39 2 Mr. Brougham's general Theorems 
the last presents to our view a curve before unknown, (at 
least to me,) as possessing the singular property of a constant 
tangent. 
PROP. XIX. PROBLEM. Fig. 9. 
A common logarithmic being given, to describe a conic hy- 
perbola, such, that if from its intersection with the given curve 
a straight line be drawn to a given point, it shall cut a given 
area of the logarithmic in a given ratio. The analysis leads to 
this 
Construction. Let BME be the logarithmic, G its modula ; AB 
the ordinate at its origin A; let C be the given point; ANOB 
the given area ; M : N the given ratio : draw BQ parallel to AN ; 
find D a fourth proportional to M, the rectangle BQ . OQ, and 
M-f-N. From AD cut off a part AL, equal to AC together 
with twice G ; at L, make LH perpendicular to AD, and, be- 
tween the asymptotes AL, HL, with a parameter * twice 
(D+T .AB.G) describe a conic hyperbola: it is the curve 
required. 
PROP. XX. PROBLEM. Fig. 10. 
To draw through the focus of a given ellipse, a straight line 
that shall cut the area of the ellipse in a given ratio. 
Construction. Let AB be the transverse axis, EF the semi- 
conjugate ; E, of consequence, the centre ; C and L the foci. 
On AB describe a semicircle. Divide the quadrant AK in the 
given ratio in which the area is to be cut, and describe the 
cycloid GMR, such, that the ordinate PM may be always a 
Or constant rectangle. 
