Conic Sections. 261 



n=:o; in which case *= — -, as is evident by putting p(;— Oin (5). 



If the axis of revolution is parallel to the principal axis of the re- 

 voWmgcurve,cp=0,s'm.cp=0,cos.cp=l,k=d,h^^a^&^-b^(^pd'\-cd^), 



b 



6 = -b^(p-\-2cd),e = 2a^&,f=0,m~ -cb^ ,n = a^ ,?ind'n'=-'y{pd+ 



cd^) - L Substituting these values, equation (3) is reduced to y"^ 



(-^=b-(^tZ+c^'+(i?+2ccZ)cos.wx+c.cos.w2x' )^) '- (a'/(P^+ 



cd'^) — (s-\-s\nM-)^ I (6). The most simple as well as useful class of 



cases embraced in (6) is that in which the axis of revolution is sup- 

 posed to coincide with the principal axis of ARH. According to 



this hypothesis ^=0, and (6) becomes y^ =—\vd-\-cd^ •\-{p -\- 



2C<Z)C0S.WP(+C.C0S.6J25^2 I __(p(^_|_c^2^ _ y/(prf-f C^2^gijj j^,^_ 



cb^ —(a^+cb^)s]nM^ 

 sin.w^xS or3/2= -^ ^ . 



([b^{pi-2cd)cos.u-2ab\/{pd+cd^ymM]x \ 



[ d^-{a^^+cb^)sm.^-^ +>^' j (^) 5 ^^^i^^ is evi- 



dently a conic section, and characterizes the section of a plane with 

 a cone, sphere, the spheroids, hyperboloid, or paraboloid, according 

 as ARH is a right line, circle, ellipse, hyperbola, or parabola. If, 

 for brevity, we put cb^ — [a^ -]-cb^)s'm.u^ =G (8), and b'^{p-{-2cd) 

 cos.u — 2ab\/{pd-{-cd^)=^q (9), the equation of the section becomes 



G [qx \ 



y^=-j[Q + X^j (10)- Conceiving A (Fig. 2,) to be that prin- 

 cipal vertex of the solid which is nearest to R, it is evident that 

 {p-\-'2cd)'\s always positive ; for (c) is negative only in the sphere 

 and spheroid, in which cases c= — l,jp = a, and a > 2c?. When the 

 plane RV is a tangent to the solid at R, SL : RL." I cos. w : sin. w. 

 But, according to well known properties of the conic sections, the 



2{pd+cd^-) . b 



subtangent SL = X2~/7 — ' ^"d the ordinate RL=-v'(p£?+C(Z^). 



Substituting these values in the proportion above and reducing, we 

 have b"{p-{-2cd)cos.u—2ab\/(^pd+cd^)s\n.u = 0. As RV de- 

 scends cos. w arid sin. w continue positive, the former increasing and 

 the latter decreasing till they become respectively (1) and (0) when 

 RV coincides with RF. When RV is situated between RF and 



