A right circular cone has base radius $ r$ and height $ h$.  The cone lies on its side on a flat table.  As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered  at the point where the vertex touches the table.  The cone first returns to its original position on the table after making $ 17$ complete rotations.  The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime.  Find $ m \plus{} n$.
1. **Determine the slant height of the cone:**
   The slant height \( l \) of a right circular cone with base radius \( r \) and height \( h \) is given by the Pythagorean theorem:
   \[
   l = \sqrt{r^2 + h^2}
   \]

2. **Calculate the path length traced by the cone:**
   As the cone rolls on the table, the point where the cone's base meets the table traces a circular arc. The radius of this circular arc is equal to the slant height \( l \) of the cone. Therefore, the length of the path traced by the cone is the circumference of this circle:
   \[
   \text{Path length} = 2\pi l = 2\pi \sqrt{r^2 + h^2}
   \]

3. **Relate the path length to the base circumference:**
   The cone makes 17 complete rotations to return to its original position. Each rotation corresponds to the circumference of the base of the cone, which is \( 2\pi r \). Therefore, the total path length is also equal to 17 times the circumference of the base:
   \[
   \text{Path length} = 17 \times 2\pi r = 34\pi r
   \]

4. **Set up the equation and solve for \( h \):**
   Equate the two expressions for the path length:
   \[
   2\pi \sqrt{r^2 + h^2} = 34\pi r
   \]
   Divide both sides by \( 2\pi \):
   \[
   \sqrt{r^2 + h^2} = 17r
   \]
   Square both sides to eliminate the square root:
   \[
   r^2 + h^2 = (17r)^2
   \]
   \[
   r^2 + h^2 = 289r^2
   \]
   Subtract \( r^2 \) from both sides:
   \[
   h^2 = 288r^2
   \]

5. **Express \( \frac{h}{r} \) in the required form:**
   Divide both sides by \( r^2 \):
   \[
   \frac{h^2}{r^2} = 288
   \]
   Take the square root of both sides:
   \[
   \frac{h}{r} = \sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2}
   \]

6. **Identify \( m \) and \( n \) and find \( m + n \):**
   The value of \( \frac{h}{r} \) is \( 12\sqrt{2} \), where \( m = 12 \) and \( n = 2 \). Therefore:
   \[
   m + n = 12 + 2 = 14
   \]

The final answer is \( \boxed{14} \).