Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/203

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Hence, by Rolle's Theorem (§105), must vanish for at least two values of , one lying between and , say , and the other lying between and say ; that is,

Again, for the same reason, must vanish for some value of between and , say ; hence

Therefore the elements of the circle passing through the points must satisfy the three equations

Now let the points and approach as a limiting position; then will all approach as a limit, and the elements of the osculating circle are therefore determined by the three equations

or, dropping the subscripts, which is the same thing,

(A)  
(B) differentiating (A).
(C) differentiating (B).

Solving (B) and (C) for and , we get ,

(D)
Radius and center of curvature.
Radius and center of curvature.

hence the coördinates of the center of curvature are

(E)