380 Mr . Brougham’s general Theorems 
a given ratio, that part of a straight line passing through a 
given point which is intercepted between a point in the curve 
not given, but which may be found, and the ordinate to the 
point where the first mentioned line meets the curve. 
Let X be the point to be found, NA the line passing through 
the given point N, and M any point whatever in the curve ; 
join XM, andf draw the ordinate MP; then AC is to CP in a 
given ratio. 
Corollary. This property suggests a very simple and accurate 
method of describing a conic hyperbola, and then finding its 
centre, asymptotes, and axes ; or, any of these being given, of 
finding the curve, and the remaining parts. 
PROP. II. PORISM. 
A conic hyperbola being given, a point may be found, such, 
that if from it there be drawn straight lines to all the intersec- 
tions of the given curve, with an infinite number of parabolas, 
or hyperbolas, of any given order whatever, lying between 
straight lines, of which one passes through a given point, and 
the other may be found, the straight lines so drawn, from the 
point found, shall be tangents to the parabolas, or hyperbolas. 
This is in fact two propositions ; there being a construction 
for the case of parabolas, and another for that of hyperbolas. 
PROP. III. PORISM. 
If, through any point whatever of a given ellipse, a straight 
line be drawn parallel to the conjugate axis, and cutting the 
ellipse in another point ; and if at the first point a perpendi- 
cular be drawn to the parallel, a point may be found, such, that 
if from it there be drawn straight lines, to the innumerable 
