C 1*9 1 
of the Lines of the Third Order, according to ''vhat 
Sir Ifaac has thought fit to conftitute a different 
Species. 
The two Species I mean, are to be reckoned 
amongft the Hyperbolo- parabolical Curves, having 
one Diameter, and one Afymptote, at N°. 8. of 
Newtons Treaiife, or Page 104. of Mr. Sterlings'^ 
whofe Equation is xyy—^bx^-^cx-;^d'-) which 
will give, not Four, as in thefe Authors, but Six 
Species of thefe Curves : For, 
I. If the Equation bx'^-^c oc ^ 0^ Ii3s two im* 
poflible Roots, the Equation xyy—bx^^cx -\-dy 
will (as they fay) give two Hyperbolo-parabolical 
Figures equally diftant on each fide the Diameter 
A B. See the 57th Figure in Newtons Treatife, and 
this is his 53d Species, and Sterlings 57th. 
IL If the Equation b x"^ — c x d^o^ has two 
equal Roots both with the Sign ^ the Equation 
xyy — bx'^*—*cx-\-dy will (as they fay) give two 
Hyperbolo-parabolical Curves crofling each other at 
the Point t in the Diameter. See Fig. the 58th in 
Newton s and this is his 54th Species, and Sterlings 
5 8th. 
III. But if the 
Equation bx‘^\cx 
-}~d=o, has two 
poflible unequalne- 
gative Roots Ap and 
Am^ the Curve gi- 
ven by the Equation 
y-'i^bx'^-\-cx 
dy will confift of 
two Hyperbolo-pa- 
rabolica! 
