Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/158

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90. Derivative of the arc in rectangular coördinates. Let s be the length[1] of the arc AP measured from a fixed point A on the curve.

Arc APQ. Denote the increment of s (= arc PQ) by . The definition of the length of arc depends on the assumption that, as Q approaches P,

If we now apply the theorem in §89 to this, we get

(G) In the limit of the ratio of chord PQ and a second infinitesimal, chord PQ may be replaced by arc PQ (= ).

From the above figure

(H)

Dividing through by , we get

(I) .

Now let Q approach P as a limiting position; then and we have

.

[Since , (G).]

(24)

Similarly, if we divide (H) by and pass to the limit, we get

(25)

Also, from the above figure,

Now as Q approaches P as a limiting position , and we get

(26) ,

[Since from (G) , and .]

  1. Defined in § 209.