To find the length of any curve

To find the length, l, of the curve AB (Fig. 113) when the equation of the curve is known. This is equivalent to finding the length of a straight line of the same length as the curve if the curve were flattened out or rectified, hence the process is called the rectification of curves. Let the coordinates of A be (x0, y0), and of B, (xn, yn). Take any two points, P, Q, on the curve. Make the construction shown in the figure. Then, by Euclid, i., 47, if P and Q are sufficiently close, we have, very nearly 

In order to apply this result it is only necessary to differentiate the equation of the curve and substitute the values of dx and dy, so obtained,-in equation (1). By integrating this equation, we obtain a general expression between. the assigned limits, we get the length of the given portion of the curve. 

If the equation is expressed in polar coordinates, the length of a small element, dl, is deduced in a similar manner. Thus, dl - J(drf + r d6f.. .. (2) 

We shall want to refer back to this result when we discuss hyperbolic functions, and also to show that 

The reader may have noticed the remarkable analogy between the chemist s atom, the physicist s particle, and molecule, and the mathematician s differential When the chemist wishes 

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