[ 45 ° ] 
M R parallel to the plane of the fedion, and meeting 
the bafe in R ; and the lines M R and P O will be 
parallel. Then thro’ the vertical A M, and the line 
M R, make a plane to pafs, which will be parallel to 
the fedion, interfering the bafe in AR, and the 
lines A R and B O will alfo be parallel. Draw BL 
parallel to DA, or N R, palling A R in L, and 
then BO = LR, BL=RO = AZ, and AL = BZ. 
And the line A R being given in pofition, the lines 
B L and A L will be given. The angles alfo MAR 
and N BO being given, the ratio of A R to RM, 
and of B O to O N, will be given. Suppofe ADz 
=B L=a, A L =c the abfciffe B 0=x, the ordinate 
P 0==y, and A R : R M; \a:m i and BO;0 N: :a:n , 
from which we have ON=— RM=^i^~j 
a * a 
A y • 
NR=tf~] — And then, in the triangles on the 
Cl ( 
moving plane, which are fimilar, M R N, P O N, 
the analogy NO:PO;:NR:RM v/ill produce the 
equation y x a y—x^n m-\-x nmc j which Ihews 
the curve B P to be an hyperbola; and the figure is 
convex to the bafe. 
4 . The fe£tion being made parallel to the bafe, it 
will be the fame curve. Fig. 3 . 
Having thro’ the vertical diredrix made a plane 
A M R to pafs perpendicular to the bafe, let the fec- 
tion B P O, and the bafe parallel to it, meet that 
plane in the lines BO and AR, and alfo the moving 
plane N M R in the lines P O and N R. From the 
point A draw A parallel to the diredrix D N of 
the bafe, and A D parallel to NR, and from B the 
line B C parallel to O R, meeting A R in C ; and 
then 
