Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/182

From Wikisource
Jump to navigation Jump to search
This page needs to be proofread.
Substituting (D) in (C), we get
(E) . Also
(F) . From (30), §91
Dividing (E) by (F) gives
  .
But . Hence
(41) .

Illustrative Example 1. Find the curvature of the parabola at the upper end of the latus rectum.

Solution. .
Substituting in (40), ,
giving the curvature at any point. At the upper end of the latus rectum (p, 2p)
  .[1] Ans.

Illustrative Example 2. Find the curvature of the logarithmic spiral at any point.

Solution. .
Substituting in (41), Ans.
  1. While in our work it is generally only the numerical value of K that is of importance, yet we can give a geometric meaning to its sign. Throughout our work we have taken the positive sign of the radical . Therefore K will be positive or negative at the same time as that is (§85), according as the curve is concave upwards or concave dx downwards.