11. A point moves so that the sum of the squares of its distances from a number of given points is constant. its locus is a sphere. 12. Prove that A sphere touches each of two given straight lines which do not meet; find the locus of its centre. CHAPTER VI. THE ELLIPSE. 43. 1. IF we define a conic section as "the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line" (Todhunter, Art. 123), we shall find the equation to be (Ex. 5, Art. 35) where a3p2 = e2 (a3 — Sap)3. SP=ePQ, vector SD = a, SP = p. .(1), When e is less than 1, the curve is the ellipse, a few of whose properties we are about to exhibit. the major axis of the ellipse, which we shall as usual abbreviate and if vector C'S be designated by a', CP by p', we have whence, by substituting in (1), the equation assumes the form a2p'2 + (Sa'p')2 = a* (1 − e2) ; which we may now write, CS being a and CP p, 3. This equation might have been obtained at once by referring the ellipse to the two foci, as Newton does in the Principia, Book 1. Prop. 11; the definition then becomes If now we write op for a2p + aSap where p is a vector which coincides with p only in the cases in which either a coincides with p or when Sap = 0, i. e. in the cases of the principal axes; the equation of the ellipse becomes The same equation is, of course, applicable to the hyperbola, e being greater than 1. 44. The following properties of op will be very frequently employed. The reader is requested to bear them constantly in mind. They need no other demonstration than what results from simple inspection of the value of op 45. To find the equation of the tangent to the ellipse. The tangent is defined to be the limit to which the secant approaches as the points of section approach each other. Let CP-p, CQ-p', then vector PQ CQ-CP=p'-p=ẞ say; = B is therefore a vector along the secant. Now Sp'op' S (p+B) & (p+B) = =S(p + B) (pp + B) (44. 1) = Sp&p + Sp&ß + SB¢p + Sß&ß. But or (44. 3) Sp'op'=1=Spop; .. Spoß+Sẞop + SB&B = 0, 25βφρ + βφβ = 0. Now Bop involves the first power of ẞ whilst ßøß involves the second, and the definition requires that the limit of the sum of the two as ẞ gets smaller and smaller should be the first only, even if that should be zero: i. e. when ẞ is along the tangent, we must have 2.SBpp = 0. Let then T be any point in the tangent, CT = π, then COR. 1. pp is a vector along the perpendicular to the tangent (32. 3), that is, op is a normal vector, or parallel to a normal vector at the point p. COR. 2. The equation of the tangent may also be written (44. 3) Spp1. 46. We may now exhibit the corresponding equations in terms of the Cartesian co-ordinates, as some of the results are best known in that form. Let CM=x, MP = y as usual; then, retaining the notation of Art. 31 with i, j as unit vectors parallel and perpendicular respectively to CA, vector CM= xi, MP=yj, CS= aei; |