3G8 



KNOWLEDGE 



[Feb. 24, 1882. 



0UV i%lati)rmatiral Column. 



PKUPOSITION IN CONIC SECTIONS. 



Let KVK' (Figs. 1, 2, 3) be o cone, touched by the spliere aSa' 

 ill a circle &ca! , Joreshortened into a straight line in the Jig.; and 

 let ASN he a plane section of the cone, also foreshortened into a 

 straight line, touching the sphere tn S. Let a'a and KA produced 

 meet in X (they m-ust mett unless AN is parallel to a»' or the section 

 AA' a circle). Suppose the section AS rotated around the straight 

 li7ie AN until the conic sectio/i occupies the plane of the paper, as 

 shown by the curve PAF', the points tchich icere at N being brought 

 to V and P. Join SP, draiv Fil parallel to NX, XM perp. to NX, to 

 meet in M. It is requirtd to show that the ratio of SP to PMis 

 constant. 



Draw MKNK' throngh N parallel to a'a. Now, in the conic 

 section AN we see SP (foreshortened) as NS, a tangent from N to 

 the sphere aSa', and N» is another tangent to the same sphere. 

 But tangents from the same point to a sphere are equal. Hence 

 NS (foreshortened) or SP = N)i (foreshortened), which obviously 

 = uK, and PM = NX. Hence 



SP : PJI = aK : NX = aA : AX, a constant ratio. Q.E.D. 



If Z S.VK > Z KVK', fig. 1, so that AN produced cuts VK' 

 (say in A'), the ratio aA : AX is less than unity, and the section is 

 the ellipse. 



If ^ SAK = Z KVK', fig. 2, so that AN 

 is parallel to VK', the ratio aX : AX is unity, 

 and the section is the parabola. 



If Z SAK <- Z KVK', fig. 3, so that NAf' 

 produced cuts K' V produced (say in A'), the, 

 ratio (lA : AX is greater than unity, and the 

 curve is the hyperbola. 



Fig. 3. 



In the cases of the ellipse and hyperbola, 

 we can take another sphere tH6' touching 

 the cone circularly and the plane of section 

 in H. For the ellipse, the second sphere 

 touches the cone on the same side as the other 

 sphere, and the plane of section on the other 





