$& a Mr. Brougham’s general Theorems 
intersection of the two curves, moving round C, shall describe 
a given ellipse. 
PROP. VI. THEOREM. 
A common logarithmic being given, and a point without it, 
a parabola, hyperbola, and ellipse, may be described, which shall 
intersect the logarithmic and each other in the same points; 
the parabola shall cut the logarithmic orthogonally ; and, if 
straight lines be drawn from the given point to the common 
intersections of the four curves, these lines shall be normals to 
the logarithmic. 
PROP. VII. PORISM. 
Two points in a circle being given, (but not in one diameter,;) 
another circle may be described, such, that if from any point 
thereof to the given points straight lines be drawn, and a line 
touching the given circle, the tangent shall be a mean pro- 
portional between the lines so inflected. 
Or, more generally, the square of the tangent shall have a 
given ratio to the rectangle under the inflected lines. 
PROP. vm. pqrism. Fig. 3. 
Two straight lines AB, AP, (not parallel,) being given in 
position, a conic parabola MN may be found, such, that if, from 
any point thereof M, a perpendicular MP be drawn to the one 
of the given lines nearest the curve, and this perpendicular be 
produced till it meets the other line in B, and if from B a line 
be drawn to a given point C, M P shall have to PB together 
with CB, a given ratio. 
