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



Let s be the length of the arc AP measured from a fixed point A on the curve.

Denote the increment of s (= arc PQ) by $$\Delta s$$. 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

From the above figure

Dividing through by $$(\Delta x)^2$$, we get

Now let Q approach P as a limiting position; then $$\Delta x \dot= 0$$ and we have

Similarly, if we divide (H) by $$(\Delta y)^2$$ and pass to the limit, we get

Also, from the above figure,

Now as Q approaches P as a limiting position $$\theta \dot= \tau$$, and we get