262 



Conic Sections. 



RL, sin. w is negative and cos. w positive till RV coincides with 

 RF. As RV departs from RL, sin. w and cos. w are each nega- 



. , Fig. 2. 



C" K" 



tive, and (g-) remains positive till it vanishes when the plane becomes 

 a tangent at R. These considerations render it evident that (</) is 

 always positive except when it vanishes in the case above referred 

 to. , Wherefore (10) characterizes a right line when q—0 andG=0; 

 an ellipse when (G) is negative, including the circle when G= — a^ ; 

 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 — ± -y/ ( ^ — t)? which being imaginary in- 

 dicates the impossibility of the proposed section. Let the hyper- 

 bolic section of a sphere or spheroid be proposed. These conditions 

 require (G) to be positive and c= — l; and consequently sin.w = 



± k/ — - — Tg- I , which is imaginary in the prolate spheroid 



where ayb, greater than unity in the oblate spheroid where a<6, 

 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.6j> radius. If the parabolic section of an 

 hyperboloid be proposed, we have c=l and G=-0, and therefore 



sin.6j^= 2 1 /a ' ^^ 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- 



