[ + 5 * ] 
then the lines AD, AC, and B C, will be given, 
and the angles R A Q^_M A R, being alfo given, the 
proportion of A R to QJRL, and of B O to MO, 
will be known. MakeAD=#, AC = c, the ab- 
fcilfe B O = x, the ordinate P O —y, and A R : R 
:: a : q from which we have MO;MR : • X • X 
-f- c 3 QJl == X -' And becaufe, in the fimilar 
Cl 
triangles M P O, and M N R, in the moving plane, 
we have MO:PO: :MR:RN, there will refult 
the equation y x a yca~x 2 ‘q-\-xcq-\-xa z ' J 
which denotes the curve B P to be an hyperbola. 
j. If the fedtion is made fo as to meet the two di- 
rectrices, the curve will be alfo an hyperbola. Fig. 4 . 
Let the fedtion B P O meet the diredtrices in B, m, 
and interfedt the plane of the bafe in the line B O j 
and make the moving plane N M R to be cut by the 
fedtion in P O, and to meet the vertical diredtrix in 
M. Then from the point M draw M R parallel to 
PO; and thro’ the lines AM, MR, imagine, as 
before, a plane to pafs, interfedting the bafe in the 
line A R r, and meeting the line B O in r, and the 
fedtion in m r. And from A draw A D parallel to 
N R, and A Qjto D N ; and thro’ r make n r paral- 
lel to NR, and to meet A QJn q 3 and D N in n. 
Draw alfo from the point B the line B C parallel to 
A D, and meeting A Q^in C. The lines then A <7, 
A r, m r, r q f B r, B n, r n 3 will be given. Make 
therefore AD==tf=BC, AQj^QR ::a:q y and AQj^RM 
: :a : m 3 Br:Bn::a:b, Br:rn::a:n , A C=DB 
e=c j the abfciffe B 0=x ; the ordinate O P==y. From 
which we have BN=-> ON=— > AQ = ff 
a a 
b x 
