250 Mr. Lubbock on some Properties 



ViVx — /3i (>.2 +i/i) + P «! = ; and similarly 



y-6y-i-^-2 (i/3 +3/2) + p «, = 



i/oi/4-/3i (3/5-3/4) +7'«i = 



i/G 3/5-/3^ ( J/(;+ .ys) + i> «2 = 



^1^6-^3(^1 + J/ti) + P aa = 

 eliminating 3/1, ^3 and 3/5. 



(/3.2-^3)i/4i/-2-(P«-2-/3o^3)j/4 + (i^«3-/3,/33)j/, + p [xj., 

 -/3o«3) = 



(/3i-/3,) (i/5-i/4-(;^«i-/3,/3,)i/fi+ (i^«,-^i /3,)j/4+ P («i /3.2 

 -/3,«,) = 



{/33-i3,)3/2i/6— (P«3-/3i^3)j/2+ (P«l-/33^l)i/6+i' («3/3l 



-/33 «i) = 

 adding together these equations, and making for simplicity 

 /3, = 0, which does not affect the generality of the solution, be- 

 cause the coordinate axes a- and t/ are supposed to be inclined 

 at any given angle 



/52i/4 (i/'2-J/6) + ^3 i/2 Q/ti-yd + ^2 ^3 i.'A-Jjd + P («-2 ^6-^2 «3 



+ «i/3,-P3a,) = 

 and by the equations above 



a _ y'>y*-yiy\ _ q g _ y6y:,—y^yi a _. ym-ytys 



and substituting these values of /Sg and /3g in the thi'ee first 

 terms of the last equation they disappear, and we have 

 a.2 /Sg—jSg a3 +«i /Sg—ZSgaj = 0, which equation of condition 

 between the coorduiates of the points «j /3j, a^/^a' "3 l^s shows 

 that they are in the same straight line. 



The same kind of proof might be applied step by step when 

 the equation of the conic section is more complicated, but it 

 is simpler to extend it at once by the theory of projections. 



Let the parabola (j/^ = 1^ ^\ s = D) be the base of a cone 

 whose vertex coincides with the origin ; the equation to the 

 cone is Dj/' = pzx, 



let this cone be intersected by a plane of which the equation is 



(z — D) sin S—x cos 5 = 0. 

 6 being the angle which this plane makes with the plane zi/. 

 The equations to the curve of intersection of this plane, and 

 the conical surface referred to axes Oz' and Or/', (O being 

 the origin,) coinciding with the intersection of this plane- 

 and the planes zx, zy will be found by substituting x cos 9 

 + D for z, and x sin fl for x in the equation D^^ ^ p z x^ 

 which substitution gives Dj/"^ = p {x cos 9 + D) x sin fl. If 

 this equation be identical with the equation y = p' x + q' x'^ 



2}'=p 



