Conic Sections. 263 



rive, from a familiar property of the hyperbola, a^ ; Js • ;cos.w'^ I 

 sm.co'^.'.sin.w'2 — — -— — — . If sin.co^ > ^ , z.^ (G) is negative, and 



if sin.cj'^ < o I i2 (^) is positive. Wherefore we conclude that 



the section of an hyperboloid parallel to an asymptote is a para- 

 bola; that, if the section makes a greater angle with the axis of the 

 hyperboid than the asymptote does, it is an ellipse or circle ; and 

 that, if the section makes a less angle, it is an hyperbola. To de- 

 termine the sections of the paraboloid, we put c=0, and therefore 

 G= — a-sin.6d^, which is zero when w=0, and negative in all other 

 cases. Hence it is inferred that the section of a paraboloid parallel 

 to its axis is a parabola, and that in all other positions it is a circle or 

 an elHpse. In the sphere and spheroids c= — 1 and G= — b^ — 

 (a^ — 6')sin.w2 ; which, being always negative indicates that all the 

 sections of the sphere and spheroids are circles or ellipses. The 

 sections of a cone are determined with equal facility. The sections 

 of a cylinder may be derived directly from (3) or (4) by putting 

 p=0, c = l, and considering the axis of revolution C'K' (Fig. 2,) 

 to be parallel to ARH, which in this case is a right line. All the 

 parabolic sections of the sohds under consideration being charac- 



terized by the equation y^= —^, have for their parameter 



6- (p4-2c^)cos.w — 2ab\/(pd-{-cd^ )sin.w 



;— . All other sections, ex- 



cepting the right line are characterized by (10) without any change 



q G B^ 



of general form. Putting p=d=A and — =±-r^ (the +or— be- 

 ing used according as (G) is positive or negative) (10) becomes 

 B^ B^ B^ 



-^. Ax±^X^=2/^ or ;p(Ax±x^)=y% the equation of a cir- 

 cle, ellipse, or hyperbola whose principal axis is A=zh 

 b^ (p-{-2cd)cos.u — 2ab(pd-\-cd^)sm.u 



c62~(«2+c6)sin.c.2 ' ^"^ '^^ conjugate axis B= 



Av'dzG s/±G q q 



a ~ a ' dzG^av'dzG"" 

 6^ (p+2cf^)cos.w — '2ab\/{pd-\-cd^ )sin.w 



^±[(c6^-(a^H-c6^)sin.c7)] * therefore also A^ : 



q^ q^ 



B^::^ J ^iT^^QA t a^::±[c62-(a^+c62)sin.w2]; which is an 



