INFINITESIMAL CALCULUS 27 stants, and accordingly the circle admits in general of a contact of the second degree with a curve at any point. The parabola has four independent constants, and consequently admits of a contact of the third order ; and so on. Again, introducing the additional condition /(")(#) = <^")(x), a finite number of points is seen to exist at which the osculating curve has a contact one degree higher ; thus a tangent may have contact of the second order, an osculating circle contact of the third order, and so on. In the case of a right line, we have <P(X)=/(X], (a) =/(*), 0V) =/"(*), where <j>(x) = ax + b, . , 0V) 0- This agrees with the condition found for a point of inflexion in 76. The problem of contact admits of being considered also from a geometrical point of view, i.e., from the consideration of the number of consecutive points of intersection of two curves. 79. The discussion of evolutes and involutes originated with Huygens, in his celebrated work, Horologiwn Oscillatorium (1673), published before the invention of the calculus. Huygens s investi gation is purely geometrical. The definition of the osculating circle was first given by Leibnitz, in the Ada Eruditorum, 1686, where he pointed out its great importance in the study of curves. Newton, in his Principia, makes frequent use of the theory of the radius of curvature, and of its connexion with evolutes. Newton also observed that the radius of curvature becomes infinite at a point of inflexion, and vanishes at a cusp called by him punctum rcctitudinis, and punctum curvaturse injlnitx, respec tively. See Opiise., i. pp. 121, 122, ed. Cast. It is worthy of remark that Sluze, in his Mcsolabuin scu dux mediss proportionates, &c. (1659), pointed out a general method for the determination of points of inflexion (puncta ficxus cont/rarii}, by reducing it to a question of maxima and minima, viz., to finding when the intercept made by the tangent, measured along any axis from a fixed point on it, is a maximum or a minimum. This method he applied successfully to the conchoid of Nicomedes. (1) It is easily found as above that the radius of curvature at any point on the curve 3a 2 7/ = # 4 is equal to p= - (2) The following expression for the radius of curvature in polar coordinates, 9 can be easily deduced. (3) If 14 = ; this becomes 1 + P = (4) Hence at a point of inflexion we have
(5) The origin is a point of inflexion on the curve represented by the equation ?i 3 + 1 = 0. (6) The length of the radius of curvature at the origin in the curve r = a sin n0 is a. (7) If on the tangent to a curve a constant length be measured from the point of contact, the normal to the locus of the points thus taken passes through the corresponding centre of curvature of the proposed curve. (8) In the ellipse iL + |_ 1= 1, if we take x = a cos 0, y=b sin 0, the coordinates a, & of the centre of curvature of any point are given by the equations (9) At a cusp (compare 73) the radius of curvature is zero for both branches. (10) In some cases two branches of the same curve may have a contact of the second or of a higher order. For instance, it is easy to show that at the origin two branches of the curve if- - 2x 3 2/ + x* - x 5 = have equal finite radii of curvature. Envelopes. 80. If we suppose a series of different values given to o in the equation f(x,y,a)-0, then for each value we get a distinct curve, and the above equation may be regarded as representing an indefinite number of curves, a single determinate curve corresponding to each distinct value of o, provided a enters into the equation in a rational form only. If now we regard the parameter a as varying continuously and consider the two curves /(> , V, ) = 0, f(x, y, q + Aq) = 0, then the coordinates of their points of intersection satisfy each of these equations, and therefore also satisfy the equation f(x, y, a + Act) -/(a, y, o) A = 0. _ Now, in the limit, when A is infinitely small, the latter equa tion becomes df(x, y, q)
- ~* - J .
da. Hence the locus of the points of ultimate, intersection for tlm entire system of curves represented by /(a;, y, a) = is obtained by eliminating a between the equations f(x, y, a) = and ^^^-),o. This locus is called the envelope of the system, and it can be easily seen that it is touched by every curve of the system. For instance, suppose L, M, N to be given functions of x and y, and o a parameter, to find the envelope of the system of curves represented by the equation Here f(x, y, a) = La 2 + 2M + N ; Consequently the envelope is the curve represented by the equation LN = M 2 . For example, if L, M, N represent right lines, the envelope of the moving line is the conic LN M 2 . In general, if the equation of the moving curve be of the form p a"+iv- 1 +rv- 2 + . +P=O, where P , P lf P 2 . . . T n are given functions of x and y, the envelope is obtained by the elimination of a between the proposed equation and its derived equation 0. It is accordingly represented by the condition that the equation in a should have equal roots ; this condition is called the discriminant of the equation. For examples see Salmon s Higher Plane Curves Arts. 85, 86. 81. In many cases the equation of the moving curve is of the form f(x, y, a, j8) = 0, where the parameters a, y3 are connected by an equation In this case we regard /3 as a function of a, and thus we get by differentiation df df dft - d<p d(f> dfi da d& da. ~ da d& da T^" == A consequently, if we make df _ d(f> da. da and the required envelope is obtained by the elimination of a, ft, A. between these and the two given equations. For example, let it be proposed to find the envelope of a line of given length (a), whose extremities move along two fixed rect angular axes. Here, taking the fixed lines for coordinate axes, and denoting the intercepts by a and #, we have Hence = a , -^ = A ; a" p" from which we get a? and the equation of the envelope is i 4 a a 2;3 4- 7/8 == a->. This envelope was discussed by John Bernoulli in the Ada Erud., 1692. Again, to find the equation to the evolute of an ellipse, regarded as the envelope of its normals. Here we have the equations 2 b- = ft 2 - b 2 , and -*-* + 75- = 1 , a /3 a- b 2 where a, j8 are the coordinates of a point on the ellipse. Hence
and we easily obtain as the required equation
