curvature to the involute at the point P, and a tangent to B' A B the evolute at the point C. Let A'C=σ, and A'B' = c. Then Hence it is obvious that, if a thread, of which the length is c, be fixed with one end at A', so as to touch the curve at this point, and be wound about the curve A'B' by a hand taking hold of the string at P, its extremity P will trace out the involute AB. Next take σp = c. Let B'B' = c, BC=σ, the origin of the arcs being now some fixed point B. Then B'C CP: = c = B'B', a result which points out the very same geometrical property as when we adopted the positive sign. From the geometrical property which we have established have arisen the names evolute and involute. To find the length of any Arc of the Evolute of a Curve. 140. By the preceding Article we know that σ + p = c. Leto, P,, and σ, P2, be corresponding values of σ, p: then Hence, to find the length of an arc of the evolute corresponding to any proposed arc of the involute, we must take the difference between the radii of curvature at the two extremities of the latter arc, and this will be the length of the former; provided that, for the whole interval, p either always decreases or always increases as σ increases. Ex. To find the length of the whole evolute of an ellipse The radii of curvature at the extremities of the axes major or the length of the whole evolute is equal to CHAPTER VII. CONTACT OF CURVES. 141. Definition of Order of Contact. Let y' = f(x'), y' = F(x), be the equations to two curves. Suppose (x, y) to be a point common to both curves. The two curves are said to have a contact of the first order at the point (x, y), if of the second order, if F' (x) = ƒ'(x); F'(x) = ƒ'(x), _F" (x)=ƒ"(x); of the third order, if F" (x) = ƒ'(x), F" (x)=ƒ"(x), F"" (x) = ƒ"" (x), and so on; the contact being of the rth order if F" (x) = ƒ'(x), _F" (x)= ƒ" (x), F'"' (x) = ƒ'" (x),... F(x) = ƒ1 (x). The higher the order of Contact, the closer the Contact. 142. Let the curve y' = p(x') have, at the point (x, y), a contact of the mth order with the curve y' = F(x′), and of the nth order with the curve y' = f(x), and suppose m to be greater than n. Then, by the theory of vanishing fractions, when h becomes less than any assignable magnitude, F (x + h) − p (x + h) _ F' (x + h) − p' (x + h) - = f'(x + h) - p' (x + h) F" (x + h) - p′′ (x + h) = &c. ƒ" (x + h) - p" (x + h) or, corresponding to a small increment h of x, the difference between the ordinates of the curves y' = F(x'), y' = p(x'), is indefinitely small in comparison with the difference between the ordinates of the curves y' = f(x'), y' = p(x'). Thus the contact is infinitely closer when of the mth than when of the nth order. Order of Contact dependent upon the number of Parameters. 143. Let the general equation to a family of curves be u' being a function of x', y', the coordinates of any point whatever in any one of the curves, and of n parameters a1, α2, Az,.... а„. Differentiating the equation to the curve n-1 times successively with regard to x' as the independent variable, we get n 1 equations involving a', y', and the n - 1 differential coeffi • Since we have n equations, (1) and (2), involving n parameters it is evident that we may assign any values we please to these n+1 quantities, the values of the n parameters being deter mined accordingly. We may therefore obtain the equation to an individual of the family of curves denoted by the equation (1) which shall have a contact of the (n - 1)th order with any proposed curve at any point (x, y), by assigning to the quantities (3) the values of the corresponding quantities in the proposed curve, and obtaining the values of the n parameters accordingly. Thus suppose that u' = F(x', y', a,,α, α.... α) an) and that f(x) is the ordinate in the proposed curve at the point of contact: then v1, 2, 3,.... v, denoting certain functions. of x, we shall have F(x', y', v1, V2, V39 will represent an individual of the family of curves represented by the equation (1), which shall have a contact of the (n - 1)th order with the proposed curve at the point (x, y). Ex. 1. The general equation to a circle is (x' − a)3 ·+ (y' – ß)3 = p3, a, ß, p, being disposable constants, upon the particular magnitudes of which the dimensions and position of the circle depend. Let it be proposed to determine the values of a, ß, p, that the circle may have a contact of the second order with any proposed curve y′′ = p(x') at a point (x, y) of the curve. Differentiating the equation to the circle twice with regard to x', we have |