in the higher Geometry. 385 
between straight lines, whereof one is an asymptote, and the 
other may be found, an hyperbola of any order be described, 
(except the conic,) from a given origin in the given asymptote 
perpendicular to the axis of the parabolas, the hyperbola thus 
described shall cut, in a gilen ratio, an area (of the parabolas) 
which may be always found. 
If from G (as origin) in AB, one of LM's asymptotes, there 
be described an hyperbola IG, of any order whatever, except 
the first, and passing through M, a point where LM cuts any 
of the parabolas AM, of any order whatever, drawn from A 
a point to be found, and lying between AB and AC, an area 
ACD may be always found, (that is, for every case of AM and 
IC',) which shall be constantly cut by IG, in the given ratio of 
M : N ; that is, the area AMN : NMDC : : M : N. 
I omit the analysis, which leads to the following construc- 
tion and composition. 
Construction. Let m -j- n be the order of the parabolas, and p -f- q 
that of the hyperbolas. Find <p a fourth proportional to m -f- n y 
q — p and m -j- 2 n ; divide GB in A, so that AR : AG : : q : 
P + (p ; then find tt a fourth proportional to M -{- N, <p and 
q — p, and y a fourth proportional to q, AG, and q — p; and, 
lastly, 9 a fourth proportional to the parameter* of LM, tt and 
M. If, with a parameter equal to x 0 — °f the rect- 
angle t . AN, and between the asymptotes AB, AC, a conic 
hyperbola be described, it shall cut the parabola in a point, the 
co-ordinates to which contain an area that shall be cut by IG 
in the ratio of M : N. 
Demonstration. Because AG is divided in R, so that AR : AG 
* i. e. The constant rectangle or space to which AP . SM is equal. 
MDCCXCVIII. 3 D 
