262 Conic Sections. 
RL, sin. w is negative and cos. w positive till RV coincides with 
RF. As RV departs from RL, sin. » and cos. w are each nega- 
ig. 2. 
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tive, and (q) remains positive nt it vanishes when the plane becomes 
a tangent at R. These considerations render it evident that (q) 1 
aleckys positive except when it vanishes in the case above referred 
to. Wherefore (10) characterizes aright line when g=0 and G=0; 
an ellipse when (G) is negative, including the circle when G= — a? 5 
an hyperbola when (G) is positive ; and a parabola when G=0. 
The following examples will answer the purpose of illustration. 
Let it be required to determine whether a paraboloid is susceptible 
of an hyperbolic section. In this case (G) being positive and c=0, 
(8) is reduced to sin.w=+ AE 
dicates the impossibility of the proposed section. Let the hyper- 
bolic section of a sphere or spheroid be proposed. These conditions 
require 6) © be positive and c=—1; and consequently sinw= 
G . 
which being imaginary i- 
4/ i= 7 eS i, which is imaginary in the prolate spheroid 
where a>b, greater than unity in the oblate spheroid where a< 4, 
and infinite in the sphere where a=b: the section therefore is im- 
possible, in the first case because sin. w is imaginary, and in the last 
two cases because sin.w> radius. If the parabolic section of an 
oo mi proposed, we have c=1 and G=0, and _ therefore 
— psa’ <Bi? an equation which implies no absurdity, and indi- 
cates consequently the possibility of the section. If w’ denote the 
angle which the asymptote of the solid makes with its axis, we de- 
