Illustration. Find the envelope of a straight line AB on which the intercept between X and Y is a constant c. The equation of AB is a 0 0 --= F(x, y; a).
V Another_A'B' of the system is , 1-0 (a'-=a +da).
a V c'a" and another A" B" is + The Intersections 1' and I" of these two with AB are definite. As A' and A" close down upon and coalesce in A, I' and I", always definite, close down upon and coalesce in their common limit I, the instantaneous pivot about which AB starts to turn. Differentiating and eliminating a we find the 4-cusped Hypocycloid, xii+94= c%, as the envelope, or path of the instan taneous pivot I.
Hence Pliicker's double conception of a curve as path of a point gliding along a straight line that turns about the point, and envelope of a straight line that turns about a point that glides on the straight line. The relation connecting the corresponding mag nitudes, arc-length s (of the path) and angle a (through which the straight line turns), is the intrinsic equation of the da curve. The is named average curvature . of ds; its limit da is named instantaneous ds curvature (it) at P. Plainly da = dr; hence dr dr d tan r dx IL= ds d tan r dx ds + (N.B. The Differential Triangle PDQ formed by dx, dy, ds, yields at once dx 1 dy = yx Ye, v - In the circle IC is the constant - hence the curvature of the circle is the reciprocal of the radius. For any point of any curve the recip rocal of this curvature is called the radius of curvature, p; hence this p at any point is the radius of a circle of equal curvature, hence called circle of curvature.
To illustrate. Draw PT and PN, tangent and normal to the curve; about K' and K" on PN, with radii p' > p and p"
Otherwise, through P, and Q' and Q" on opposite sides of P, draw a circle. Let mid-. normals to PQ', PQ" meet PN at S' S", and each other at S"'; as Q' and Q" approach and coalesce in S', S , S" all approach and coalesce in their common limit, S (or K). Hence the osculatory circle= circle through three consecutive points of a curve, and centre of curvature =intersection of two consecutive normals. The co-ordinates (u and v) of K are given by x (I + v=y+ 1 -(1 Elimi nating x and y between these equations and the equation of the curve, we get the equa tion connecting u and v for every K, i.e., the
equation of the Evolute, or locus of the centres of curvature of the original curve (the Involute). Since v.. yx I = 0, the tangents at cor responding P and K are perpendicular, the nor mal to the involute is tangent to the evolute. Also, it is easy to prove that the arc-length in Evolute can differ only by a constant from the radius of curvature (p) in Involute. Hence a point of a cord held tight while being un wound from the evolute must trace an involute; hence the former name. To any involute there is only one evolute, hut to any evolute there are infinitely many (so-called parallel) invo lutes, an excellent illustration of a one valued determination with many-valued inverse, and also of the definiteness of dif ferentiation as compared with the indefinite ness of its inverse, Integration (see below). Some curves reproduce themselves in their evolutes, notably Cycloid and Logarithmic Spiral, which latter inspired the engraving and epitaph on the tomb of Jacob Bernoulli (1654-1705) : Eadem mutata resurgo. The general theory of the Contact of curves, Asymptotes, etc., beautifully exemplifies this Calculus, but cannot be treated here (see Curves, Higher Plane).
f for x=a both terms of a fraction V (x) vanish, then y loses definite - 0 (x)Ca) ness, taking the unmeaning form, y' (Cf. Algebra).
xm - The fundamental example is y= ' x a for x = a. However, we may still seek Lim. 0(x) for x=:-..a+0, though it would be arbitrary to assume this limit as the value of 0(a) (Dar 0(a) boux). If x a be removable from both terms, we may cancel it for x+ a, and then x" a' seek the limit for x =a. Thus x a x'-Fax-l-as for x+a; this last='3a' for x=a; x'a' hence Lim. a --3as, but not 3a' x a Lim. 0(x) for x= a, unless arbitrarily. Now X-10 sto(x) Lim. for 0(a) = 0 = 0(a) (L'Hospital). Hence the ordinary rule: Take the limit of the quotient of the D's of the terms at the critical value. Or, expand the terms in the neighborhood of a, simplify, cancel the vanish ing common factor, and evaluate for x Similarly, with proper modification, we treat ; and co do , reduced to 0 Indeterminate exponentials like 1", 0" etc., are first reduced to 0 - by passing to Logarithms.