z — c) 3 , or say u=z — z 3 ; the curve 
is thus given by the two equations x = z — z 3 , y = 1 — Sz 2 . 
17. As a second example, take the differential equation 
We have here the curve 
27m 4 = 0. 
y 3 — y 2 + 4& 2 — 27a? 4 = 0, 
which is a trinodal quartic curve, as shown in the figure. There is a node at the 
origin with the tangents y 2 — 4a? 2 = 0, and, writing the equation in the form 
(3y + 1) (3y- 2) 2 - (27a; 2 - 2) 2 = 0, 
Fig. 2. 
we have for the other two nodes a? = + , y = f 
o Y o 
The lowest points are given by 
2 
^ = ±3V3’ y = ~^ viz * the line y = 
have y 3 — y 2 = 0, that is, y 2 = 0, the node 
is a horizontal tangent. Writing x = 0, we 
at the origin, and y— 1, the height of the