260 Conic Sections. 



RVT being each perpendicular to the plane ARHM, their common 

 section VT is also perpendicular to RV and HK ; and consequently 

 HK^ -YK^=yT'^y\ But VK=RQH-VP=*4-sin.(c.-9)x. 

 Substituting in the last equation this value of VK, and for HK the 

 value (m') in (2), we shall have 



/ e+/cos.(M-(p)x { (e+fcos. {t^-(p)x Y 

 y" = [- -^ ^\\ 2n I - 



A4-^cos.(^ — (p)x+ w^cos.(m — (p) 2^^^ "^ i\ _ L_{_sin.(w-9)xj (3); 



which by an obvious reduction becomes ?/^ =^j— ^ f -e-/cos.(w-(p))(;d= 



[e2_4nA+2(e/-4ji<5)cos.(w-(p)x + (/--4mw)cos.(w-(p)2-)^2-j2 \ _ 



['r4-sin.(w— 9)x]2, (4); which are the equations, referred to rec- 

 tangular coordinates of the section of a plane with the surface of a 

 solid formed by the revolution of any conic section about an axis 

 situated in its plane. 



The equations (2), (3), (4), are liable to a failing case which, 

 though of rare occurrence, it may be well to notice and make pro- 



a' 



vision for. This happens when w=0 or sin. 9^ •=-73—77, and is oc- 

 casioned by regarding (1) as a quadratic equation, which it evidently 

 ceases to be when w=0. In this case 



/i+5cos.(w-(p)x+mcos.(w-(p)2x'' ,r-^ A .1 



n'=- e+/cos.(..-9)x ' ^^^ ' ^"'^ consequently 



r = [ e+/cos.(.-9)r~ / -[*+sm.(^-9)x]" (4 )• 



In the application of these general equations to proposed cases it 

 is sufficient merely to remark that (K), (^), (u), (9), are positive in 

 the situations in which they are represented in fig. (1), and change 

 their signs according to the familiar principles of trigonometry. It 

 is evident from an inspection of the figure that ("tt) and (d) are not 

 arbitrary lines, being dependent upon the equation of ARH and the 

 given magnitudes [Jc), (&), (9). To determine (tt), it is necessary 

 only to put (*) in the place of (m') and zero in that of [f) in (2) 

 which expresses the relation between the rectangular coordinates of 

 ARH relative to the axis CK and origin of abscissae Q. This sub- 



e / e^ h\^ 

 stitution gives <= -Q^ilTT"""") 5 which fails as above when 



