[ 449 ] 
2. If the fedion be made by a plane B P O parallel 
to the diredr ix of the bafe DN, and in any given 
angle to the bafe, it will be an hyperbola, Fig 2. 
Let the fedion P B O meet the bafe in the line 
B O, which will be parallel to D N, and make the 
moving plane N M R interfed the bafe in N R, the 
vertical diredrix in M, and the fedion in P O ; by 
which D B=NO. From the point M draw MR 
parallel to P O, and imagine a plane to pafs through 
the diredrix A M and the line M R, meeting the 
bafe in AR ; which will be given in pofition. Then 
thro’ the point A draw A (^parallel to D N, inter- 
fedingNR in Q^and making N Q=AD; and the 
two triangles A R AMR, will have all their 
angles given, and the proportion of their hdes. And 
therefore the ratio of A. Q^to Q R, and of A Q to 
MR, will be given. Make AD=^, BD=b, AQjQR 
: ; a : y, and AQiMR::^:^ the abfcifle B 0 =at, 
and the ordinate PO = v; from which QR = ^, 
y a 
NR=tf-f— MR=~ And then from the fimi- 
a , a . 
lar triangles NOP, N M R, the analogy N O : O P 
: ;NR:MR will give the equation y x q-Fy a 2 =xbm, 
and the curve B P is an hyperbola. 
3. If the fedion PB be made by a plane parallel 
to the vertical diredrix A M, it will be an hyperbola. 
Fig. 1. 
Let the moving plane M R N, parallel to D A, 
interfed the bafe in R N, and the vertical direc- 
trix in M, and make the plane of the fedion B P O 
cut the moving plane NMR in P O, and the bafe 
in the line BO, meeting D A in Z, and the bafe di- 
redrix DN in B; from the point M draw the line 
Vo l. LI, Mmm MK 
