384 Mr. Brougham's general Theorems 
first mentioned lines shall cut, in a given ratio, the areas of all 
the triangles whose bases and altitudes are the co-ordinates to 
a second conic hyperbola, which may be found, at the points 
where it cuts the lines drawn from the given point. 
PROP. XI. PORISM. 
A conic hyperbola being given, a straight line may be found, 
such, that if another move along it in a given angle, and pass 
through the intersections of the curve with all the parabolas, 
(or hyperbolas,) of any given order whatever, lying between 
straight lines to be found, the moving line shall cut, in a given 
ratio, the areas of the curves described, contained by the peri- 
pheries and co-ordinates to another conic hyperbola, that may 
be found, at the points where this cuts the curves described. 
PROP. XII. PORISM. 
A conic hyperbola being given, a straight line may be found, 
along which if another move in a given angle, and pass through 
any point whatever of the hyperbola, and if this point of sec- 
tion be joined with another that may be found, the moving line 
shall cut, in a given ratio, the triangles whose bases and alti- 
tudes are the co-ordinates to a conic hyperbola, which may be 
found, at the points where it meets the lines drawn from the 
point found. 
Scholium. I proceed to give one or two examples, wherein areas 
are cut in a given ratio, not by straight lines, but by curves. 
PROP. XIII. PORISM. Fig. 5. 
A conic hyperbola being given, if through any of its innu- 
merable intersections with all the parabolas of any order, lying 
