And 
[ 452 ] 
bx p Q _ gca-\-qbx ^ p m c a+m b x 
then, in the iimilar triangles NPO, MRN, having 
NO:PO;:NR:MR, the equation will be ya+-\- 
y c a z q-\-y xb a q=^n me a x-\-n m b x z . And the curve 
is an hyperbola ; and in the cafe of this Fig. 4 . it 
will be convex to the plane of the bafe. But when 
B N is negative in the cafe of Fig. f. the equation, 
retaining the fame fymbols, will be y q c a z — 
yxbqa—mncax — nmbx z ; and the hyperbolic 
curve will be concave towards the bafe. 
6. If the vertical diredtrix AM is made parallel 
to the plan of the bafe, but the plane palling thro’ it 
not parallel to the other diredtrix, then the fedtion, 
meeting the two dire&rices, will alfo be the fame 
curve. Fig . 6. 
For in this cafe the line MR is a conftant quan- 
tity ; and therefore, if the common fedtion a R of 
the plane thro’ the vertical A M, with the bafe, meet 
a D parallel to N R in a , other being as before, 
making MR=w, and T>a=a ; from the analogy 
NO:PO:;NR:MR in the triangles N O P, M R N, 
we fhall have m n a x—y a^+y a c q -\-y x q b ; by 
which the curve is known to be an hyperbola. 
And in all thofe fedtions, where the common 
interfedtion of the plane, palling thro’ the vertical 
with the bafe, is not parallel to the other diredtrix, 
the curve is an hyperbola. 
7. But now, if we fuppofe the common interfec- 
tion A R of the plane palling thro’ the vertical, with 
the bafe, to be parallel to the diredtrix D N of the 
bafe, and both diredtrices to be cut by the fedtion, 
the curve will be a parabola. Fig. 7 . 
For 
