Problem 575
Show that the length of the polar subnormal
Show that the length of the polar subnormal of the Archimedean spiral $\rho=a\varphi$ is constant.
Explanation of Method
A single differentiation, and the defining constant of the spiral reappears as its subnormal.
Step 1: The polar subnormal
For $\rho=f(\varphi)$, the length of the polar subnormal is $S_N=|\rho'(\varphi)|$.
Step 2: Differentiate
\[ \frac{d\rho}{d\varphi} = a. \tag{1} \]
Step 3: Verification
\[a=4\text{: } d\rho/d\varphi=4\text{ at every }\varphi\text{ tested }(\varphi=0.2,\,1.5,\,6.0)\ \checkmark\]
Result
Final answer:
\[\boxed{\,S_N = |a| = a = \text{const}\,}\]
independent of $\varphi$. $\blacksquare$